Modular Functions and Modular Forms

J. S. Milne

Abstract. These are the notes for Math 678, University of Michigan, Fall 1990, exactly as they were handed out during the course except for some minor revisions and corrections.

Please send comments and corrections to me at jmilne@umich.edu.

Contents

Introduction 1 Riemann surfaces 1 The general problem 1 Riemann surfaces that are quotients of D. 1 Modular functions. 2 Modular forms. 3 Plane affine algebraic curves 3 Plane projective curves. 4 Arithmetic of Modular Curves. 5 Elliptic curves. 5 Elliptic functions. 5 Elliptic curves and modular curves. 6 Books 7 1. Preliminaries 8 Continuous group actions. 8 Riemann surfaces: classical approach 10 Riemann surfaces as ringed spaces 11 Differential forms. 12 Analysis on compact Riemann surfaces. 13 Riemann-Roch Theorem. 15 The genus of X. 17 Riemann surfaces as algebraic curves. 18 2. Elliptic Modular Curves as Riemann Surfaces 19

\begin{matrix}c \\ ◯ \end{matrix}1997 J.S. Milne. You may make one copy of these notes for your own personal use.

ii J. S. MILNE

The upper-half plane as a quotient of SL_{2}(ℝ) 19 Quotients of ℍ 21 Discrete subgroups of SL_{2}(ℝ) 22 Classification of linear fractional transformations 24 Fundamental domains 26 Fundamental domains for congruence subgroups 29 Defining complex structures on quotients 30 The complex structure on Γ(1)\ℍ^{∗} 30 The complex structure on Γ\ℍ^{∗} 31 The genus of X(Γ) 32 3. Elliptic functions 35 Lattices and bases 35 Quotients of ℂ by lattices 35 Doubly periodic functions 35 Endomorphisms of ℂ/Λ 36 The Weierstrass ℘-function 37 The addition formula 39 Eisenstein series 39 The field of doubly periodic functions 40 Elliptic curves 40 The elliptic curve E(Λ) 40 4. Modular Functions and Modular Forms 42 Modular functions 42 Modular forms 43 Modular forms as k-fold differentials 44 The dimension of the space of modular forms 45 Zeros of modular forms 47 Modular forms for Γ(1) 48 The Fourier coefficients of the Eisenstein series for Γ(1) 49 The expansion of Δ and j 51 The size of the coefficients of a cusp form 52 Modular forms as sections of line bundles 52 Poincaré series 54 The geometry of ℍ 56 Petersson inner product 57 Completeness of the Poincaré series 58 Eisenstein series for Γ(N) 58 5. Hecke Operators 62 Introduction 62

MODULAR FUNCTIONS AND MODULAR FORMS iii

Abstract Hecke operators 65 Lemmas on 2 × 2 matrices 67 Hecke operators for Γ(1) 68 The ℤ-structure on the space of modular forms for Γ(1) 72 Geometric interpretation of Hecke operators 75 The Hecke algebra 76 6. The Modular Equation for Γ_{0}(N) 81 7. The Canonical Model of X_{0}(N) over ℚ 86 Review of some algebraic geometry 86 Curves and Riemann surfaces 88 The curve X_{0}(N) over ℚ 90 8. Modular Curves as Moduli Varieties 92 The general notion of a moduli variety 92 The moduli variety for elliptic curves 93 The curve Y_{0}(N)_{ℚ} as a moduli variety 94 The curve Y (N) as a moduli variety 95 9. Modular Forms, Dirichlet Series, and Functional Equations 96 The Mellin transform 96 Weil’s theorem 98 10. Correspondences on Curves; the Theorem of Eichler-Shimura 100 The ring of correspondences of a curve 100 The Hecke correspondence 101 The Frobenius map 101 Brief review of the points of order p on elliptic curves 102 The Eichler-Shimura theorem 102 11. Curves and their Zeta Functions 104 Two elementary results 104 The zeta function of a curve over a finite field 105 The zeta function of a curve over ℚ 106 Review of elliptic curves 107 The zeta function of X_{0}(N): case of genus 1 108 Review of the theory of curves 109 The zeta function of X_{0}(N): general case 111 The Conjecture of Taniyama and Weil 111 Notes 113 Fermat’s last theorem 113 Application to the conjecture of Birch and Swinnerton-Dyer 114 12. Complex Multiplication for Elliptic Curves 115 Abelian extensions of ℚ 115

0 J. S. MILNE

Orders in K 116 Elliptic curves over ℂ 117 Algebraicity of j 117 The integrality of j 118 Statement of the main theorem (first form) 120 The theory of 𝔞-isogenies 120 Reduction of elliptic curves 121 The Frobenius map 122 Proof of the main theorem 122 The main theorem for orders 123 Points of order m 124 Adelic version of the main theorem 124 Index 125

Introduction

It is easy to define modular functions and forms, but less easy to say why they are important, especially to number theorists. Thus I will begin with a rather long overview of the subject.

Riemann surfaces. Let X be a connected Hausdorff topological space. A coordinate neighbourhood of P ∈ X is a pair (U, z) with U an open neighbourhood of P and z a homeomorphism of U onto an open subset of the complex plane. A complex structure on X is a compatible family of coordinate neighbourhoods that cover X. A Riemann surface is a topological space together with a complex structure.

For example, any open subset X of ℂ is a Riemann surface, and the unit sphere can be given a complex structure with two coordinate neighbourhoods, namely the complements of the north and south poles mapped onto the complex plane in the standard way. With this complex structure it is called the Riemann sphere. We shall see that a torus can be given infinitely many different complex structures.

Let X be a Riemann surface, and let V be an open subset of X. A function f : V → ℂ is said to be holomorphic if, for all coordinate neighbourhoods (U, z) of X, f ∘ z^{−1} is a holomorphic function on z(U). Similarly, one can define the notion of a meromorphic function on a Riemann surface.

The general problem. We can state the following grandiose problem: study all holomorphic functions on all Riemann surfaces. In order to do this, we would first have to find all Riemann surfaces. This problem is easier than it looks.

Let X be a Riemann surface. From topology, we know that there is a simply connected topological space X̃ (the universal covering space of X) and a map p : X̃ → X which is a local homeomorphism. There is a unique complex structure on X̃ for which p : X̃ → X is a local isomorphism of Riemann surfaces. If Γ is the group of covering transformations of p : X̃ → X, then X = Γ\X̃.

Theorem 0.1. A simply connected Riemann surface is isomorphic to (exactly) one of the following three:

(a) ℂ;

(b) the open unit disk D =^{df} {z ∈ ℂ | |z| < 1};

(c) the Riemann sphere.

Proof. This is the famous Riemann mapping theorem.

The main focus of this course will be on Riemann surfaces with D as their universal covering space, but we shall also need to look at those with ℂ as their universal covering space; the third type will not occur.

Riemann surfaces that are quotients of D. In fact, rather than working with D, it will be more convenient to work with the complex upper half plane:

ℍ = {z ∈ ℂ | ℑ(z) > 0}.

The map z ↦ \frac{z−i}{z+i} is an isomorphism of ℍ onto D (in the jargon the complex analysts use, ℍ and D are conformally equivalent). We want to study Riemann surfaces of the form Γ\ℍ, where Γ is a discrete group acting on ℍ. How do we find such Γ’s? There

is an obvious big group acting on ℍ, namely, SL_{2}(ℝ). For α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℝ)

and z ∈ ℍ, define

α(z) = \frac{az + b}{cz + d}.

Then

ℑ(αz) = ℑ (\frac{az + b}{cz + d}) = ℑ (\frac{(az + b)(cz̄ + d)}{|cz + d|^{2}}) = \frac{ℑ(adz + bcz̄)}{|cz + d|^{2}}.

But

ℑ(adz + bcz̄) = (ad − bc) · ℑ(z) = ℑ(z),

because det(α) = 1. Hence ℑ(αz) = ℑ(z)/|cz + d|^{2} for α ∈ SL_{2}(ℝ). In particular,

z ∈ ℍ ⟹ α(z) ∈ ℍ.

Later we shall see that there is an isomorphism

SL_{2}(ℝ)/{±I} → Aut(ℍ)

(bi-holomorphic automorphisms of ℍ). There are some obvious discrete groups in SL_{2}(ℝ), for example, Γ = SL_{2}(ℤ). This is called the (full) elliptic modular group. For any N ≥ 0, we define

Γ(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) | a ≡ 1, b ≡ 0, c ≡ 0, d ≡ 1 mod N\}

and call it the principal congruence subgroup of level N; in particular, Γ(1) = SL_{2}(ℤ). There are many discrete subgroups in SL_{2}(ℝ), but those of most interest to number theorists are the ones containing a principal congruence subgroup as a subgroup of finite index.

Let Y (N) = Γ(N)\ℍ and endow it with the quotient topology. Let p : ℍ → Y (N) be the quotient map. There is a unique complex structure on Y (N) such that a function f on an open subset U of Y (N) is holomorphic if and only if f ∘ p is holomorphic on p^{−1}(U). Thus f ↦ f ∘p defines a one-to-one correspondence between holomorphic functions on U ⊂ Y (N) and holomorphic functions on p^{−1}(U) invariant under Γ(N), i.e., such that g(γz) = g(z) for all γ ∈ Γ(N).

The Riemann surface Y (N) is not compact, but there is a natural way of compactifying it by adding a finite number of points. The compact Riemann surface is denoted by X(N). For example, Y (1) is compactified by adding a single point.

Modular functions. A modular function f(z) of level N is a meromorphic function on ℍ invariant under Γ(N) and “meromorphic at the cusps”. Because it is invariant under Γ(N), it can be regarded as a function on Y (N), and the second condition means that it remains meromorphic when considered as a function on X(N), i.e., it has at worst a pole at each point of X(N) \ Y (N).

In the case of the full modular group, it is easy to make explicit the condition “meromorphic at the cusps” (in this case, cusp). To be invariant under the full modular group means that

f (\frac{az + b}{cz + d}) = f(z) for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ).

Since ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ SL_{2}(ℤ), we have that f(z + 1) = f(z). Any function satisfying this

condition can be written in the form f(z) = f^{∗}(q), q = e^{2πiz}. As z ranges over the upper half plane, q(z) ranges over a punctured disk in ℂ (open disk with centred at 0, with 0 removed). To say that f(z) is meromorphic at the cusp means that f^{∗}(q) is meromorphic on the whole disk; hence that f has an expansion

f(z) = ∑_{n≥−N_{0}} a_{n}q^{n}.

Modular forms. To construct a modular function, we have to construct a meromorphic function on ℍ that is invariant under the action of Γ(N). This is difficult. It is easier to construct functions that transform in a certain way under the action of Γ(N); the quotient of two such functions of same type will then be a modular function.

This is analogous to the following situation. Let ℙ^{1}(k) = (k × k \ {(0, 0)})/k^{×}. We seek rational functions f(X, Y) = g(X, Y)/h(X, Y) such that (a, b) ↦ f(a, b) is a function on ℙ^{1} (with a few points removed). Thus we need f(X, Y) to be invariant under the action of k^{×}, i.e., such that f(aX, aY) = f(X, Y), all a ∈ k^{×}. Recall that a homogeneous form of degree d is a polynomial h(X, Y) such that h(aX, aY) = a^{d}h(X, Y) for all a ∈ k^{×}. If g and h are homogeneous forms of the same degree, then g/h will be a rational function on ℙ^{1}.

The relation of homogeneous forms to rational functions on ℙ^{1} is exactly the same as the relation of modular forms to modular functions.

Definition 0.2. A holomorphic function f(z) on ℍ is a modular form of level N and weight 2k if

(a) f(αz) = (cz + d)^{2k} · f(z), all α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ(N);

(b) f(z) is “holomorphic at the cusps”.

For the full modular group, note that (a) again implies that f(z + 1) = f(z), and so f can be written as a function of q = e^{2πiz}; condition (b) then says that this function is holomorphic at 0, so that

f(z) = ∑_{n≥0} a_{n}q^{n}, q = e^{2πiz}.

Plane affine algebraic curves. . Let k be a field. A plane affine algebraic curve C over k is defined by a nonzero polynomial f(X, Y) ∈ k[X, Y]. The points of C with coordinates in a field K ⊃ k are the zeros of f(X, Y) in K × K; we write C(K) for this set. Let k[C] = k[X, Y]/(f(X, Y)), and call it the ring of regular functions on C. When f(X, Y) is irreducible (this is the most interesting case so far as we are concerned), we write k(C) for the field of fractions of k[C], and call it the field of rational functions on C.

We say that C is nonsingular if f(X, Y), \frac{∂f}{∂X}, \frac{∂f}{∂Y} have no common zero in the algebraic closure of k. A point where all three vanish is called a singular point on the curve.

Example 0.3. Let C be the curve defined by Y^{2} = 4X^{3} − aX − b, i.e., by the polynomial

f(X, Y) = Y^{2} − 4X^{3} + aX + b.

Assume char k ≠ 2. The derivatives of f are 2Y and −12X^{2} + a. Thus a singular point on C is a pair (x, y) such that y = 0 and x is a repeated root of 4X^{3} − aX − b. Therefore C is nonsingular if and only if the roots of 4X^{3} − aX − b are all simple, which is true if and only if its discriminant Δ = a^{3} − 27b^{2} ≠ 0.

Proposition 0.4. Let C be a nonsingular affine algebraic curve over ℂ; then C(ℂ) has a natural structure as a Riemann surface.

Proof. Let P be a point in C(ℂ), and suppose (∂f/∂Y)(P) ≠ 0. Then the implicit function theorem shows that the projection (x, y) ↦ x : C(ℂ) → ℂ defines a homeomorphism of an open neighbourhood of P onto an open neighbourhood of x(P) in ℂ. This we take to be a coordinate neighbourhood of P.

Plane projective curves. A plane projective curve C over k is defined by a nonconstant homogeneous polynomial F (X, Y, Z). Let ℙ^{2}(k) = (k^{3} \ {(0, 0, 0)})/k^{×}, and write (a : b : c) for the equivalence class of (a, b, c) in ℙ^{2}(k). As F (X, Y, Z) is homogeneous, F (cx, cy, cz) = c^{m} · F (x, y, z) for every c ∈ k^{×}, where m = deg(F (X, Y, Z)). Thus it makes sense to say F (x, y, z) is zero or nonzero for (x : y : z) ∈ ℙ^{2}(k). The points of C with coordinates in a field K ⊃ k are the zeros of F (X, Y, Z) in ℙ^{2}(K). Write C(K) for this set. Let

k[C] = k[X, Y, Z]/(F (X, Y, Z)),

and call it the homogeneous coordinate ring of C. When F (X, Y, Z) is irreducible, so that k[C] is an integral domain, we write k(C) for the subfield of the field of fractions of k[C] of elements of degree zero (i.e., quotients of elements of the same degree), and we call it the field of rational functions on C.

A plane projective curve C is the union of three affine curves C_{X}, C_{Y}, C_{Z} defined by the polynomials F (1, Y, Z), F (X, 1, Z), F (X, Y, 1) respectively, and we say that C is nonsingular if all three affine curves are nonsingular. There is a natural complex structure on C(ℂ), and the Riemann surface C(ℂ) is compact.

More generally, one can define a nonsingular projective algebraic curve C in projective space ℙ^{n} of any dimension. Its points C(ℂ) again form a Riemann surface.

Theorem 0.5. Every compact Riemann surface S is of the form C(ℂ) for some nonsingular projective algebraic curve C, and C is uniquely determined up to isomorphism. Moreover, ℂ(C) is the field of meromorphic functions on S.

The statement is not true for noncompact Riemann surfaces, for example, ℍ is not of the form C(ℂ) for C an algebraic curve, and neither is the graph of the exponential function z ↦ e^{z}.

Arithmetic of Modular Curves. The theorem shows that we can regard X(N) as an algebraic curve, defined by some homogeneous polynomial(s) with coefficients in ℂ. The central fact underlying the arithmetic of the modular curves (and hence of modular functions and modular forms) is that this algebraic curve is defined, in a natural way, over ℚ[ζ_{N}], where ζ_{N} = exp(2πi/N), i.e., the polynomials defining X(N) (as an algebraic curve) can be taken to have coefficients in ℚ[ζ_{N}], and there is a natural way of doing this.

This statement has as a consequence that it makes sense to speak of the points of X(N) with coordinates in any field containing ℚ[ζ_{N}]. In the remainder of the introduction, I will explain what the points of Y (1) are in any field containing ℚ.

Elliptic curves. An elliptic curve E over a field k (of characteristic zero) is a plane projective curve given by an equation:

Y^{2}Z = 4X^{3} − aXZ^{2} − bZ^{3}, Δ =^{df} a^{3} − 27b^{2} ≠ 0.

When we replace X with X/c^{2} and Y with Y/c^{3}, some c ∈ k^{×}, and multiply through by c^{6}, the equation becomes

Y^{2}Z = 4X^{3} − ac^{4}XZ^{2} − bc^{6}Z^{3},

and so we should not distinguish the curve defined by this equation from that defined by the first equation. Note that

j(E) =^{df} 1728a^{3}/Δ

is invariant under this change. In fact one can show (with a suitable definition of isomorphism) that two elliptic curves E and E′ are isomorphic over an algebraically closed field if and only if j(E) = j(E′).

Elliptic functions. What are the quotients of ℂ? A lattice in ℂ is a subset of the form

Λ = ℤω_{1} + ℤω_{2}

with ω_{1} and ω_{2} complex numbers that are linearly independent over ℝ. The quotient ℂ/Λ is (topologically) a torus. Let p : ℂ → ℂ/Λ be the quotient map. The space ℂ/Λ has a unique complex structure such that a function f on an open subset U of ℂ/Λ is holomorphic if and only if f ∘ p is holomorphic on p^{−1}(U).

To give a meromorphic function on ℂ/Λ we have to give a meromorphic function f on ℂ invariant under the action of Λ, i.e., such that f(z + λ) = f(z) for all λ ∈ Λ. Define

℘(z) = \frac{1}{z^{2}} + ∑_{λ∈Λ,λ≠0} (\frac{1}{(z − λ)^{2}} − \frac{1}{λ^{2}} ) .

This is a meromorphic function on ℂ, invariant under Λ, and the map

[z] ↦ (℘(z) : ℘′(z) : 1) : ℂ/Λ → ℙ^{2}(ℂ)

defines an isomorphism of the Riemann surface ℂ/Λ onto the Riemann surface E(ℂ), where E is the elliptic curve,

Y^{2}Z = 4X^{3} − g_{2}XZ − g_{3}Z^{3}

for certain g_{2} and g_{3}.

Elliptic curves and modular curves. We have a map Λ ↦ E(Λ) =^{df} ℂ/Λ from lattices to elliptic curves. When is E(Λ) ≈ E(Λ′)? If Λ′ = cΛ for some c ∈ ℂ, then

[z] ↦ [cz] : ℂ/Λ → ℂ/Λ′

is an isomorphism of Riemann surfaces, and in fact one can show

E(Λ) ≈ E(Λ′) ⟺ Λ′ = cΛ, some c ∈ ℂ^{×}.

By scaling with an element of ℂ^{×}, we can normalize our lattices so that they are of the form

Λ(τ) = ℤ · 1 + ℤ · τ, some τ ∈ ℍ.

Note that Λ(τ) = Λ(τ′) if and only if there is a matrix ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) such that

τ′ = \frac{aτ+b}{cτ+d}. Thus we have a map

τ ↦ E(τ) : ℍ → {elliptic curves over ℂ}/≈,

and the above remarks show that it gives an injection

Γ(1)\ℍ ↪ {elliptic curves over ℂ}/≈ .

One shows that the function τ ↦ j(E(τ)) : ℍ → ℂ is holomorphic, and has only a simple pole at the cusp; in fact

j(τ) = q^{−1} + 744 + 196884q + 21493760q^{2} + ⋯, q = e^{2πiτ}.

It is therefore a modular function for the full modular group. One shows further that it defines an isomorphism j : Y (N) → ℂ. The surjectivity of j implies that every elliptic curve over ℂ is isomorphic to one of the form E(τ), some τ ∈ ℍ. Therefore

Γ(1)\ℍ ↔^{1:1} {elliptic curves over ℂ}/≈ .

The algebraic curve Y (1)_{ℚ} over ℚ naturally attached to Y (1) has the property that its points with coordinates in L, L a field containing ℚ, are given by

Y (1)(L) = {elliptic curves over L}/∼,

where E ∼ E′ if E and E′ become isomorphic over the algebraic closure of L. Moreover, this property determines Y (1)_{ℚ}.

From this, one sees that arithmetic facts about elliptic curves correspond to arithmetic facts about special values of modular functions and modular forms. For example, let E be an elliptic curve over a number field L; then, when regarded as an elliptic curve over ℂ, E is isomorphic to E(τ) for some τ ∈ ℂ, and we deduce that

j(τ) = j(E(τ)) = j(E) ∈ L,

i.e., the transcendental function j takes a value at τ which is algebraic! For example, if ℤ + ℤτ is the ring of integers in a quadratic imaginary field K, one can prove in this fashion that, not only is j(τ) algebraic, but it in fact generates the Hilbert class field of K (largest abelian extension of K unramified over K at all primes, including the infinite primes).

MODULAR FUNCTIONS AND MODULAR FORMS 7

Books. Ahlfors, L.V., Complex Analysis, McGraw-Hill, 1953. Cartan, H., Elementary Theory of Analytic Functions of One or Several Complex Variables, Addison Wesley, 1963. I much prefer this to Ahlfors’s book. Fulton, W., Algebraic Curves, Benjamin, 1969. Griffiths, P. A., Introduction to Algebraic Curves, AMS, 1989. Gunning, R., Lectures on Modular Forms, Princeton U. P., 1962. One of the first, and perhaps still the best, treatments of the basic material. Koblitz, N., Introduction to Elliptic Curves and Modular Forms, Springer, 1984. He studies elliptic curves, and uses modular curves to help with this; we do the opposite. Nevertheless, there will be a large overlap between this course and the book. Lang, S., Introduction to Modular Forms, Springer, 1976. The direction of this book is quite different from the course. Miyake, T., Modular Forms, Springer, 1976. This is a very good source for the analysis one needs to understand the arithmetic theory, but he doesn’t do much arithmetic. Ogg, A., Modular Forms and Dirichlet Series, Benjamin, 1969. A useful book, but the organization is a little strange. Schoeneberg, B., Elliptic Modular Functions, Springer, 1974. Again, he concentrates on the analysis rather than the arithmetic. Serre, J.-P., Cours d’Arithmétique, Presses Univ. de France, 1970. The last chapter is a beautiful, but brief, introduction to modular forms. Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Princeton U.P., 1971. An important book, but quite difficult. These notes may serve as an introduction to Shimura’s book, which covers much more. Silverman, J., The Arithmetic of Elliptic Curves, Springer, 1986. Silverman, J., Advanced Topics in the Arithmetic of Elliptic Curves, Springer, 1994.

References of the form Math xxx are to course notes available at http://www.math.lsa.umich.edu/∼jmilne

Part I: The Basic Theory

In this part, we develop the theory of modular functions and modular forms, and the Riemann surfaces on which they live.

1. Preliminaries

In this section we review some definitions and results concerning continuous group actions and Riemann surfaces. Following Bourbaki, we require (locally) compact spaces to be Hausdorff. We often use [x] to denote the equivalence class containing x.

Continuous group actions. Recall that a group G with a topology is a topological group if the maps

(g, g′) ↦ gg′ : G × G → G, g ↦ g^{−1} : G → G

are continuous. Let G be a topological group and let X be a topological space. An action of G on X,

(g, x) ↦ gx : G × X → X,

is continuous if this map is continuous. Then, for each g ∈ G, x ↦ gx : X → X is a homeomorphism (with inverse x ↦ g^{−1}x). An orbit under the action is the set Gx of translates of an x ∈ X. The stabilizer of x ∈ X (or the isotropy group at x) is

Stab(x) = {g ∈ G | gx = x}.

If X is Hausdorff, then Stab(x) is closed (it is the inverse image of x under of g ↦ gx : G → X). There is a bijection

G/Stab(x) → Gx, g · Stab(x) ↦ gx;

in particular, when G acts transitively on X, there is a bijection

G/Stab(x) → X.

Let G\X be the set of orbits for the action of G on X. It is the set of equivalence classes for the obvious equivalence relation, and so acquires the quotient topology: if p denotes the map x ↦ Gx : G → G\X, then U ⊂ G\X is open if and only if p^{−1}(U) is open in G. Note that p : X → G\X is both continuous and open. (It is continuous by definition; in fact, we have given G\X the finest topology for which p is a continuous map. Let U be an open subset of X; we want to show that p(U) is open. But p^{−1}(p(U)) = ⋃_{g∈G} gU, which is clearly open.)

Let H be a subgroup of G. Then H acts on G on the left and on the right, and H\G and G/H are the spaces of right and left cosets.

Lemma 1.1. The space G/H is Hausdorff if and only if H is closed in G.

Proof. Write p for the map G → G/H, g ↦ gH. If G/H is Hausdorff, then the point eH in G/H is closed, and so H = p^{−1}(eH) is closed (e =identity element).

Conversely, suppose H is a closed subgroup, and let aH and bH be distinct elements of G/H. Since G is a topological group, the map

f : G × G → G, (g, g′) ↦ g^{−1}g′,

is continuous, and so f^{−1}(H) is closed. As aH ≠ bH, (a, b) ∉ f^{−1}(H), and so there is an open neighbourhood of (a, b), which we can take to be of the form U × V, that is disjoint from f^{−1}(H). Now the images of U and V in G/H are disjoint open neighbourhoods of aH and bH.

As we noted above, when G acts transitively on X, there is a bijection G/Stab(x) → X for any x ∈ X. Under some mild hypotheses, this will be a homeomorphism.

Proposition 1.2. Suppose G acts continuously and transitively on X. If G and X are locally compact and Hausdorff, and there is a countable basis for the topology of G, then the map

[g] ↦ gx : G/Stab(x) → X

is a homeomorphism.

Proof. We know the map is a bijection, and it is obvious from the definitions that it is continuous, and so we only have to show that it is open. Let U be an open subset of G, and let g ∈ U; we have to show that gx is an interior point of Ux.

Consider the map G × G → G, (h, h′) ↦ ghh′. It is continuous and maps (e, e) into U, and so there is a neighbourhood V of e, which we can take to be compact^{1}, such that V × V is mapped into U; thus gV^{2} ⊂ U. After replacing V with V ∩ V^{−1}, we can assume V^{−1} = V. (Here V^{−1} = {h^{−1} | h ∈ V }; V^{2} = {hh′ | h, h′ ∈ V }.)

As e ∈ V, G = ⋃ gV (union over g ∈ G). Each set gV is a union of open sets in the countable basis, and we only need to take enough g’s in order to get each basic open set contained in a gV at least once. Therefore, there is a countable set of elements g_{1}, g_{2}, … ∈ G such that G = ⋃ g_{n}V.

As g_{n}V is compact, its image g_{n}V x in X is compact, and as X is Hausdorff, this implies that g_{n}V x is closed. The following lemma shows that at least one of the g_{n}V x’s has an interior point. But y ↦ g_{n}y : X → X is a homeomorphism mapping V x onto g_{n}V x, and so V x has interior point, i.e., there is a point hx ∈ V x and an open subset W of X such that hx ∈ W ⊂ V x. Now

gx = gh^{−1} · hx ∈ gh^{−1}W ⊂ gV^{2}x ⊂ Ux

which shows that gx is an interior point of Ux.

Lemma 1.3. Let X be a nonempty locally compact (hence Hausdorff) space, and suppose X = ⋃ V_{n} where (V_{n}) is a countable family of closed subsets. Then at least one of the V_{n} has an interior point.

Proof. The hypotheses imply that X is regular: the points of X are closed, and for any point x not in a closed set A, there are disjoint open sets U and V containing x and A respectively.

Suppose no V_{n} has an interior point. Take U_{1} to be any nonempty open subset of X whose closure Ū_{1} is compact. As V_{1} has empty interior, U_{1} is not contained in V_{1}, and because U_{1} is regular, there is a nonempty open subset U_{2} of U_{1} such that Ū_{2} ⊂ U_{1} − V_{1}. Continuing in this fashion, we obtain nonempty open sets U_{3}, U_{4} ... such that Ū_{n+1} ⊂ U_{n} − V_{n}. The Ū_{n} form a decreasing sequence of nonempty compact sets, and so ∩Ū_{n} ≠ ∅, which contradicts X = ⋃ V_{n}.

1not necessarily open

Riemann surfaces: classical approach. Let X be a connected Hausdorff topological space. A coordinate neighbourhood for X is pair (U, z) with U an open subset of X and z a homeomorphism of U onto an open subset of the complex plane ℂ. Two coordinate neighbourhoods (U_{i}, z_{i}) and (U_{j}, z_{j}) are compatible if the function

z_{i} ∘ z_{j}^{−1} : z_{j}(U_{i} ∩ U_{j}) → z_{i}(U_{i} ∩ U_{j})

is holomorphic with nowhere vanishing derivative (the condition is vacuous if U_{i} ∩ U_{j} = ∅). A family of coordinate neighbourhoods (U_{i}, z_{i})_{i∈I} is a coordinate covering if X = ⋃ U_{i} and (U_{i}, z_{i}) is compatible with (U_{j}, z_{j}) for all pairs (i, j) ∈ I × I. Two coordinate coverings are said to be equivalent if their union is also a coordinate covering. This defines an equivalence relation on the set coordinate coverings, and we call an equivalence class of coordinate coverings a complex structure on X. A space X together with a complex structure is a Riemann surface.

Let U = (U_{i}, z_{i})_{i∈I} be a coordinate covering of X. A function f : U → ℂ on an open subset U of X is said to be holomorphic relative to U if

f ∘ z^{−1} : z(U ∩ U_{i}) → ℂ

is holomorphic for all i ∈ I. When U′ is an equivalent coordinate covering, f is holomorphic relative to U if and only if it is holomorphic relative to U′, and so it makes sense to say that f is holomorphic relative to a complex structure on X: a function f : U → ℂ on an open subset U of a Riemann surface X is holomorphic if it is holomorphic relative to one (hence every) coordinate covering defining the complex structure on X. Recall that a meromorphic function on an open subset U of ℂ is a holomorphic function on f on U − Ξ, for some discrete subset Ξ of U, which has at worst a pole at each point of Ξ (i.e., for each a ∈ Ξ, there exists an m such that (z − a)^{m}f(z) is holomorphic in some neighbourhood of a). A meromorphic function on an open subset U of a Riemann surface is defined exactly the same way.

Example 1.4. Any open subset U of ℂ is a Riemann surface with a single coordinate neighbourhood (U itself, with the identity function z). The holomorphic and meromorphic functions on U with this structure of a Riemann surface are just the usual holomorphic and meromorphic functions.

Example 1.5. Let X be the unit sphere

S : x^{2} + y^{2} + z^{2} = 1

in ℝ^{3}. Let P be the north pole (0, 0, 1). Stereographic projection from P gives a map

(x, y, z) ↦ \frac{x + iy}{1 − z} : X − P → ℂ.

Take this to be a coordinate neighbourhood for X. Similarly, stereographic projection from the south pole S gives a second coordinate neighbourhood. These two coordinate neighbourhoods define a complex structure on X, and X together with this complex structure is called the Riemann sphere.

Example 1.6. Let X be the torus ℝ^{2}/ℤ^{2}. We shall see that there are infinitely many different complex structures on X.

A map f : X → X′ from one Riemann surface to a second is holomorphic if for each point P of X, there are coordinate neighbourhoods (U, z) of P and (U′, z′) of f(P) such that z′ ∘ f ∘ z^{−1} : z(U) → z(U′) is holomorphic. An isomorphism of Riemann surfaces is a bijective holomorphic map whose inverse is also holomorphic.

Riemann surfaces as ringed spaces. Fix a field k. Let X be a topological space, and suppose that for each open subset U of X, we are given a set O(U) of functions U → k. Then O is called a sheaf of k-algebras on X if

(a) f, g ∈ O(U) ⟹ f ± g, fg ∈ O(U); the function x ↦ 1 is in O(U);

(b) f ∈ O(U), V ⊂ U ⟹ f|V ∈ O(V);

(c) let U = ⋃ U_{i} be an open covering of an open subset U of X, and for each i, let f_{i} ∈ O(U_{i}); if f_{i}|U_{i} ∩ U_{j} = f_{j}|U_{i} ∩ U_{j} for all i, j, then there exists an f ∈ O(U) such that f|U_{i} = f_{i} for all i.

When Y is an open subset of X, we obtain a sheaf of k-algebras O|Y on Y by restricting the map U ↦ O(U) to the open subsets of Y, i.e., for all open U ⊂ Y, define (O|Y)(U) = O(U).

From now on, by a ringed space we shall mean a pair (X, O_{X}) with X a topological space and O_{X} a sheaf of ℂ-algebras—we often omit the subscript on O. A morphism φ : (X, O) → (X′, O′) of ringed spaces is a continuous map φ : X → X′ such that, for all open subsets U′ of X′,

f ∈ O′(U′) ⟹ f ∘ φ ∈ O(φ^{−1}(U′)).

An isomorphism φ : (X, O) → (X′, O′) of ringed spaces is a homeomorphism such that φ and φ^{−1} are morphisms. Thus a homeomorphism φ : X → X′ is an isomorphism of ringed spaces if, for every U open in X with image U′ in X′, the map

f ↦ f ∘ φ : O′(U′) → O(U)

is bijective.

For example, on any open subset V of the complex plane ℂ, there is a sheaf O_{V} with

O_{V} (U) = {holomorphic functions f : U → ℂ},

all open U ⊂ V. We call such a pair (V, O_{V}) a standard ringed space.

The following statements (concerning a Hausdorff topological space X) are all easy to prove.

1.7. Let U = (U_{i}, z_{i}) be a coordinate covering of X, and, for any open subset U of ℂ, let O(U) be the set of functions f : U → C that are holomorphic relative to U. Then U ↦ O(U) is a sheaf of ℂ-algebras on X.

1.8. Let U and U′ be coordinate coverings of X; then U and U′ are equivalent if and only they define the same sheaves of holomorphic functions.

Thus, a complex structure on X defines a sheaf of ℂ-algebras on X, and the sheaf uniquely determines the complex structure.

1.9. A sheaf O_{X} of ℂ-algebras on X arises from a complex structure if and only if it satisfies the following condition:

(∗) there is an open covering X = ⋃ U_{i} of X such that each (U_{i}, O_{X}|U_{i})

is isomorphic to a standard ringed space.

Thus to give a complex structure on X is the same as to give a sheaf of ℂ-algebras satisfying (∗).

Example 1.10. Let n ∈ ℤ act on ℂ as z ↦ z +n. Topologically, ℂ/ℤ is cylinder. We can give it a complex structure as follows: let p : ℂ → ℂ/ℤ be the quotient map; for any point P ∈ ℂ/ℤ, choose a Q ∈ f^{−1}(P); there exist neighbourhoods U of P and V of Q such that p is a homeomorphism V → U; take any such pair (U, p^{−1} : U → V) to be a coordinate neighbourhood. The corresponding sheaf of holomorphic functions has the following description: for any open subset U of ℂ/ℤ, a function f : U → ℂ is holomorphic if and only if f ∘ p is holomorphic (check!). Thus the holomorphic functions f on U ⊂ ℂ/ℤ can be identified with the holomorphic functions on p^{−1}(U) invariant under the action of ℤ, i.e., such that f(z +n) = f(z) for all n ∈ ℤ (it suffices to check that f(z + 1) = f(z), as 1 generates ℤ as an abelian group).

For example, q(z) = e^{2πiz} defines a holomorphic function on ℂ/ℤ. It gives an isomorphism ℂ/ℤ → ℂ^{×} (complex plane with the origin removed)—in fact, this is an isomorphism of both of Riemann surfaces and of topological groups. The inverse function ℂ^{×} → ℂ/ℤ is (by definition of log) (2πi)^{−1} · log .

Before Riemann (and, unfortunately, also after), mathematicians considered functions only on open subsets of the complex plane ℂ. Thus they were forced to talk about “multi-valued functions” and functions “holomorphic at points at infinity”. This works reasonably well for functions of one variable, but collapses into total confusion in several variables. Riemann recognized that the functions were defined in a natural way on spaces that were only locally isomorphic to open subsets of ℂ, that is, on Riemann surfaces, and emphasized the importance of studying these spaces. In this course we follow Riemann—it may have been more natural to call the course “ Elliptic Modular Curves” rather than “ Modular Functions and Modular Forms”.

Differential forms. We adopt a naive approach to differential forms on Riemann surfaces.

A differential form on an open subset U of ℂ is an expression of the form f(z)dz where f is a meromorphic function on U. With any meromorphic function f(z) on U, we associate the differential form df =^{df} \frac{df}{dz}dz. Let w : U → U′ be a mapping from U to another open subset U′ of ℂ; we can write it z′ = w(z). Let ω = f(z′)dz′ be a differential form on U′. Then w^{∗}(ω) is the differential form f(w(z))\frac{dw(z)}{dz}dz on U.

Let X be a Riemann surface, and let (U_{i}, z_{i}) be a coordinate covering of X. To give a differential form on X is to give differential forms ω_{i} = f(z_{i})dz_{i} on z_{i}(U_{i}) for each i that agree on overlaps in the following sense: let z_{i} = w_{ij}(z_{j}), so that w_{ij} is the conformal mapping z_{i} ∘ z_{j}^{−1} : z_{j}(U_{i} ∩ U_{j}) → z_{i}(U_{i} ∩ U_{j}); then w_{ij}^{∗}(ω_{i}) = ω_{j}, i.e.,

f_{j}(z_{j})dz_{j} = f_{i}(w_{ij}(z_{j})) · w′_{ij}(z_{j})dz_{j}.

Contrast this with functions: to give a meromorphic function f on X is to give meromorphic functions f_{i}(z_{i}) on z_{i}(U_{i}) for each i that agree on overlaps in the sense that

f_{j}(z_{j}) = f_{i}(w_{ij}(z_{j})) on z_{j}(U_{i} ∩ U_{j}).

A differential form is said to be of the first kind (or holomorphic) if it has no poles on X, of the second kind if it has residue 0 at each point of X where it has a pole, and of the third kind if it is not of the second kind.

Example 1.11. The Riemann sphere S can be thought of as the set of lines through the origin in ℂ^{2}. Thus a point on S is determined by a point (other than the origin) on the line. In this way, the Riemann sphere is identified with

ℙ^{1}(ℂ) = (ℂ × ℂ \ {(0, 0)})/ℂ^{×}.

We write (x_{0} : x_{1}) for the equivalence class of (x_{0}, x_{1}); thus (x_{0} : x_{1}) = (cx_{0} : cx_{1}) for c ≠ 0.

Let U_{0} be the subset where x_{0} ≠ 0; then z_{0} : (x_{0} : x_{1}) ↦ x_{1}/x_{0} is a homeomorphism U_{0} → ℂ. Similarly, if U_{1} is the set where x_{1} ≠ 0, then z_{1} : (x_{0} : x_{1}) ↦ x_{0}/x_{1} is a homeomorphism U_{1} → ℂ. The pair (U_{0}, z_{0}), (U_{1}, z_{1}) is a coordinate covering of S. Note that on U_{0} ∩ U_{1}, z_{0} and z_{1} are both defined, and z_{1} = z_{0}^{−1}; in fact, z_{0}(U_{0} ∩ U_{1}) = ℂ \ {0} = z_{1}(U_{0} ∩ U_{1}) and the map w_{01} : z_{1}(U_{0} ∩ U_{1}) → z_{0}(U_{0} ∩ U_{1}) is z ↦ z^{−1}.

A meromorphic function on S is defined by a meromorphic function f_{0}(z_{0}) of z_{0} ∈ ℂ and a meromorphic function f_{1}(z_{1}) of z_{1} ∈ ℂ such that for z_{0}z_{1} ≠ 0, f_{1}(z_{1}) = f_{0}(z_{1}^{−1}). In other words, it is defined by a meromorphic function f(z)(= f_{1}(z_{1})) such that f(z^{−1}) is also meromorphic on ℂ. (It is automatically meromorphic on ℂ \ {0}.) In all good complex analysis courses it is shown that the meromorphic functions on S are exactly the rational functions of z, namely, the functions P (z)/Q(z), P, Q ∈ ℂ[X], Q ≠ 0 (see 1.14 below).

A meromorphic differential form on S is defined by a differential form f_{0}(z_{0})dz_{0} on ℂ and a differential form f_{1}(z_{1})dz_{1} on ℂ, such that

f_{1}(z_{1}) = f_{0}(z_{1}^{−1}) · \frac{−1}{z_{1}^{2}} for z_{1} ≠ 0.

Analysis on compact Riemann surfaces. We merely sketch what we need. For details, see for example R. Gunning, Lectures on Riemann Surfaces, Princeton, 1966, or P. Griffiths, Introduction to Algebraic Curves, AMS, 1989. Note that a Riemann surface X (considered as a topological space) is orientable: each open subset of the complex plane has a natural orientation; hence each coordinate neighbourhood of X has a natural orientation, and these agree on overlaps because conformal mappings preserve orientation. Also note that a holomorphic mapping f : X → S (the Riemann sphere) can be regarded as a meromorphic function on X, and that all meromorphic functions are of this form. The only functions holomorphic on the whole of a compact Riemann surface are the constant functions.

Proposition 1.12. (a) A meromorphic function f on a compact Riemann

surface has the same number of poles as it has zeros (counting multiplicities).

(b) Let ω be a differential form on a compact Riemann surface; then the sum of the residues of ω at its poles is zero.

Proof. (Sketch) We first prove (b). Recall that if ω = fdz is a differential form on an open subset of ℂ and C is any closed path in ℂ not passing through any poles of f, then

∫_{C} ω = 2πi(∑_{poles} Res_{p}ω)

(sum over the poles p enclosed by C). Fix a finite coordinate covering (U_{i}, z_{i}) of the Riemann surface, and choose a triangulation of the Riemann surface such that each triangle is completely enclosed in some U_{i}; then 2πi(∑ Res_{p}ω) is the sum of the integrals of ω over the various paths, but these cancel out.

Statement (a) is just the special case of (b) in which ω = df/f.

When we apply (a) to f − c, c some fixed number, we obtain the following result.

Corollary 1.13. Let f be a nonconstant meromorphic function on a compact Riemann surface X. Then there is an integer n > 0 such that f takes each value exactly n times (counting multiplicities).

Proof. The number n is equal to the number of poles of f.

The integer n is called the valence of f. A constant function is said to have valence 0. If f has valence n, then it defines a function X → S (Riemann sphere) which is n to 1 (counting multiplicities). In fact, there will be only finitely many ramification points, i.e., points where it is not exactly n to 1 (when one doesn’t count multiplicities).

Proposition 1.14. Let S be the Riemann sphere. The meromorphic functions are precisely the rational functions of z, i.e., the field of meromorphic functions on S is ℂ(z).

Proof. Let g(z) be a meromorphic function on S. After possibly replacing g(z) with g(z − c), we may suppose that g(z) has neither a zero nor a pole at ∞ (= north pole). Suppose that g(z) has a pole of order m_{i} at p_{i}, i = 1, …, r, a zero of order n_{i} at q_{i}, i = 1, …, s, and no other poles or zero. The function

g(z)\frac{∏(z − p_{i})^{m_{i}}}{∏(z − q_{i})^{n_{i}}}

has no zeros or poles at a point P ≠ ∞, and it has no zero or pole at ∞ because (see 1.12) ∑ m_{i} = ∑ n_{i}. It is therefore constant, and so

g(z) = constant × \frac{∏(z − q_{i})^{n_{i}}}{∏(z − p_{i})^{m_{i}}} .

Remark 1.15. The proposition shows that the meromorphic functions on S are all algebraic: they are just quotients of polynomials. Thus the field M(S) of meromorphic functions on S is equal to the field of rational functions on ℙ^{1} as defined by algebraic geometry. This is dramatically different from what is true for meromorphic functions on the complex plane. In fact, there exists a vast array of holomorphic functions on ℂ—see Ahlfors for a classification of them (IV.3.3 of the first edition).

Proposition 1.16. Let f be a nonconstant meromorphic function with valence n on a compact Riemann surface X. Then any meromorphic function g on X is a root of a polynomial of degree ≤ n with coefficients in ℂ(f).

Proof. (Sketch) Regard f as a mapping X → S (Riemann sphere) and let c be a point of S such that f^{−1}(c) has exactly n elements {P_{1}(c), ..., P_{n}(c)}. Let z ∈ X be such that f(z) = c; then

0 = ∏_{i}(g(z) − g(P_{i}(c))) = g^{n}(z) + r_{1}(c)g^{n−1}(z) + ⋯ + r_{n}(c)

where the r_{i}(c) are symmetric functions in the g(P_{i}(c)). When we let c vary (avoiding the c where f(z) − c has multiple zeros), each r_{i}(c) becomes a meromorphic function on S, and hence is a rational function of c = f(z).

Theorem 1.17. Let X be a compact Riemann surface. There exists a nonconstant meromorphic function f on X, and the set of such functions forms a finitely generated field M(X) of transcendence degree 1 over ℂ.

The first statement is the fundamental existence theorem (Gunning, ibid., p107). Its proof is not easy (it is implied by the Riemann-Roch Theorem), but for all the Riemann surfaces in this course, we will be able to write down a nonconstant meromorphic function.

It is obvious that the meromorphic functions on X form a field M(X). Let f be a nonconstant such function, and let n be its valence. Then (1.16) shows that every other function is algebraic over ℂ(f), and in fact satisfies a polynomial of degree ≤ n. Therefore M(f) has degree ≤ n over ℂ(f), because if it had degree > n then it would contain a subfield L of finite degree n′ > n over ℂ(f), and the Primitive Element Theorem (Math 594f, 5.1) tells us that then L = ℂ(f)(g) for some g whose minimum polynomial has degree n′.

Example 1.18. Let S be the Riemann sphere. For any meromorphic function f on S with valence 1, M(S) = ℂ(f).

Remark 1.19. The meromorphic functions on a compact complex manifold X of dimension m > 1 again form a field that is finitely generated over ℂ, but its transcendence degree may be < m. For example, there are compact complex manifolds of dimension 2 with no nonconstant meromorphic functions.

Riemann-Roch Theorem. The Riemann-Roch theorem describes how many functions there are on a compact Riemann surface with given poles and zeros.

Let X be a compact Riemann surface. The group of divisors Div(X) on X is the free (additive) abelian group generated by the points on X; thus an element of Div(X) is a finite sum ∑ n_{i}P_{i}, n_{i} ∈ ℤ. A divisor D = ∑ n_{i}P_{i} is positive (or effective) if every n_{i} ≥ 0; we then write D ≥ 0.

Let f be a nonzero meromorphic function on X. For any point P ∈ X, let ord_{P} (f) = m, −m, or 0 according as f has a zero of order m at P, a pole of order m at P, or neither a pole nor a zero at P. The divisor of f is

div(f) = ∑ ord_{p}(f) · P.

This is a finite sum because the zeros and poles of f form discrete sets, and we are assuming X to be compact.

The map f ↦ div(f) : ℂ(X)^{×} → Div(X) is a homomorphism, and its image is called the group of principal divisors. Two divisors are said to be linearly equivalent if their difference is principal. The degree of a divisor ∑ n_{i}P_{i} is ∑ n_{i}. The map D ↦ deg(D) is a homomorphism Div(X) → ℤ whose kernel contains the principal divisors. Thus it makes sense to speak of the degree of a linear equivalence class of divisors.

It is possible to attach a divisor to a differential form ω: let P ∈ X, and let (U_{i}, z_{i}) be a coordinate neighbourhood containing P; the differential form ω is described by a differential f_{i}dz_{i} on U_{i}, and we set ord_{p}(ω) = ord_{p}(f_{i}). Then ord_{p}(ω) is independent of the choice of the coordinate neighbourhood U_{i} (because w_{ij} and its derivative have no zeros or poles), and we define

div(ω) = ∑ ord_{p}(ω) · P.

Again, this is a finite sum. Note that, for any meromorphic function f,

div(fω) = div(f) + div(ω).

If ω is one nonzero differential form, then any other is of the form fω for some f ∈ M(X), and so the linear equivalence class of div(ω) is independent of ω; we write K for div(ω), and 𝔨 for its linear equivalence class.

For a divisor D, define

L(D) = {f ∈ M(X) | div(f) + D ≥ 0} ∪ {0}.

This is a vector space over ℂ, and if D′ = D + (g), then f ↦ fg^{−1} is an isomorphism L(D) → L(D′). Thus the dimension ℓ(D) of L(D) depends only on the linear equivalence class of D.

Theorem 1.20 (Riemann-Roch). Let X be a compact Riemann surface. Then there is an integer g ≥ 0 such that for any divisor D,

ℓ(D) = deg(D) + 1 − g + ℓ(K − D).

Proof. See Gunning 1962, §7, or Griffiths 1989, for a proof in the context of Riemann surfaces, and Fulton 1969, Chapter 8, for a proof in the context of algebraic curves. One approach to proving it is to verify it first for the Riemann sphere S (see below), and then to regard X as a finite covering of S.

Note that in the statement of the Riemann-Roch Theorem, we could replace the divisors with equivalence classes of divisors.

Corollary 1.21. A canonical divisor K has degree 2g − 2, and ℓ(K) = g.

Proof. Put D = 0 in (1.20). The only functions with div(f) ≥ 0 are the constant functions, and so the equation becomes 1 = 0 + 1 − g + ℓ(K). Hence ℓ(K) = g. Put D = K; then the equation becomes g = deg(K) + 1 − g + 1, which gives deg(K) = 2g − 2.

Let K = div(ω). Then f ↦ fω is an isomorphism from L(D) to the space of holomorphic differential forms on X, which therefore has dimension g.

The term in the Riemann-Roch formula that is difficult to evaluate is ℓ(K − D). Thus it is useful to note that if deg(D) > 2g − 2, then L(K − D) = 0 (because, for f ∈ M(X)^{×}, deg(D) > 2g − 2 ⟹ deg(div(f) + K − D) < 0, and so div(f) + K − D can’t be a positive divisor). Hence:

Corollary 1.22. If deg(D) > 2g − 2, then ℓ(D) = deg(D) + 1 − g.

Example 1.23. Let X be the Riemann sphere, and let D = mP_{∞}, where P_{∞} is the “point at infinity”. Then L(D) is the space of meromorphic functions on ℂ with at worst a pole of order m at infinity and no poles elsewhere. These functions are the polynomials of degree ≤ m, and they form a vector space of m + 1, in other words,

ℓ(D) = deg(D) + 1,

and so the Riemann-Roch theorem shows that g = 0. Consider the differential dz on ℂ, and let z′ = 1/z. The dz = −1/z′^{2}dz′, and so dz extends to a meromorphic differential on X with a pole of order 2 at ∞. Thus deg(div(ω)) = −2, in agreement with the above formulas.

Exercise 1.24. Prove (1.20) for the Riemann sphere. (Hint: use partial fractions.)

The genus of X. Let X be a compact Riemann surface. It can be regarded as a topological space, and so we can define homology groups H_{0}(X, ℚ), H_{1}(X, ℚ), H_{2}(X, ℚ). It is known that H_{0} and H_{2} each have dimension 1, and H_{1} has dimension 2g. It is a theorem that this g is the same as that occurring in the Riemann-Roch theorem (see below). Hence g depends only on X as a topological space, and not on its complex structure. The Euler-Poincaré characteristic of X is

χ(X) =^{df} dim H_{0} − dim H_{1} + dim H_{2} = 2 − 2g.

Since X is oriented, it can be triangulated. When one chooses a triangulation, then one finds (easily) that

2 − 2g = V − E + F,

where V is the number of vertices, E is the number of edges, and F is the number of faces.

Example 1.25. Triangulate the sphere by projecting out from a regular tetrahedron whose vertices are on the sphere. Then V = 4, E = 6, F = 4, and so g = 0.

Example 1.26. Consider the map z ↦ z^{e} : D → D, where D is the unit open disk. This map is exactly e : 1 except at the origin, which is a ramification point of order e. Consider the differential dz′ on D. The map is z′ = w(z) = z^{e}, and so the inverse image of the differential dz′ is dz′ = dw(z) = ez^{e−1}dz. Thus w^{∗}(dz′) has a zero of order e − 1 at 0.

Theorem 1.27 (Riemann-Hurwitz Formula). Let f : Y → X be a holomorphic mapping of compact Riemann surfaces that is m : 1 (except over finitely many points). For each point P of X, let e_{P} be the multiplicity of P in the fibre of f; then

2g(Y) − 2 = (2g(X) − 2)m + ∑(e_{P} − 1).

Proof. Choose a differential ω on X such that ω has no pole or zero at a ramification point of X. Then f^{∗}ω has a pole and a zero above each pole and zero of ω (of the same order as that of ω); in addition it has a zero of order e − 1 at each ramification point in Y (cf. the above example). Thus

deg(f^{∗}ω) = m deg(ω) + ∑(e_{P} − 1),

and we can apply (1.21).

Remark 1.28. One can also prove this formula topologically. Triangulate X in such a way that each ramification point is a vertex for the triangulation, and pull the triangulation back to Y. There are the following formulas for the numbers of faces, edges, and vertices for the triangulations of Y and X :

F (Y) = m · F (X), E(Y) = m · E(X), V (Y) = m · V (X) − ∑(e_{P} − 1).

Thus

2 − 2g(Y) = (2 − 2g(X)) − ∑(e_{P} − 1),

in agreement with (1.27).

We have verified that the two notions of genus agree for the Riemann sphere S (they both give 0). But for any Riemann surface X, there is a nonconstant function f : X → S (by 1.17) and we have just observed that the formulas relating the genus of X to that of S is the same for the two notions of genus, and so we have shown that the two notions give the same value for X.

Riemann surfaces as algebraic curves. Let X be a compact Riemann surface. Then (see 1.17) M(X) is a finitely generated field of transcendence degree 1 over ℂ, and so there exist meromorphic functions f and g on X such that M(X) = ℂ(f, g). There is a nonzero irreducible polynomial Φ(X, Y) such that

Φ(f, g) = 0.

Then z ↦ (f(z), g(z)) : X → ℂ^{2} maps an open subset of X onto an open subset of the algebraic curve defined by the equation:

Φ(X, Y) = 0.

Unfortunately, this algebraic curve will in general have singularities. A better approach is the following. Suppose that the genus of X is ≥ 2 (and X is not hyperelliptic), and choose a basis ω_{0}, ..., ω_{n}, (n = g −1) for the space of holomorphic differential forms on X. For P ∈ X, we can represent each ω_{i} in the form f_{i} · dz in some neighbourhood of P. After possibly replacing each ω_{i} with fω_{i}, f a meromorphic function defined near P, the f_{i}’s will all be defined at P, and at least one will be nonzero at P. Thus (f_{0}(P) : …: f_{n}(P)) is a well-defined point of ℙ^{n}(ℂ), independent of the choice of f. It is known that the map φ

P ↦ (f_{0}(P) : ... : f_{n}(P)) : X → ℙ^{n}(ℂ)

is a homeomorphism of X onto a closed subset of ℙ^{n}(ℂ), and that there is a finite set of homogeneous polynomials in n + 1 variables whose zero set is precisely φ(X). Moreover, the image is a nonsingular curve in ℙ^{n}(ℂ) (Griffiths 1989, IV.3). If X has genus < 2, or is hyperelliptic, a slight modification of this method again realizes X as an algebraic curve in ℙ^{n} for some n.

2. Elliptic Modular Curves as Riemann Surfaces

In this section, we define the Riemann surfaces Y (N) = Γ(N)\ℍ and their natural compactifications, X(N). Recall that ℍ is the complex upper half plane

ℍ = {z ∈ ℂ | ℑ(z) > 0}.

The upper-half plane as a quotient of SL_{2}(ℝ). We saw in the Introduction that there is an action of SL_{2}(ℝ) on ℍ as follows:

SL_{2}(ℝ) × ℍ → ℍ, (α, z) ↦ α(z) = \frac{az + b}{cz + d}, α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

Because ℑ(αz) = ℑ(z)/|cz + d|^{2}, ℑ(z) > 0 ⟹ ℑ(αz) > 0. When we give SL_{2}(ℝ) and ℍ their natural topologies, this action is continuous.

The special orthogonal group (or “circle group”) is defined to be

SO_{2}(ℝ) = \{( \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ) | θ ∈ ℝ\} .

Note that SO_{2}(ℝ) is a closed subgroup of SL_{2}(ℝ), and so SL_{2}(ℝ)/ SO_{2}(ℝ) is a Hausdorff topological space (by 1.1).

Proposition 2.1. (a) The group SL_{2}(ℝ) acts transitively on ℍ, i.e., for any

elements z, z′ ∈ ℍ, there exists an α ∈ SL_{2}(ℝ) such that αz = z′.

(b) The action of SL_{2}(ℝ) on ℍ induces an isomorphism

SL_{2}(ℝ)/{±I} → Aut(ℍ) (biholomorphic automorphisms of ℍ)

(c) The stabilizer of i is SO_{2}(ℝ).

(d) The map

SL_{2}(ℝ)/ SO_{2}(R) → H, α · SO_{2}(R) ↦ α(i)

is a homeomorphism.

Proof. (a) Let z ∈ ℍ; it suffices to show that there exists an α ∈ SL_{2}(ℝ) such that α(i) = z—if z′ is a second point, then α′(i) = z′ for some α′ ∈ SL_{2}(ℝ), and

α′α^{−1}(z) = z′. Write z = x + iy; then α =^{df} \sqrt{y}^{−1} ( \begin{matrix}y & x \\ 0 & 1 \end{matrix} ) ∈

SL_{2}(ℝ), and α(i) = z.

(b) If ( \begin{matrix}a & b \\ c & d \end{matrix} )·z = z then cz^{2} + (d − a)z − b = 0. If this is true for all z ∈ ℍ (any

three z’s would do), then the polynomial must have zero coefficients, and so c = 0,

d = a, and b = 0. Thus ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ), and this has determinant 1 if and

only if a = ±1. Thus only ±I act trivially on ℍ.

Let γ be an automorphism ℍ. We know from (a) that there is an α ∈ SL_{2}(ℝ) such that α(i) = γ(i). After replacing γ with α^{−1} ∘ γ, we can assume that γ(i) = i. Recall that the map ρ : ℍ → D, z ↦ \frac{z−i}{z+i} is an isomorphism from ℍ onto the open unit disk, and it maps i to 0. Use ρ to transfer γ into an automorphism γ′ of D fixing

Lemma 2.2. The automorphisms of D fixing 0 are the maps of the form z ↦ λz, |λ| = 1.

Proof. This is an easy consequence of the Schwarz Lemma (Cartan 1963, III.3), which says the following:

Let f(z) be a holomorphic function on the disk |z| < 1 and suppose that

f(0) = 0, |f(z)| < 1 for |z| < 1.

Then (i) |f(z)| ≤ |z| for |z| < 1; (ii) if |f(z_{0})| = |z_{0}| for some z_{0} ≠ 0, then there is a λ such that f(z) =

λz (and |λ| = 1).

Let γ be an automorphism of D fixing 0. When we apply (i) to γ and γ^{−1}, we find that |γ(z)| = |z| for all z in the disk, and so we can apply (ii) to find that f is of the required form.

The lemma tells us that there is a θ ∈ ℝ such that ρ ∘ γ ∘ ρ^{−1}(z) = e^{2θi} · z

for all z, and the following exercise shows that γ(z) = ( \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ) · z. Thus

γ ∈ SO_{2}(ℝ) ⊂ SL_{2}(R).

(c) We have already proved this, but it is easy to give a direct proof. We have

\frac{ai + b}{ci + d} = i ⟺ ai + b = −c + di ⟺ a = d, b = −c.

Therefore the matrix is of the form ( \begin{matrix}a & −b \\ b & a \end{matrix} ) with a^{2} + b^{2} = 1, and so is in SO_{2}(ℝ).

(d) This is a consequence of the general result (1.2).

Exercise 2.3. Let ψ : ℂ^{2} × ℂ^{2} → ℂ be the Hermitian form

(( \begin{matrix}z_{1} \\ z_{2} \end{matrix} ) , ( \begin{matrix}w_{1} \\ w_{2} \end{matrix} )) ↦ z̄_{1}w_{1} − z_{2}w̄_{2}.

and let SU(1, 1) (special unitary group) be the subgroup of elements α ∈ SL_{2}( ℂ) such that ψ(α(z), α(w)) = ψ(z, w).

(a) Show that SU(1, 1) = \{( \begin{matrix}u & v \\ v̄ & ū \end{matrix} ) | u, v ∈ ℂ, |u|^{2} − |v|^{2} = 1\}.

(b) Define an action of SU(1, 1) on the unit disk as follows:

( \begin{matrix}u & v \\ v̄ & ū \end{matrix} ) · z = \frac{uz + v}{v̄z + ū}.

Show that this defines an isomorphism SU(1, 1)/{±I} → Aut(D).

(c) Show that, under the standard isomorphism ρ : ℍ → D, the action of

the element ( \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ) of SL_{2}(ℝ) on ℍ corresponds to the action of

( \begin{matrix}e^{iθ} & 0 \\ 0 & e^{−iθ} \end{matrix} ) on D.

Quotients of ℍ. Let Γ be a group acting on a topological space X. If Γ\X is Hausdorff, then the orbits are closed, but this condition is not sufficient to ensure that the quotient space is Hausdorff. The action is said to be discontinuous if for every x ∈ X and infinite sequence (γ_{i}) of distinct elements of Γ, the set {γ_{i}x} has no cluster point; it is said to be properly discontinuous if, for any pair of points x and y of X, there exist neighbourhoods U_{x} and U_{y} of x and y such that the set {γ ∈ Γ | γU_{x} ∩ U_{y} ≠ ∅} is finite.

Proposition 2.4. Let G be a locally compact group acting on a topological space X such that for one (hence every) point x_{0} ∈ X, the stabilizer K of x_{0} in G is compact and gK ↦ gx_{0} : G/K → X is a homeomorphism. The following conditions on a subgroup Γ of G are equivalent:

(a) Γ acts discontinuously on X;

(b) Γ acts properly discontinuously on X;

(c) for any compact subsets A and B of X, {γ ∈ Γ| γA ∩ B ≠ ∅} is finite;

(d) Γ is a discrete subgroup of G.

Proof. (d) ⟹ (c) (This is the only implication we shall use.) Write p for the map, gK ↦ gx_{0} : G → X. Let A be a compact subset of X. I claim that p^{−1}(A) is compact. Write G = ⋃ V_{i} where the V_{i} are open with compact closures V̄_{i}. Then A ⊂ ⋃ p(V_{i}), and in fact we need only finitely many p(V_{i})’sto cover A. Then p^{−1}(A) ⊂ ⋃ V_{i}K ⊂ ⋃ V̄_{i}K (finite union), and each V̄_{i}K is compact (it is the image of V̄_{i} × K under the multiplication map G × G → G). Thus p^{−1}(A) is a closed subset of a compact set, and so is compact. Similarly, p^{−1}(B) is compact.

Suppose γA ∩ B ≠ ∅ and γ ∈ Γ. Then γ(p^{−1}A) ∩ p^{−1}B ≠ ∅, and so γ ∈ Γ ∩ (p^{−1}B) · (p^{−1}A)^{−1}. But this last set is the intersection of a discrete set with a compact set and so is finite.

(The implications (c) ⟹ (b) ⟹ (a) are trivial, and (b) ⟹ (c) is easy. For (c) ⟹ (d), let V be any neighbourhood of 1 in G whose closure V̄ is compact. For any x ∈ X, Γ ∩ V ⊂ {γ ∈ Γ| γx ∈ V̄ · x}, which is finite, because both {x} and V̄ · x are compact. Thus Γ ∩ V is discrete, which shows that e is an isolated point of Γ.)

The next result makes statement (c) more precise.

Proposition 2.5. Let G, K, X be as in (2.4), and let Γ be a discrete subgroup of G.

(a) For any x ∈ X, {g ∈ Γ| gx = x} is finite.

(b) For any x ∈ X, there is a neighbourhood U of x with the following property: if γ ∈ Γ and U ∩ γU ≠ ∅, then γx = x.

(c) For any points x and y ∈ X that are not in the same Γ-orbit, there exist neighbourhoods U of x and V of y such that γU ∩ V = ∅ for all γ ∈ Γ.

Proof. (a) We saw in the proof of (2.4) that p^{−1}(compact) is compact, where p(g) = gx. Therefore p^{−1}(x) is compact, and the set we are interested in is p^{−1}(x)∩Γ.

(b) Let V be a compact neighbourhood of x. Because (2.4c) holds, there is a finite set {γ_{1}, ..., γ_{n}} of elements of Γ such that V ∩ γ_{i}V ≠ ∅. Let γ_{1}, ..., γ_{s} be the γ_{i}’s fixing x. For each i > s, choose disjoint neighbourhoods V_{i} of x and W_{i} of γ_{i}x, and put

U = V ∩ (∩_{i>s}V_{i} ∩ γ_{i}^{−1}W_{i}).

For i > s, γ_{i}U ⊂ W_{i} which is disjoint from V_{i}, which contains U.

(c) Choose compact neighbourhoods A of x and B of y, and let γ_{1}, ..., γ_{n} be the elements of Γ such that γ_{i}A ∩ B ≠ ∅. We know γ_{i}x ≠ y, and so we can find disjoint neighbourhoods U_{i} and V_{i} of γ_{i}x and y. Take

U = A ∩ γ_{1}^{−1}U_{1} ∩ ... ∩ γ_{n}^{−1}U_{n}, V = B ∩ V_{1} ∩ ... ∩ V_{n}.

Corollary 2.6. Under the hypotheses of (2.5), the space Γ\X is Hausdorff.

Proof. Let x and y be points of X not in the same Γ-orbit, and choose neighbourhoods U and V as in (2.5). Then the images of U and V in Γ\X are disjoint neighbourhoods of Γx and Γy.

A group Γ is said to act freely on a set X if Stab(x) = e for all x ∈ X.

Proposition 2.7. Let Γ be a discrete subgroup of SL_{2}(ℝ) such that Γ (or Γ/{±I} if −I ∈ Γ) acts freely on ℍ. Then there is a unique complex structure on Γ\ℍ with the following property: a function f on an open subset U of Γ\ℍ is holomorphic if and only if f ∘ p is holomorphic.

Proof. The uniqueness follows from the fact (see 1.8) that the sheaf of holomorphic functions on a Riemann surface determines the complex structure. Let z ∈ Γ\ℍ, and choose an x ∈ p^{−1}(z). According to (2.5b), there is a neighbourhood U of x such that γU is disjoint from U for all γ ∈ Γ, γ ≠ e. The map p|U : U → p(U) is a homeomorphism, and we take all pairs of the form (p(U), (p|U)^{−1}) to be coordinate neighbourhoods. It is easy to check that they are all compatible, and that the holomorphic functions are as described. (Alternatively, one can define O(U) as in the statement of the proposition, and verify that U ↦ O(U) is a sheaf of ℂ-algebras satisfying (1.9(*).)

Unfortunately SL_{2}(ℤ)/{±I} doesn’t act freely.

Discrete subgroups of SL_{2}(ℝ). To check that a subgroup Γ of SL_{2}(ℝ) is discrete, it suffices to check that e is isolated in Γ. A discrete subgroup of SL_{2}(ℝ) (or SL_{2}(ℝ)) is called a Fuchsian group. Discrete subgroups of SL_{2}(ℝ) abound, but those of interest to number theorists are rather special.

Congruence subgroups of the elliptic modular group. Clearly SL_{2}(ℤ) is discrete, and a fortiori, Γ(N) is discrete. A congruence subgroup of SL_{2}(ℤ) is a subgroup containing Γ(N) for some N. For example,

Γ_{0}(N) =^{df} \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) | c ≡ 0 mod N)\}

is a congruence subgroup of SL_{2}(ℤ). By definition, the sequence

1 → Γ(N) → SL_{2}(ℤ) → SL_{2}(ℤ/Nℤ)

is exact (SL_{2}(A) makes sense for any commutative ring—it is the group of 2 × 2 matrices with coefficients in A having determinant 1). I claim that the map SL_{2}(ℤ) → SL_{2}(Z/NZ) is surjective. To prove this, we have to show that if A ∈ M_{2}(ℤ) and det(A) ≡ 1 mod N, then there is a B ∈ M_{2}(ℤ) such that B ≡ A mod N and

det(B) = 1. Let A = ( \begin{matrix}a & b \\ c & d \end{matrix} ); the condition on A is that

ab − cd − Nm = 1

for some m ∈ ℤ. Hence gcd(c, d, N) = 1, and we can find an integer n such that gcd(c, d + nN) = 1 (apply the Chinese Remainder Theorem to find an n such that d+nN ≡ 1 mod p for every prime p dividing c but not dividing N and n ≡ 0 mod p for every prime p dividing both c and N). We can replace d with d + nN, and so assume that gcd(c, d) = 1. Consider the matrix

B = ( \begin{matrix}a + eN & b + fN \\ c & d \end{matrix} )

for some integers e, f. Its determinant is ad − bc + N(ed − fc) = 1 + (m + ed − fc)N. Since gcd(c, d) = 1, there exist integers e, f such that m = fc + ed, and with this choice, B is the required matrix.

Note that the surjectivity of SL_{2}(ℤ) → SL_{2}(ℤ/Nℤ) is implies that SL_{2}(ℤ) is dense in SL_{2}(ℤ̂), where ℤ̂ = \underleftarrow{lim}_{N}ℤ/NZ =completion of ℤ for the topology of subgroups of finite index = ∏ ℤ_{ℓ}.

Discrete groups coming from quaternion algebras. For any rational numbers a, b, let B = B_{a,b} be the ℚ-algebra with basis {1, i, j, k} and multiplication given by

i^{2} = a, j^{2} = b, ij = k = −ji.

Then B ⊗ ℝ is an algebra over ℝ with the same basis and multiplication table, and it is isomorphic either to M_{2}(ℝ) or the usual (Hamiltonian) quaternion algebra—we suppose the former.

For α = c + di + ej + fk ∈ B, define Nm(α) = c^{2} − d^{2} − e^{2} − f^{2} ∈ ℚ. Under the isomorphism B ⊗ ℝ → M_{2}(ℝ), the norm corresponds to the determinant, and so the isomorphism induces an isomorphism

{α ∈ B ⊗ ℝ | Nm(α) = 1} →^{≈} SL_{2}(ℝ).

An order in B is a subring O that is finitely generated over ℤ (hence free of rank 4). Define

Γ_{a,b} = {α ∈ O | Nm(α) = 1}.

Under the above isomorphism this is mapped to a discrete subgroup of SL_{2}(ℝ), and we can define congruence subgroups of Γ_{a,b} as for SL_{2}(ℤ).

For a suitable choice of (a, b), B = M_{2}(ℚ) (ring of 2 × 2 matrices with coefficients in ℚ), and if we choose O to be M_{2}(ℤ), then we recover the elliptic modular groups.

If B is not isomorphic to M_{2}(ℚ), then the families of discrete groups that we get are quite different from the congruence subgroups of SL_{2}(ℤ): they have the property that Γ\ℍ is compact.

There are infinitely many nonisomorphic quaternion algebras over ℚ, and so the congruence subgroups of SL_{2}(ℤ) form just one among an infinite sequence of families of discrete subgroups of SL_{2}(ℝ).

[These groups were found by Poincaré in the 1880’s, but he regarded them as automorphism groups of the quadratic forms Φ_{a,b} = −aX^{2} − bY^{2} + abZ^{2}. For a description of how he found them, see p52, of his book, Science and Method.]

Exercise 2.8. Two subgroups Γ and Γ′ of a group are said to be commensurable if Γ ∩ Γ′ is of finite index in both Γ and Γ′.

(a) Commensurability is an equivalence relation (only transitivity is nonobvious).

(b) If Γ and Γ′ are commensurable subgroups of a topological group G, and Γ is discrete, then so also is Γ′.

(c) If Γ and Γ′ are commensurable subgroups of SL_{2}(ℝ) and Γ\ℍ is compact, so also is Γ′\ℍ.

Arithmetic subgroups of the elliptic modular group. A subgroup of SL_{2}(ℚ) is arithmetic if it is commensurable with SL_{2}(ℤ). For example, every subgroup of finite index in SL_{2}(ℤ), hence every congruence subgroup, is arithmetic. The congruence subgroups are sparse among the arithmetic subgroups: if we let N(m) be the number of congruence subgroups of SL_{2}(ℤ) of index < m, and let N′(m) be the number of subgroups of index < m, then N(m)/N′(m) → 0 as m → ∞.

Remark 2.9. This course will be concerned with quotients of ℍ by congruence groups in the elliptic modular group SL_{2}(ℤ), although the congruence groups arising from quaternion algebras are of (almost) equal interest to number theorists. There is some tantalizing evidence that modular forms relative to other arithmetic groups may also have interesting arithmetic properties, but we shall ignore this.

There are many nonarithmetic discrete subgroups of SL_{2}(ℝ). The ones of most interest (to analysts) are those of the “first kind”—they are “large” in the sense that Γ\ SL_{2}(ℝ) (hence Γ\ℍ) has finite volume relative to a Haar measure on SL_{2}(ℝ).

Among matrix groups, SL_{2} is anomalous in having so many discrete subgroups. For other groups there is a wonderful theorem of Margulis (for which he got the Fields medal), which says that, under some mild hypotheses (which exclude SL_{2}), any discrete subgroup Γ of G(ℝ) such that Γ\G(ℝ) has finite measure is arithmetic, and for many groups one knows that all arithmetic subgroups are congruence (see Prasad’s talk at the International Congress in Kyoto).

Classification of linear fractional transformations. The group SL_{2}(ℂ) acts on ℂ^{2}, and hence on the set ℙ^{1}(C) of lines through the origin in ℂ^{2}. When we identify a line with its slope, ℙ^{1}(ℂ) becomes identified with ℂ ∪ {∞}, and we get an action of GL_{2}(ℂ) on ℂ ∪ {∞} :

( \begin{matrix}a & b \\ c & d \end{matrix} ) z = \frac{az + b}{cz + d}, ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∞ = \frac{a}{c}.

These mappings are called the linear fractional transformations of ℙ^{1}(ℂ) = ℂ ∪ {∞}.

They map circles and lines in ℂ into circles or lines in ℂ. The scalar matrices ( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ) act as the identity transformation. By the theory of Jordan canonical forms, any nonscalar α is conjugate to a matrix of the following type:

(i) ( \begin{matrix}λ & 1 \\ 0 & λ \end{matrix} ) (ii) ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} ) , λ ≠ μ.

according as it has repeated eigenvalues or distinct eigenvalues. In the first case, α is conjugate to a transformation z ↦ z + λ^{−1}, and in the second to z ↦ cz, c ≠ 1. In case (i), α is called parabolic, and case (ii), it is called elliptic if |c| = 1, hyperbolic if c is real and positive, and loxodromic otherwise.

When α ∈ SL_{2}(ℂ), the four cases can be distinguished by the trace of α :

α is parabolic ⟺ Tr(α) = ±2;
α is elliptic ⟺ Tr(α) is real and | Tr(α)| < 2;
α is hyperbolic ⟺ Tr(α) is real and | Tr(α)| > 2;
α is loxodromic ⟺ Tr(α) is not real.

We now investigate the elements of these types in SL_{2}(ℝ).

Parabolic transformations. Suppose α ∈ SL_{2}(ℝ), α ≠ ±I, is parabolic. Then it has exactly one eigenvector, and that eigenvector is real. Suppose the eigenvector is

( \begin{matrix}e \\ f \end{matrix} ); if f ≠ 0, then α has a fixed point in ℝ; if f = 0, then ∞ is a fixed point (the

transformation is then of the form z ↦ z + c). Thus α has exactly one fixed point in ℝ ∪ {∞}.

Elliptic transformations. Suppose α ∈ SL_{2}(ℝ), α ≠ ±I, is elliptic. Its characteristic polynomial is X^{2} + bX + 1 with |b| < 2; hence Δ = b^{2} − 4 < 0, and so α has two complex conjugate eigenvectors. Thus α has exactly one fixed point z in ℍ and a second fixed point, namely, z̄, in the lower half plane.

Hyperbolic transformations. Suppose α ∈ SL_{2}(ℝ) and α is hyperbolic. Its characteristic polynomial is X^{2} + bX + 1 with |b| > 2; hence Δ = b^{2} − 4 > 0, and so α has two distinct real eigenvectors. Thus α has two distinct fixed points in ℝ ∪ {∞}.

Let Γ be a discrete subgroup of SL_{2}(ℝ). A point z ∈ ℍ is called an elliptic point if it is the fixed point of an elliptic element γ of Γ; a point s ∈ ℝ ∪ {∞} is called a cusp if there exists a parabolic element γ ∈ Γ with s as its fixed point.

Proposition 2.10. If z is an elliptic point of Γ, then {γ ∈ Γ | γz = z} is a finite cyclic group.

Proof. There exists an α ∈ SL_{2}(ℝ) such that α(i) = z, and γ ↦ α^{−1}γα defines an isomorphism

{γ ∈ Γ | γz = z} ≈ SO_{2}(ℝ) ∩ (α^{−1}Γα),

and this last group is finite. The correspondences θ ↔ e^{2πiθ} ↔ ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) are

isomorphisms

ℝ/ℤ ↔ {z ∈ ℂ | |z| = 1} ↔ SO_{2}(ℝ).

Therefore SO_{2}(ℝ)_{tors} ≈ ℚ/ℤ, and every finite subgroup of ℚ/ℤ is cyclic (each is of the form n^{−1}ℤ/ℤ where n is the least common denominator of the elements of the group).

Remark 2.11. Let Γ(1) be the full modular group SL_{2}(ℤ). I claim the cusps of Γ(1) are exactly the points of ℚ ∪ {∞}, and each is Γ(1)-equivalent to ∞. Certainly

∞ is the fixed point of the parabolic matrix T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). Suppose m/n ∈ ℚ; we

can assume m and n to be relatively prime, and so there are integers r and s such

that rm − sn = 1; let γ = ( \begin{matrix}m & s \\ n & r \end{matrix} ); then γ(∞) = m/n, and m/n is fixed by the

parabolic element γT γ^{−1}. Conversely, every parabolic element α of Γ(1) is conjugate to ±T, say α = ±γT γ^{−1}, γ ∈ GL_{2}(ℚ). The point fixed by α is γ∞, which belongs to ℚ ∪ {∞}.

We now find the elliptic points of Γ(1). Let γ be an elliptic element in Γ(1). The characteristic polynomial of γ is of degree 2, and its roots are roots of 1 (because γ has finite order). The only roots of 1 lying in a quadratic field have order dividing 4 or 6. From this, it easy to see that every elliptic point of ℍ relative to Γ(1) is Γ(1)-equivalent to exactly one of i or ρ = (1 + i\sqrt{3})/2. (See also 2.12 below.)

Now let Γ be a subgroup of Γ(1) of finite index. The cusps of Γ are the cusps of Γ(1), namely, the elements of ℚ ∪ {∞} = ℙ^{1}(ℚ), but in general they will fall into more than one Γ-orbit. Every elliptic point of Γ is an elliptic point of Γ(1); conversely, an elliptic point of Γ(1) is an elliptic point of Γ if an only if it is fixed by an element of Γ other than ±I.

Fundamental domains. Let Γ be a discrete subgroup of SL_{2}(ℝ). A fundamental domain for Γ is a connected open subset D of ℍ such that no two points of D are equivalent under Γ and ℍ = ⋃ γD̄, where D̄ is the closure of D . These conditions are equivalent respectively, to the statements: the map D → Γ\ℍ is injective; the map D̄ → Γ\ℍ is surjective. Every Γ has a fundamental domain, but we shall prove this only for the subgroups of finite index in Γ(1).

Let S = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). Thus

\begin{matrix}Sz = \frac{−1}{z}, & T z = z + 1 \\ S^{2} ≡ 1 mod ± I, & (ST)^{3} ≡ 1 mod ± I. \end{matrix}

To apply S to a z with |z| = 1, first reflect in the x-axis, and then reflect through the origin (because S(e^{iθ}) = −(e^{−iθ})).

Theorem 2.12. Let D = {z ∈ ℍ | |z| > 1, |ℜ(z)| < 1/2}.

(a) D is a fundamental domain for Γ(1) = SL_{2}(ℤ); moreover, two elements z and

z′ of D̄ are equivalent under Γ(1) if and only if (i) ℜ(z) = ±1/2 and z′ = z ± 1, (then z′ = T z or z = T z′), or (ii) |z| = 1 and z′ = −1/z = Sz.

(b) Let z ∈ D̄; if the stabilizer of z ≠ {±I}, then (i) z = i, and Stab(i) =<S>, which has order 2 in Γ(1)/{±I}), or (ii) z = ρ = exp(2πi/6), and Stab(ρ) =< T S >, which has order 3 in

Γ(1)/{±I}), or (iii) z = ρ^{2}, and Stab(ρ^{2}) =<ST>, which has order 3 in Γ(1)/{±I}).

(c) The group Γ(1)/{±I} is generated by S and T.

Proof. Let Γ′ be the subgroup of Γ(1) generated by S and T. We shall show that Γ′ · D = ℍ.

Lemma 2.13. For a fixed z ∈ ℍ and N ∈ ℕ, there are only finitely many pairs of integers (c, d) such that

|cz + d| ≤ N.

Proof. Write z = x + iy. If (c, d) is such a pair, then

|cz + d|^{2} = (cx + d)^{2} + c^{2}y^{2},

so that

c^{2}y^{2} ≤ (cx + d)^{2} + c^{2}y^{2} ≤ N.

As z ∈ ℍ, y > 0, and so |c| ≤ N/y, which implies that there are only finitely many possibilities for c. For any such c, the equation

(cx + d)^{2} + c^{2}y^{2} ≤ N

shows that there are only finitely many possible values of d.

Recall that, if γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ), then ℑ(γz) = ℑ(z)/|cz + d|^{2}. Fix a z ∈ ℍ, and

choose γ ∈ Γ′ such that |cz + d| is a minimum—the lemma implies that such a γ exits. Then ℑ(γz) is a maximum among elements in the orbit of z.

For some n, z′ =^{df} T^{n}(γz) will have

−1/2 ≤ ℜ(z′) ≤ 1/2.

I claim that |z′| ≥ 1. If not, then

ℑ(Sz′) = ℑ(−1/z′) = ℑ (\frac{−x′ + iy′}{|z′|^{2}} ) = \frac{ℑ(z′)}{|z′|^{2}} > ℑ(z′) = ℑ(γz),

which contradicts our choice of γz. We have shown that Γ′ · D̄ = ℍ.

Suppose z, z′ ∈ D̄ are Γ-conjugate. Then either ℑ(z) ≥ ℑ(z′) or ℑ(z) ≤ ℑ(z′), and

we shall assume the latter. Suppose z′ = γz with γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ), and let z = x + iy.

Then our assumption implies that

(cx + d)^{2} + cy^{2} = |cz + d|^{2} ≤ 1.

This is impossible if c ≥ 2 (because y ≥ 1/2), and so we need only consider the cases c = 0, 1, −1.

c = 0 : Then d = ±1, γ = ± ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ), and γ is translation by b. Because z and

γz ∈ D̄, this implies that b = ±1, and we are in case (a(i)).

c = 1 : As |z + d| ≤ 1 we must have d = 0, unless z = ρ = \frac{1}{2} + i\frac{\sqrt{3}}{2}, in which case

d = 0 or −1, or z = ρ^{2}, in which case d = 0 or 1. If d = 0, then γ = ± ( \begin{matrix}a & −1 \\ 1 & 0 \end{matrix} ),

and γz = a − \frac{1}{z}. If a = 0, then we are in case (a(ii)). If a ≠ 0, then a = 1 and z = ρ^{2}, or a = −1 and z = ρ.

c = −1 : This case can be treated similarly (or simply change the signs of a, b, c, d in the last case).

This completes the proof of (a) and (b) of the theorem.

We now prove (c). Let γ ∈ Γ. Choose a point z_{0} ∈ D. Because Γ · D̄ = ℍ, there is an element γ′ ∈ Γ′ and a point z ∈ D̄ such that γ′z = γz_{0} ∈ D̄. Then z_{0} is Γ(1)-equivalent to (γ^{′−1}γ)z_{0} ∈ D̄; because z_{0} ∈ D, part (a) shows that z_{0} = (γ^{′−1}γ)z_{0}. Hence γ^{′−1}γ ∈ Stab(z_{0}) ∩ Γ(1) = {±I}, and so γ′ and γ are equal in Γ(1)/{±1}.

Remark 2.14. We showed that the group Γ(1)/{±I} has generators S and T with relations S^{2} = 1 and (ST)^{3} = 1. One can show that this is a full set of relations, and that Γ(1)/{±I} is the free product of the cyclic group of order 2 generated by S and the cyclic group of order 3 generated by ST.

Aside 2.15. Our computation of the fundamental domain has applications for quadratic forms and sphere packings.

Consider a binary quadratic form:

q(x, y) = ax^{2} + bxy + cy^{2}, a, b, c ∈ ℝ.

Assume q is definite, i.e., its discriminant Δ = b^{2} − 4ac < 0. Two forms q and q′ are equivalent if there is a matrix A ∈ SL_{2}(ℤ) taking q into q′ by the change of variables,

( \begin{matrix}x′ \\ y′ \end{matrix} ) = A ( \begin{matrix}x \\ y \end{matrix} ) .

In other words, the forms

q(x, y) = (x, y) · Q · ( \begin{matrix}x \\ y \end{matrix} ) , q′(x, y) = (x, y) · Q′ · ( \begin{matrix}x \\ y \end{matrix} )

are equivalent if Q = A^{tr} · Q′ · A.

Every definite binary quadratic form can be written q(x, y) = a(x − ωy)(x − ω̄y) with ω ∈ ℍ. The association q ↔ ω is a one-to-one correspondence between the definite binary quadratic forms with a fixed discriminant Δ and the points of ℍ. Moreover, two forms are equivalent if and only if the points lie in the same SL_{2}(ℤ)- orbit. A definite binary quadratic form is said to be reduced if ω is in

\{z ∈ ℍ | −\frac{1}{2} ≤ ℜ(z) < 1 and |z| > 1, or |z| = 1 and − \frac{1}{2} ≤ ℜ(z) ≤ 0\}.

More explicitly, q(x, y) = ax^{2} + bxy + cy^{2} is reduced if and only if either

−a < b ≤ a < c or 0 ≤ b ≤ a = c.

Theorem 2.12 implies:

Every definite binary quadratic form is equivalent to a reduced form; two reduced forms are equivalent if and only if they are equal.

We say that a quadratic form is integral if it has integral coefficients.

There are only finitely many equivalence classes of integral definite binary quadratic forms with a given discriminant.

Each equivalence class contains exactly one reduced form ax^{2} + bxy + cy^{2}. Since

4a^{2} ≤ 4ac = b^{2} − Δ ≤ a^{2} − Δ

we see that there are only finitely many values of a for a fixed Δ. Since |b| ≤ a, the same is true of b, and for each pair (a, b) there is at most one integer c such that b^{2} − 4ac = Δ.

For more details, see W. LeVeque, Topics in Number Theory, II, Addison-Wesley, 1956, Chapter 1.

We can apply this to lattice sphere packings in ℝ^{2}. Such a packing is determined by the lattice of centres of the spheres (here disks). The object, of course, is to make the packing as dense as possible. With a lattice Λ in ℝ^{2} and a choice of a basis {f_{1}, f_{2}} for Λ, we can associate the quadratic form

q(x_{1}, x_{2}) = ∥f_{1}x_{1} + f_{2}x_{2}∥^{2}.

The problem of finding dense sphere packings translates into finding quadratic forms q with

γ(q) =^{df} min{q(x) | x ∈ ℤ^{2}, x ≠ 0}^{2}/disc(q)

as large as possible. Note that changing the choice of basis for Λ translates into acting on q with an element of SL_{2}(ℤ), and so we can confine our attention to reduced quadratic forms. It is then easy to show that the quadratic form with γ(q) minimum

is that corresponding to ρ. The corresponding lattice has basis ( \begin{matrix}2 \\ 0 \end{matrix} ) and ( \begin{matrix}1 \\ √ & \overline{3} \end{matrix} )

(just as you would expect), and the quadratic form is 4(x^{2} + xy + y^{2}).

For more on sphere packings, see Math 679, Section 21.

Fundamental domains for congruence subgroups. First we have the following general result.

Proposition 2.16. Let Γ be a discrete subgroup of SL_{2}(ℝ), and let D be a fundamental domain for Γ. Let Γ′ be a subgroup of Γ of finite index, and write Γ as a disjoint union of right cosets of Γ′ :

Γ = Γ′γ_{1} ∪ ... ∪ Γ′γ_{m}

Then D′ =^{df} ⋃ γ_{i}D is a fundamental domain for Γ′ (possibly nonconnected).

Proof. Let z ∈ ℍ. Then z = γz′ for some z′ ∈ D̄, γ ∈ Γ, and γ = γ′γ_{i} for some γ′ ∈ Γ′. Thus z = γ′γ_{i}z ∈ Γ′ · (γ_{i}D̄).

If γD′ ∩ D′ ≠ ∅, then it would contain a transform of D. But then γγ_{i}D = γ_{j}D for some i ≠ j, which would imply that γγ_{i} = γ_{j}, and this is a contradiction.

Proposition 2.17. It is possible to choose the γ_{i} so that the closure of D′ is connected; the interior of the closure of D′ is then a connected fundamental domain for Γ.

Proof. Omit.

Remark 2.18. Once one has obtained a fundamental domain for Γ, as in (2.16), it is possible to read off a system of generators and relations for Γ.

Defining complex structures on quotients. Before defining ℍ^{∗} and the complex structure on the quotient Γ\ℍ^{∗} we discuss two simple examples.

Example 2.19. Let D be the open unit disk, and let Δ be a finite group acting on D. The Schwarz lemma implies that Aut(D) = {z ∈ ℂ | |z| = 1} ≈ ℝ/ℤ, and it follows that Δ is a finite cyclic group. Let z ↦ ζz be its generator and suppose that ζ^{m} = 1. Then z^{m} is invariant under Δ, and so defines a function on Δ\D. It is a homeomorphism from Δ\D onto D, and therefore defines a complex structure on Δ\D.

Let p be the quotient map D → Δ\D. The map f ↦ f ∘ p is a bijection from the holomorphic functions on U ⊂ Δ\D to the holomorphic functions of z^{m} on p^{−1}(U) ⊂ D; but these are precisely the holomorphic functions on p^{−1}(U) invariant under the action of Δ.

Example 2.20. Let X = {z ∈ ℂ | ℑ(z) > c} (some c). Fix an integer h, and let n ∈ ℤ act on X as z ↦ z + nh. Add a point “∞” and define a topology on X^{∗} = X ∪ {∞} as follows: a fundamental system of neighbourhoods of a point in X is as before; a fundamental system of neighbourhoods for ∞ is formed of sets of the form {z ∈ ℂ | ℑ(z) > N}. We can extend the action of ℤ on X to a continuous action on X^{∗} by requiring ∞ + nh = ∞ for all n ∈ ℤ. Consider the quotient space Γ\X^{∗}. The function

q(z) = \{ \begin{matrix}e^{2πiz/h} & z ≠ ∞, \\ 0 & z = ∞ \end{matrix}

is a homeomorphism Γ\X^{∗} → D from Γ\X^{∗} onto the open disk of radius e^{−2πc/h} and centre 0. It therefore defines a complex structure on Γ\X^{∗}.

The complex structure on Γ(1)\ℍ^{∗}. We first define the complex structure on Γ(1)\ℍ. Write p for the quotient map ℍ → Γ(1)\H. Let P be a point of Γ(1)\ℍ, and let Q be a point of ℍ mapping to it.

If Q is not an elliptic point, we can choose a neighbourhood U of Q such that p is a homeomorphism U → p(U). We define (p(U), p^{−1}) to be a coordinate neighbourhood of P.

If Q is equivalent to i, we may as well take it to equal i. The map z ↦ \frac{z−i}{z+i} defines an isomorphism of an open disk D with centre i onto an open disk D′ with centre 0, and the action of S on D is transformed into the automorphism σ : z ↦ −z of D′ (because it fixes i and has order 2). Thus <S>\D is homeomorphic to <σ>\D′, and we give <S>\D the complex structure making this a bi-holomorphic isomorphism. More explicitly, \frac{z−i}{z+i} is a holomorphic function defined in a neighbourhood of i, and

S = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) maps it to

\frac{−z^{−1} − i}{−z^{−1} + i} = \frac{−1 − iz}{−1 + iz} = \frac{−i + z}{−i − z} = −\frac{z − i}{z + i}

Thus z ↦ (\frac{z−i}{z+i})^{2} is a holomorphic function defined in a neighbourhood of i which is invariant under the action of S; it therefore defines a holomorphic function in a neighbourhood of p(i), and we take this to the coordinate function near p(i).

The point Q = ρ^{2} can be treated similarly. Apply a linear fractional transformation that maps Q to zero, and then take the cube of the map. Explicitly, ρ^{2} is fixed by ST, which has order 3 (as a transformation). The function z ↦ \frac{z−ρ^{2}}{z−ρ̄^{2}} defines an isomorphism from a disk with centre ρ^{2} onto a disk with centre 0, and (\frac{z−ρ^{2}}{z−ρ̄^{2}} )^{3} is invariant under ST. It therefore defines a function on a neighbourhood of p(ρ^{2}), and we take this to be the coordinate function near p(ρ^{2}).

The Riemann surface Γ(1)\ℍ we obtain is not compact—to compactify it, we need to add a point. The simplest way to do this is to add a point ∞ to ℍ, as in (2.20), and use the function q(z) = exp(2πiz) to map some neighbourhood U = {z ∈ ℍ | ℑ(z) > N} of ∞ onto an open disk V with centre 0. The function q is invariant under the action of the stabilizer of <T> of ∞, and so defines a holomorphic function q :<T> \U → V, which we take to be the coordinate function near p(∞).

Alternatively, we can consider ℍ^{∗} = H ∪ ℙ^{1}(ℚ), i.e., ℍ^{∗} is the union of ℍ with the set of cusps for Γ(1). Each cusp other than^{2} ∞ is a rational point on the real axis, and is of the form σ∞ for some σ ∈ Γ(1) (see 2.11). Give σ∞ the fundamental system of neighbourhoods for which σ is a homeomorphism. Then Γ(1) acts continuously on ℍ^{∗}, and we can consider the quotient space Γ(1)\ℍ^{∗}. Clearly, Γ(1)\ℍ^{∗} = (Γ(1)\H)∪{∞}, and we can endow it with the same complex structure as before.

Proposition 2.21. The Riemann surface Γ(1)\ℍ^{∗} is a compact and of genus zero; it is therefore isomorphic to the Riemann sphere.

Proof. It is compact because D̄ ∪ {∞} is compact. We sketch four proofs that it has genus 0. First, by examining carefully how the points of D̄ are identified, one can see that it must be homeomorphic to a sphere. Second, show that it is simply connected (loops can be contracted), and the Riemann sphere is the only simply connected compact Riemann surface (Riemann Mapping Theorem 0.1). Third, triangulate it by taking ρ, i, and ∞ as the vertices of the obvious triangle, add a fourth vertex not on any side of the triangle, and join it to the first three vertices; then 2 − 2g = 4 − 6 + 4 = 2. Finally, there is a direct proof that there is a function j holomorphic on Γ\ℍ and having a simple pole at ∞—it is therefore of valence one, and so defines an isomorphism of Γ\ℍ^{∗} onto the Riemann sphere.

The complex structure on Γ\ℍ^{∗}. Let Γ ⊂ Γ(1) of finite index. We can define a compact Riemann surface Γ\ℍ^{∗} in much the same way as for Γ(1). The complement of Γ\ℍ in Γ\ℍ^{∗} is the set of equivalence classes of cusps for Γ.

First Γ\ℍ is given a complex structure in exactly the same way as in the case Γ = Γ(1). The point ∞ will always be cusp (Γ must contain T^{h} for some h, and T^{h} is a parabolic element fixing ∞). If h is the smallest power of T in Γ, then the function q = exp(2πiz/h) is a coordinate function near ∞. Any other cusp for Γ is of the form σ∞ for σ ∈ Γ(1), and z ↦ q(σ^{−1}(z)) is a coordinate function near σ∞.

We write Y (Γ) = Γ\ℍ and X(Γ) = Γ\ℍ^{∗}. We abbreviate Y (Γ(N)) to Y (N), X(Γ(N)) to X(N), Y (Γ_{0}(N)) to Y_{0}(N), X(Γ_{0}(N)) to X_{0}(N) and so on.

2We sometimes denote ∞ by i∞ and imagine it to be at the end of the imaginary axis.

The genus of X(Γ). We now compute the genus of X(Γ) by considering it as a covering of X(Γ(1)). According to (1.27)

2g − 2 = −2m + ∑(e_{P} − 1)

or

g = 1 − m + ∑(e_{P} − 1)/2.

where m is the degree of the covering X(Γ) → X(Γ(1)) and e_{P} is the ramification index at the point P. The ramification points are the images of elliptic points on ℍ^{∗} and the cusps.

Theorem 2.22. Let Γ be a subgroup of Γ(1) of finite index, and let ν_{2} =the number of inequivalent elliptic points of order 2; ν_{3} =the number of inequivalent elliptic points of order 3; ν_{∞} = the number of inequivalent cusps. Then the genus of X(Γ) is

g = 1 + m/12 − ν_{2}/4 − ν_{3}/3 − ν_{∞}/2.

Proof. Let p be the quotient map ℍ^{∗} → Γ(1)\ℍ^{∗}, and let φ be the map Γ\ℍ^{∗} → Γ(1)\H^{∗}. If Q is a point of ℍ^{∗} and P′ and P are its images in Γ\ℍ^{∗} and Γ(1)\ℍ^{∗} then the ramification indices multiply:

e(Q/P) = e(Q/P′) · e(P′/P).

If Q is a cusp, then this formula is not useful, as e(Q/P) = ∞ = e(Q/P′) (the map p is ∞ : 1 on every neighbourhood of ∞). For Q ∈ ℍ and not an elliptic point it tells us P′ is not ramified.

Suppose that P = p(i), so that Q is Γ(1)-equivalent to i. Then either e(Q/P′) = 2 or e(P′/P) = 2. In the first case, Q is an elliptic point for Γ and P′ is unramified over P; in the second, Q is not an elliptic point for Γ, and the ramification index of P′ over P is 2. There are ν_{2} points P′ of the first type, and (m − ν_{2})/2 points of the second. Hence ∑ e_{P′} − 1 = (m − ν_{2})/2.

Suppose that P = p(ρ), so that Q is Γ(1)-equivalent to ρ. Then either e(Q/P′) = 3 or e(P′/P) = 3. In the first case, Q is an elliptic point for Γ and P′ is unramified over P; in the second, Q is not an elliptic point for Γ, and the ramification index of P′ over is 3. There are ν_{3} points P′ of the first type, and (m − ν_{3})/3 points of the second. Hence ∑ e_{P′} − 1 = 2(m − ν_{3})/3.

Suppose that P = p(∞), so that Q is a cusp for Γ. There are ν_{∞} points P′ and ∑ e_{i} = m; hence ∑ e_{i} − 1 = m − ν_{∞}.

We conclude:

∑(e_{P′} − 1) = (m − ν_{2})/2 (P′ lying over φ(i))
∑(e_{P′} − 1) = 2(m − ν_{3})/3 (P′ lying over φ(ρ))
∑(e_{P′} − 1) = (m − ν_{∞}) (P′ lying over φ(∞)).

Therefore

g = 1 − m + ∑(e_{P} − 1)/2 = 1 + m/12 − ν_{2}/4 − ν_{3}/3 − ν_{∞}/2.

Example 2.23. Consider the principal congruence subgroup Γ(N). We have to compute the index of Γ(N) in Γ, i.e., the order of SL_{2}(ℤ/Nℤ). One sees easily that:

(a) GL_{2}(ℤ/Nℤ) ≈ ∏ GL_{2}(ℤ/p_{i}^{r_{i}}ℤ) if N = ∏ p_{i}^{r_{i}} (because ℤ/Nℤ ≈ ∏ ℤ/p_{i}^{r_{i}}ℤ).

(b) The order of GL_{2}(𝔽_{p}) = (p^{2} − 1)(p^{2} − p) (because the top row of a matrix in GL_{2}(𝔽_{p}) can be any nonzero element of k^{2}, and the second row can then be any element of k^{2} not on the line spanned by the first row).

(c) The kernel of GL_{2}(ℤ/p^{r}ℤ) → GL_{2}(𝔽_{p}) consists of all matrices of the form

I + p ( \begin{matrix}a & b \\ c & d \end{matrix} ) with a, b, c, d ∈ ℤ/p^{r−1}ℤ, and so the order of GL_{2}(ℤ/p^{r}ℤ) is

(p^{r−1})^{4} · (p^{2} − 1)(p^{2} − p).

(d) # GL_{2}(ℤ/p^{r}ℤ) = φ(p^{r}) · # SL_{2}(ℤ/p^{r}ℤ), where φ(p^{r}) = #(ℤ/p^{r}ℤ)^{×} = (p − 1)p^{r−1}.

On putting these statements together, one finds that

(Γ(1) : Γ(N)) = N^{3} · ∏_{p|N}(1 − p^{−2}).

Write Γ̄(N) for the image of Γ(N) in Γ(1)/{±I}. Then

(Γ̄(1) : Γ̄(N)) = (Γ(1) : Γ(N))/2,

unless N = 2, in which case it = 6.

What are ν_{2}, ν_{3}, and ν_{∞}? Assume N > 1. Then Γ(N) has no elliptic points—

the only torsion elements in Γ̄(1) are S = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), ST = ( \begin{matrix}0 & −1 \\ 1 & 1 \end{matrix} ), (ST)^{2},

and their conjugates; none of these three elements is in Γ(N) for any N > 1, and because Γ(N) is a normal subgroup, their conjugates can’t be either. The number of inequivalent cusps is μ_{N}/N where μ_{N} = (Γ̄(1) : Γ̄(N) (see 2.24). We conclude that the genus of Γ(N)\ℍ^{∗} is

g(N) = 1 + μ_{N} · (N − 6)/12N (N > 1).

For example,

\begin{aligned}N & = & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ μ & = & 6 & 12 & 24 & 60 & 72 & 168 & 192 & 324 & 360 & 660 \\ g & = & 0 & 0 & 0 & 0 & 1 & 3 & 5 & 10 & 13 & 26. \end{aligned}

There are similarly explicit formulas for the genus of X_{0}(N)—see Shimura 1971, p25.

Exercise 2.24. Let G be a group (possibly infinite) acting transitively on a set X, and let H be a subgroup of finite index in G. Fix a point x_{0} in X and let G_{0} be the stabilizer of x_{0} in G, and let H_{0} be the stabilizer of x_{0} in H. Prove that the number of orbits of H acting on X is

(G : H)/(G_{0} : H_{0}).

Deduce that the number of inequivalent cusps for Γ(N) is μ_{N}/N.

Remark 2.25. The Taniyama conjecture says that, for any elliptic curve E over ℚ, there exists a surjective map X_{0}(N) → E, where N is the conductor of E (the conductor of E is divisible only by the primes where E has bad reduction). The conjecture is suggested by studying zeta functions (see later). For any particular N, it is possible to verify the conjecture by listing all elliptic curves over ℚ with conductor N, and checking that there is a map X_{0}(N) → E. (Mathematica includes a list of elliptic curves with small conductor.) It is known (Ribet) that the Taniyama conjecture implies Fermat’s last theorem. The Taniyama conjecture has been proved for most elliptic curves by Wiles, Taylor, and Diamond (see Math 679).

An elliptic curve for which there is a map X_{0}(N) → E for some N is called a modular elliptic curve; contrast elliptic modular curves which are the curves of the form Γ\ℍ^{∗} for Γ a subgroup of finite index in Γ(1).

Aside 2.26. A bounded symmetric domain X is a bounded open connected subset of ℂ^{n}, some n, that is symmetric in the following sense: for each point x ∈ X, there is an involution s_{x} of X having x as an isolated fixed point. A complex manifold isomorphic to a bounded symmetric domain is also (loosely) referred to as a bounded symmetric domain.

For example, the unit disk D is a bounded symmetric domain—0 is the fixed point of the involution z ↦ −z, and since Aut(D) acts transitively on D this shows every other point must also be the fixed point of an involution. Every bounded symmetric domain is simply connected, and so (by the Riemann mapping theorem) every bounded symmetric domain of dimension one is isomorphic to the unit disk. As ℍ ≈ D, we also refer to ℍ as a bounded symmetric domain.

The bounded symmetric domains of all dimensions were classified by Elie Cartan (except for the exceptional ones). Just as for ℍ, the group of automorphisms Aut(X) of a bounded symmetric domain is a Lie group, which is simple if X is indecomposable (i.e., not equal to a product of bounded symmetric domains). There are bounded symmetric domains attached to groups of type A_{n}, B_{n}, C_{n}, D_{n}, E_{6}, E_{7} (here n is an integer ≥ 1).

Let X be a bounded symmetric domain. One can find many semisimple algebraic groups G over ℚ for which there exists a homomorphism G(ℝ)^{+} → Aut(X) with finite cokernel and compact kernel—the ^{+} denotes the identity component of G(ℝ) for the real topology. For example, we saw above that any quaternion algebra over ℚ that splits over ℝ gives rise to such a group for ℍ. Given such a G, one defines congruence subgroups Γ ⊂ G(ℤ) just as for SL_{2}(ℤ), and studies the quotients.

In 1964, Baily and Borel showed that each quotient Γ\X has a unique structure as an algebraic variety; in fact, they proved that Γ\X could be embedded in a natural way into a projective algebraic variety Γ\X^{∗}.

Various examples of these varieties were studied by Poincaré, Hilbert, Siegel, and many others, but Shimura began an intensive study of them in the 1960’s, and they are now called Shimura varieties.

Given a Shimura variety Γ\X^{∗}, one can attach a number field E to it, and prove that the Shimura variety is defined, in a natural way, over E. Thus one obtains a vast array of varieties defined over number fields, all with very interesting arithmetic properties. In this course, we study only the simplest case.

3. Elliptic functions

In this section, we review some of the theory of elliptic functions. For more details, see Cartan 1963, V.2.5, VI.5.3, or Silverman 1986, VI, or Koblitz 1984, I. Section 10 of my notes on Elliptic Curves (Math 679) is an expanded version of this section.

Lattices and bases. Let ω_{1} and ω_{2} be two nonzero complex numbers such that τ = ω_{1}/ω_{2} is imaginary. By interchanging ω_{1} and ω_{2} if necessary, we can ensure that τ = ω_{1}/ω_{2} lies in the upper half plane. Write

Λ = ℤω_{1} + ℤω_{2},

so that Λ is the lattice generated by ω_{1} and ω_{2}. We are interested in Λ rather than the basis {ω_{1}, ω_{2}}. If {ω′_{1}, ω′_{2}} is a second pair of elements of Λ, so that

ω′_{1} = aω_{1} + bω_{2}, ω′_{2} = cω_{1} + dω_{2}

for some a, b, c, d ∈ ℤ, then under what conditions on a, b, c, d does {ω′_{1}, ω′_{2}} form another basis for Λ with τ′ = ω′_{1}/ω′_{2} ∈ ℍ? Clearly ω′_{1} and ω′_{2} generate Λ if and

only if det ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ±1. We have τ′ = \frac{aτ+b}{cτ+d}, and the calculation on p2 shows

that ℑ(τ′) = det ( \begin{matrix}a & b \\ c & d \end{matrix} ) · ℑ(τ)/|cz + d|^{2}, and so we need that det ( \begin{matrix}a & b \\ c & d \end{matrix} ) > 1.

Therefore, the bases (ω′_{1}, ω′_{2}) of Λ with ℑ(ω′_{1}/ω′_{2}) > 0 are those of the form

( \begin{matrix}ω′_{1} \\ ω′_{2} \end{matrix} ) = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}ω_{1} \\ ω_{2} \end{matrix} ) with ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ).

Any parallelogram with vertices z_{0}, z_{0} + ω_{1}, z_{0} + ω_{1} + ω_{2}, z_{0} + ω_{2}, where {ω_{1}, ω_{2}} is a basis for Λ, is called a fundamental parallelogram for Λ.

Quotients of ℂ by lattices. Let Λ be a lattice in ℂ (by which I always mean a full lattice, i.e., a set of the form ℤω_{1} + ℤω_{2} with ω_{1} and ω_{2} linearly independent over ℝ). We can make the quotient space ℂ/Λ into a Riemann surface as follows: let Q be a point in ℂ and let P be its image ℂ/Λ; then there exist neighbourhoods V of Q and U of P such that the quotient map p : ℂ → ℂ/Λ defines a homeomorphism V → U; we take every such pair (U, p^{−1} : V → U) to be a coordinate neighbourhood. In this way we get a complex structure on ℂ/Λ having the following property: the map p : ℂ → ℂ/Λ is holomorphic, and for any open subset U of ℂ/Λ, a function f : U → ℂ is holomorphic if and only if f ∘ p is holomorphic on p^{−1}(U).

Topologically, ℂ/Λ ≈ (ℝ/ℤ)^{2}, which is a single-holed torus. Thus ℂ/Λ has genus 1. All spaces ℂ/Λ are homeomorphic, but, as we shall see, they are not all isomorphic as Riemann surfaces.

Doubly periodic functions. Let Λ be a lattice in ℂ. A meromorphic function f(z) on the complex plane is said to be doubly periodic with respect to Λ if it satisfies the functional equation:

f(z + ω) = f(z) for each ω ∈ Λ.

Equivalently,

f(z + ω_{1}) = f(z), f(z + ω_{2}) = f(z)

for {ω_{1}, ω_{2}} a basis for Λ.

Proposition 3.1. Let f(z) be a doubly periodic function for Λ, not identically zero, and let D be a fundamental parallelogram for Λ such that f has no zeros or poles on the boundary of D. Then

(a) ∑_{P ∈D} Res_{P} (f) = 0;

(b) ∑_{P ∈D} ord_{P} (f) = 0;

(c) ∑_{P ∈D} ord_{P} (f) · P ≡ 0 (mod Λ).

The first sum is over the points of D where f has a pole, and the other sums are over the points where it has a zero or pole. Each sum is finite.

Proof. Regard f as a function on ℂ/Λ, and apply Proposition 1.12 to get (a) and (b). To get (c) apply (1.12b) to z · f′(z)/f(z).

Corollary 3.2. A nonconstant doubly periodic function has at least two poles.

Proof. A doubly periodic function that is holomorphic is bounded in a closed period parallelogram (by compactness), and hence on the entire plane (by periodicity); so it is constant, by Liouville’s theorem. A doubly periodic function with a simple pole in a period parallelogram is impossible, because by (3.1a) the residue at the pole would be zero, and so the function would be holomorphic.

Endomorphisms of ℂ/Λ. Note that ℂ/Λ has a natural group structure.

Proposition 3.3. Let Λ and Λ′ be two lattices in ℂ. An element α ∈ ℂ such that αΛ ⊂ Λ′ defines a holomorphic map

φ_{α} : ℂ/Λ → ℂ/Λ′, [z] ↦ [αz],

sending [0] to [0], and every such map is of this form (for a unique α).

Proof. It is obvious that α defines such a map. Conversely, let φ : ℂ/Λ → ℂ/Λ′ be a holomorphic map such that φ([0]) = [0]. Then ℂ is the universal covering space of both ℂ/Λ and ℂ/Λ′, and a standard result in topology shows that φ lifts to a continuous map φ̃ : ℂ → ℂ such that φ̃(0) = 0:

ℂ ⟶^{φ̃} ℂ
↓  ↓
ℂ/Λ ⟶^{φ} ℂ/Λ′

Because the vertical maps are local isomorphisms, φ̃ is automatically holomorphic. For any ω ∈ Λ, the map z ↦ φ̃(z + ω) − φ̃(z) takes values in Λ′ ⊂ ℂ. As it is continuous, ℂ is connected, and Λ′ is discrete, it must be constant. Therefore φ̃′ must be a holomorphic doubly periodic function, and so it is constant, say φ̃′(z) = α. Then φ̃(z) = αz + β, and the fact that φ̃(0) = 0 implies that β = 0.

Corollary 3.4. Any holomorphic map φ : ℂ/Λ → ℂ/Λ′ such that φ(0) = 0 is a homomorphism.

Proof. Clearly [z] ↦ [αz] is a homomorphism.

Compare this with the result (Math 631, 5.37): any regular map φ : A → A′ from an abelian variety A to an abelian variety A′ such that φ(0) = 0 is a homomorphism.

Corollary 3.5. The Riemann surfaces ℂ/Λ and ℂ/Λ′ are isomorphic if and only if Λ′ = αΛ for some α ∈ ℂ^{×}.

Corollary 3.6. For any lattice Λ, End(ℂ/Λ) is either ℤ or a subring R of the ring of integers in a quadratic imaginary number field K.

Proof. Write Λ = ℤω_{1} +ℤω_{2} with τ =^{df} ω_{1}/ω_{2} ∈ ℍ, and suppose that there exists an α ∈ ℂ, α ∉ ℤ, such that αΛ ⊂ Λ. Then

αω_{1} = aω_{1} + bω_{2}
αω_{2} = cω_{1} + dω_{2},

with a, b, c, d ∈ ℤ. On dividing through by ω_{2} we obtain the equations

ατ = aτ + b
α = cτ + d.

As α ∉ ℤ, c ≠ 0. On eliminating α from the between the two equations, we find that

cτ^{2} + (d − a)τ + b = 0.

Therefore ℚ[τ] is of degree 2 over ℚ. On eliminating τ from between the two equations, we find that

α^{2} − (a + d)α + bc = 0.

Therefore α is integral over ℤ, and hence is contained in the ring of integers of ℚ[τ].

The Weierstrass ℘-function. . We want to construct some doubly periodic functions. Note that when G is a finite group acting on a set S, then it is easy to construct functions invariant under the action of G : take h to be any function h : S → ℂ, and define

f(s) = ∑_{g∈G} h(gs);

then f(g′s) = ∑_{g∈G} h(g′gs) = f(s), and so f is invariant (and all invariant functions are of this form, obviously). When G is not finite, one has to verify that the series converges—in fact, in order to be able to change the order of summation, one needs absolute convergence. Moreover, when S is a Riemann surface and h is holomorphic, to ensure that f is holomorphic, one needs that the series converges absolutely uniformly on compact sets.

Now let φ(z) be a holomorphic function ℂ and write

Φ(z) = ∑_{ω∈Λ} φ(z + ω).

Assume that as |z| → ∞, φ(z) → 0 so fast that the series for Φ(z) is absolutely convergent for all z for which none of the terms in the series has a pole. Then Φ(z) is doubly periodic with respect to Λ; for replacing z by z + ω_{0} for some ω_{0} ∈ Λ merely rearranges the terms in the sum. This is the most obvious way to construct doubly periodic functions; similar methods can be used to construct functions on other quotients of domains.

To prove the absolute uniform convergence on compact subsets of such series, the following test is useful.

Lemma 3.7. Let D be a bounded open set in the complex plane and let c > 1 be constant. Suppose that ψ(z, ω), ω ∈ Λ, is a function that is meromorphic in z for each ω and which satisfies ^{3} the condition (∗)

ψ(z, mω_{1} + nω_{2}) = O((m^{2} + n^{2})^{−c}) as m^{2} + n^{2} → ∞

uniformly in z for z in D. Then the series ∑_{ω∈Λ} ψ(z, ω), with finitely many terms which have poles in D deleted, is uniformly absolutely convergent in D.

Proof. That only finitely many terms can have poles in D follows from (*). This condition on ψ means that there are constants A and B such that

|ψ(z, mω_{1} + nω_{2})| < B(m^{2} + n^{2})^{−c}

whenever m^{2} + n^{2} > A. To prove the lemma it suffices to show that, given any ε > 0, there is an integer N such that S < ε for every finite sum S = ∑ |ψ(z, mω_{1} + nω_{2})| in which all the terms are distinct and each one of them has m^{2} + n^{2} ≥ 2N^{2}. Now S consists of eight subsums, a typical member of which consists of the terms for which m ≥ n ≥ 0. (There is some overlap between these sums, but that is harmless.) In this subsum we have m ≥ N and ψ < Bm^{−2c}, assuming as we may that 2N^{2} > A; and there are at most m + 1 possible values of n for a given m. Thus

S ≤ ∑_{m=N}^{∞} Bm^{−2c}(m + 1) < B_{1}N^{2−2c}

for a suitable constant B_{1}, and this proves the lemma.

We know from (3.1) that the simplest possible nonconstant doubly periodic function is one with a double pole at each point of Λ and no other poles. Suppose f(z) is such a function. Then f(z) − f(−z) is a doubly periodic function with no poles except perhaps simple ones at the points of Λ. Hence by the argument above, it must be constant, and since it is an odd function it must vanish. Thus f(z) is even, and we can make it unique by imposing the normalization condition f(z) = z^{−2} + O(z^{2}) near z = 0—it turns out to be convenient to force the constant term in this expansion to vanish rather than to assign the zeros of f(z). There is such an f(z)—indeed it is the Weierstrass function ℘(z)—but we can’t define it by the method at the start of this subsection because if φ(z) = z^{−2}, the series Φ(z) is not absolutely convergent. However, if φ(z) = −2z^{−3}, we can apply this method, and it gives ℘′, the derivative of the Weierstrass ℘-function. Define

℘′(z; Λ) = ℘′(z; ω_{1}, ω_{2}) = −2 ∑_{ω∈Λ} \frac{1}{(z − ω)^{3}} .

3The expression f(z) = O(φ(z)) means that |f(z)| < Cφ(z) for some constant C (independent of z) for all values of z in question.

Hence

℘(z) = \frac{1}{z^{2}} + ∑_{ω∈Λ,ω≠0} (\frac{1}{(z − ω)^{2}} − \frac{1}{ω^{2}} ) .

Theorem 3.8. Let P_{1}, ..., P_{n} and Q_{1}, ..., Q_{n} be two sets of n ≥ 2 points in the complex plane, possibly with repetitions, but such that no P_{i} is congruent to a Q_{j} modulo Λ. If ∑ P_{i} ≡ ∑ Q_{j} mod Λ, then there exists a doubly periodic function f(z) whose poles are the P_{i} and whose zeros are the Q_{j} with correct multiplicty, and f(z) is unique up to multiplication by a nonzero constant.

Proof. There is an elementary (constructive) proof. Alternatively, one can apply the Riemann-Roch theorem to ℂ/Λ.

The addition formula. Consider ℘(z + z′). It is a doubly periodic function of z, and therefore it is a rational function of ℘ and ℘′.

Proposition 3.9. There is the following formula:

℘(z + z′) = \frac{1}{4} \{\frac{℘′(z) − ℘′(z′)}{℘(z) − ℘(z′)}\}^{2} − ℘(z) − ℘(z′).

Proof. Let f(z) denote the difference between the left and the right sides. Its only possible poles (in D) are at 0, or ±z′, and by examining the Laurent expansion of f(z) near these points one sees that it has no pole at 0 or z, and at worst a simple pole at z′. Since it is doubly periodic, it must be constant, and since f(0) = 0, it must be identically zero.

Eisenstein series. Write

G_{k}(Λ) = ∑_{ω∈Λ,ω≠0} ω^{−2k}

and define G_{k}(z) = G_{k}(zℤ + ℤ).

Proposition 3.10. The Eisenstein series G_{k}(z), k > 1, converges to a holomorphic function on ℍ; it takes the value 2ζ(2k) at infinity. (Here ζ(s) = ∑ n^{−s}, the usual zeta function.)

Proof. Apply Lemma 3.7 to see that G_{k}(z) is a holomorphic function on ℍ. It remains to consider G_{k}(z) as z → i∞ (remaining in D, the fundamental domain for Γ(1)). Because the series for G_{k}(z) converges uniformly absolutely on D, lim_{z→i∞} G_{k}(z) = ∑ lim_{z→i∞} 1/(mz + n)^{2k}. But lim_{z→i∞} 1/(mz + n)^{2k} = 0 unless m = 0, and so

lim_{z→i∞} G_{k}(z) = ∑_{n∈ℤ,n≠0}1/n^{2k} = 2 ∑_{n≥1} 1/n^{2k} = 2ζ(2k).

The field of doubly periodic functions.

Proposition 3.11. The field of doubly periodic functions is just ℂ(℘(z), ℘′(z)), and

℘′(z)^{2} = 4℘(z)^{3} − g_{2}℘(z) − g_{3}

where g_{2} = 60G_{2} and g_{3} = 140G_{3}.

Proof. To prove the second statement, define f(z) to be the difference of the left and the right hand sides, and show (from its Laurent expansion) that it is holomorphic near 0 and take the value 0 there. Since it is doubly periodic and holomorphic elsewhere, this implies that it is zero. The proof of the first statement is omitted.

Elliptic curves. Let k be a field of characteristic ≠ 2, 3. By an elliptic curve over k, I shall mean a nonsingular projective curve E of genus one together with a point 0 ∈ E(k). From the Riemann-Roch theorem, one obtains regular functions x and y on E such that x has a double pole at 0 and y a triple pole at 0, and neither has any other poles. Again from the Riemann-Roch theorem applied to the divisor 6 · 0, one finds that there is a relation between 1, x, x^{2}, x^{3}, y, y^{2}, xy, which can be put in the form

y^{2} = 4x^{3} − ax − b.

The fact that E is nonsingular implies that Δ =^{df} a^{3} − 27b^{2} ≠ 0. Thus E is isomorphic to the projective curve defined by the equation,

Y^{2}Z = 4X^{3} − aXZ^{2} − bZ^{3},

and every equation of this form (with Δ ≠ 0) defines an elliptic curve. Define

j(E) = 1728a^{3}/Δ.

Then two elliptic curves E and E′ are isomorphic if and only if j(E) = j(E′). If E is an elliptic curve over ℂ, then E(ℂ) has a natural complex structure—it is a Riemann surface. (See Math 679 for proofs of these, and other statements, about elliptic curves.)

An elliptic curve has a unique group structure (defined by regular maps) having 0 as its zero.

The elliptic curve E(Λ). Let Λ be a lattice in ℂ. We have seen that

℘′(z)^{2} = 4℘(z)^{3} − g_{2}℘(z) − g_{3}.

Let E(Λ) be the projective curve defined by the equation:

Y^{2}Z = 4X^{3} − g_{2}XZ^{2} − g_{3}Z^{3}.

Proposition 3.12. The curve E(Λ) is an elliptic curve (i.e., Δ ≠ 0), and the map

ℂ/Λ → E(Λ), z ↦ (℘(z) : ℘′(z) : 1), 0 ↦ (0 : 1 : 0)

is an isomorphism of Riemann surfaces. Every elliptic curve E is isomorphic to E(Λ) for some Λ.

Proof. There are direct proofs of this result, but we shall see in the next section that z ↦ Δ(z, ℤ+ℤ) is a modular function for Γ(1) with weight 12 having no zeros in ℍ, and that z ↦ j(zℤ+ℤ) is a modular function and defines a bijection Γ(1)\ℍ → ℂ (therefore every j equals j(Λ) for some lattice ℤz + ℤ, z ∈ ℍ).

The addition formula shows that the map in the proposition is a homomorphism.

Proposition 3.13. There are natural equivalences between the following categories:

(a) Objects: Elliptic curves E over ℂ. Morphisms: Regular maps E → E′ that are homomorphisms.

(b) Objects: Riemann surfaces E of genus 1 together with a point 0. Morphisms: Holomorphic maps E → E′ sending 0 to 0′.

(c) Objects: Lattices Λ ⊂ ℂ. Morphisms: Hom(Λ, Λ′) = {α ∈ ℂ | αΛ ⊂ Λ′}.

Proof. The functor c → b is Λ ↦ ℂ/Λ. The functor a → b is (E, 0) ↦ (E(ℂ), 0), regarded as a pointed Riemann surface.

4. Modular Functions and Modular Forms

Modular functions. Let Γ be a subgroup of finite index in Γ(1). A modular function for Γ is a meromorphic function on the compact Riemann surface Γ\ℍ^{∗}. We often regard it as a meromorphic function on ℍ^{∗} invariant under Γ. Thus, from this point of view, a modular function f for Γ is a function on ℍ satisfying the following conditions:

(a) f(z) is invariant under Γ, i.e., f(γz) = f(z) for all γ ∈ Γ;

(b) f(z) is meromorphic in ℍ;

(c) f(z) is meromorphic at the cusps.

For the cusp i∞, the last condition means the following: the subgroup of Γ(1)

fixing i∞ is generated by T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )—it is free abelian group of rank 1; the

subgroup of Γ fixing i∞ is a subgroup of finite index in <T >, and it therefore is

generated by ( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} ) for some h ∈ ℕ, (h is called the width of the cusp); as f(z) is

invariant under ( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} ), f(z+h) = f(z), and so f(z) can be expressed as a function

f^{∗}(q) of the variable q = exp(2πiz/h); this function f^{∗}(q) is defined on a punctured disk, 0 < |q| < ε, and for f to be meromorphic at i∞ means f^{∗} is meromorphic at q = 0.

For a cusp τ ≠ i∞, the condition means the following: we know there is an element σ ∈ Γ(1) such that τ = σ(i∞); the function z ↦ f(σz) is invariant under σΓσ^{−1}, and f(σz) is required to be meromorphic at i∞ in the above sense.

Of course (c) has to be checked only for a set of representatives of the Γ-equivalence classes of cusps (which will be finite).

Recall that a function f(z) that is holomorphic in a neighbourhood of a point a ∈ ℂ (except possibly at a) is holomorphic at a if and only if f(z) is bounded in a neighbourhood of a. It follows that f(z) has a pole at a, and therefore defines a meromorphic function in a neighbourhood of a, if and only if (z − a)^{n}f(z) is bounded near a for some n, i.e., if f(z) = O((z −a)^{−n}) near a. When we apply this remark to a modular function, we see that f(z) is meromorphic at i∞ if and only if f^{∗}(q) = O(q^{−n}) for some n as q → 0, i.e., if and only if, for some A > 0, e^{Aiz} · f(z) is bounded as z → i∞.

Example 4.1. As Γ(1) is generated by S and T, to check condition (a) it suffices to verify that

f(−1/z) = f(z), f(z + 1) = f(z).

The second equation implies that f = f^{∗}(q), q = exp(2πiz), and condition (c) says that

f^{∗}(q) = ∑_{n≥−N_{0}} a_{i}q^{i}.

Example 4.2. Consider Γ(2). Then Γ(2) is of index 6 in Γ(1). It is possible to find a set of generators for Γ(2) just as we found a set of generators for Γ(1), and again it suffices to check condition (a) for the generators. There are three inequivalent cusps, namely, i∞, S(i∞) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 \\ 0 \end{matrix} ) = ( \begin{matrix}0 \\ 1 \end{matrix} ) = 0, and T S(i∞) = 1.

Note that S(0) = i∞. The stabilizer of i∞ in Γ(2) is generated by ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ), and

so f(z) = f^{∗}(q), q = exp(2πiz/2), and for f(z) to be meromorphic at i∞ means f^{∗} is meromorphic at 0. For f(z) to be meromorphic at 0 means that f(Sz) = f(−1/z) is meromorphic at i∞, and for f(z) to be meromorphic at 1 means that f(1 − \frac{1}{z}) is meromorphic at i∞.

Proposition 4.3. There exists a unique modular function J for Γ(1) which is holomorphic except at i∞, where it has a simple pole, and which takes the values

J(i) = 1, J(ρ) = 0.

Proof. From Proposition 2.21 we know there is an isomorphism of Riemann surfaces f : Γ(1)\ℍ^{∗} → S (Riemann sphere). Write a, b, c for the images of ρ, i, ∞. Then there exists a (unique) linear fractional transformation S → S sending a, b, c to 0, 1, ∞, and on composing f with it we obtain a function J satisfying the correct conditions.

If g is a second function satisfying the same conditions, then g ∘ f^{−1} is an automorphism of the Riemann sphere, and so it is a linear fractional transformation. Since it fixes 0, 1, ∞ it must be the identity map.

Remark 4.4. Let j(z) = 1728g_{2}^{3}/Δ, as on Section 3. Then j(z) is invariant under Γ(1) because g_{2}^{3} and Δ are both modular forms of weight 12 (we give all the details for this example later). It is holomorphic on ℍ because both of g_{2}^{3} and Δ are holomorphic on ℍ, and Δ has no zeros on ℍ. Because Δ has a simple zero at ∞, j has a simple pole at ∞ . Therefore j(z) has valence one, and it defines an isomorphism from Γ\ℍ^{∗} onto S (the Riemann sphere). In fact, j(z) = 1728J(z).

Modular forms. Let Γ be a subgroup of finite index in Γ(1).

Definition 4.5. A modular form for Γ of weight 2k is a function on ℍ such that:

(a) f(γz) = (cz + d)^{2k} · f(z), all z ∈ ℍ;

(b) f(z) is holomorphic in ℍ;

(c) f(z) is holomorphic at the cusps of Γ.

A modular form is a cusp form if it is zero at the cusps.

For example, for the cusp i∞, this last condition means the following: let h be the width of i∞ as a cusp for Γ; then implies that f(z + h) = f(z), and so f(z) = f^{∗}(q) for some function f^{∗} on a punctured disk; f^{∗} is required to be holomorphic at q = 0.

When f(z) is zero at every cusp, it is called a cusp form. Occasionally we shall refer to a function satisfying only (4.5a) as being weakly modular of weight 2k, and a function satisfying (4.5a,b,c) with “holomorphic” replaced by “meromorphic” as being a meromorphic modular form of weight 2k. Thus a meromorphic modular form of weight 0 is a modular function.

As our first examples of modular forms, we have the Eisenstein series. Let L be the set of lattices in ℂ, and write Λ(ω_{1}, ω_{2}) for the lattice ℤω_{1} + ℤω_{2} generated by independent elements ω_{1}, ω_{2} with ℑ(ω_{1}/ω_{2}) > 0. Note that Λ(ω′_{1}, ω′_{2}) = Λ(ω_{1}, ω_{2}) if and only if

( \begin{matrix}ω′_{1} \\ ω′_{2} \end{matrix} ) = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}ω_{1} \\ ω_{2} \end{matrix} ), some ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) = Γ(1).

Lemma 4.6. Let F : L → ℂ be a function of weight 2k, i.e., such that F (λΛ) = λ^{−2k} · F (Λ) for λ ∈ ℂ^{×}. Then f(z) =^{df} F (Λ(z, 1)) is a weakly modular form on ℍ of weight 2k and F ↦ f is a bijection from the functions of weight 2k on L to the weakly modular forms of weight 2k on ℍ.

Proof. Write F (ω_{1}, ω_{2}) for the value of F at the lattice Λ(ω_{1}, ω_{2}). Then because F is of weight 2k, we have

F (λω_{1}, λω_{2}) = λ^{−2k} · F (ω_{1}, ω_{2}), λ ∈ ℂ^{×},

and, because F (ω_{1}, ω_{2}) depends only on Λ(ω_{1}, ω_{2}), it is invariant under the action of SL_{2}(ℤ) :

F (aω_{1} + bω_{2}, cω_{1} + dω_{2}) = F (ω_{1}, ω_{2}), all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ), (∗).

The first equation shows that ω_{2}^{2k} · F (ω_{1}, ω_{2}) is invariant under (ω_{1}, ω_{2}) ↦ (λω_{1}, λω_{2}), λ ∈ ℂ^{×}, and so depends only on the ratio ω_{1}/ω_{2}; thus there is a function f(z) such that

F (ω_{1}, ω_{2}) = ω_{2}^{−2k} · f(ω_{1}/ω_{2}), (∗∗).

When expressed in terms of f, (*) becomes

(cω_{1} + dω_{2})^{−2k} · f(aω_{1} + bω_{2}/cω_{1} + dω_{2}) = ω_{2}^{−2k} · f(ω_{1}/ω_{2}),

or

(cz + d)^{−2k} · f(az + b/cz + d) = f(z).

This shows that f is weakly modular. Conversely, given a weakly modular f, define F by the formula (**).

Proposition 4.7. The Eisenstein series G_{k}(z), k > 1, is a modular form of weight 2k for Γ(1) which takes the value 2ζ(2k) at infinity.

Proof. Recall that we defined G_{k}(Λ) = ∑_{ω∈Λ,ω≠0} 1/ω^{2k}. Clearly, G_{k}(λΛ) = λ^{−2k}G_{k}(Λ), and therefore G_{k}(z) =_{df} G_{k}(Λ(z, 1)) = ∑_{(m,n)≠(0,0)} 1/(mz+n)^{2k} is weakly modular. That it is holomorphic on ℍ and takes the value 2ζ(2k) at i∞ is proved in Proposition 3.10.

Modular forms as k-fold differentials. The definition of modular form may seem strange, but we have seen that such functions arise naturally in the theory of elliptic functions. Here we give another explanation of the definition. For the experts, we shall show later that the modular forms of a fixed weight 2k are the sections of a line bundle on Γ\ℍ^{∗}.

Remark 4.8. Consider a differential ω = f(z) · dz on ℍ, where f(z) is a meromorphic function. Under what conditions on f is ω invariant under the action of Γ? Let γ(z) = \frac{az+b}{cz+d}; then

γ^{∗}ω = f(γz) · d\frac{az + b}{cz + d} = f(γz) · \frac{(a(cz + d) − c(az + b))}{(cz + d)^{2}} · dz = f(γz) · (cz + d)^{−2} · dz.

Thus ω is invariant if and only if f(z) is a meromorphic differential form of weight 2. We have one-to-one correspondences between the following sets:

{meromorphic modular forms of weight 2 on ℍ for Γ}

{meromorphic differential forms on ℍ^{∗} invariant under the action of Γ} {meromorphic differential forms on Γ\ℍ^{∗}}. There is a notion of a k-fold differential form on a Riemann surface. Locally it

can be written ω = f(z) · (dz)^{k}, and if w = w(z), then

w^{∗}ω = f(w(z)) · (dw(z))^{k} = f(w(z)) · w′(z)^{k} · (dz)^{k}.

Then modular forms of weight 2k correspond to Γ-invariant k-fold differential forms on ℍ^{∗}, and hence to meromorphic k-fold differential forms on Γ\ℍ^{∗}. Warning: these statements don’t (quite) hold with meromorphic replaced with holomorphic.

We say that ω = f(z) · (dz)^{k} has a zero or pole of order m at z = 0 according as f(z) has a zero or pole of order m at z = 0. This definition is independent of the choice of the local coordinate near the point in question on the Riemann surface.

The dimension of the space of modular forms. For a subgroup of finite index in Γ(1), we write M_{k}(Γ) for the space of modular forms of weight 2k for Γ, and S_{k}(Γ) for the subspace of cusp forms of weight 2k. They are vector spaces over ℂ, and we shall use the Riemann-Roch theorem to compute their dimensions.

Note that M_{0}(Γ) consists of modular functions that are holomorphic on ℍ and at the cusps, and therefore define holomorphic functions on Γ\ℍ^{∗}. Because Γ\ℍ^{∗} is compact, such a function is constant, and so M_{0}(Γ) = ℂ. The product of a modular form of weight k with a modular form of weight ℓ is a modular form of weight k + ℓ. Therefore,

M(Γ) =^{df} ⊕M_{k}(Γ)

is a graded ring. The next theorem gives us the dimensions of the homogeneous pieces.

Theorem 4.9. The dimension of M_{k}(Γ) is given by:

dim(M_{k}(Γ)) = \{ \begin{matrix}0 & if k ≤ −1 \\ 1 & if k = 0 \\ (2k − 1)(g − 1) + ν_{∞}k + ∑_{P} [k(1 − \frac{1}{e_{P}})] & if k ≥ 1 \end{matrix}

where g is the genus of X(Γ) (=^{df} Γ\ℍ^{∗}); ν_{∞} is the number of inequivalent cusps; the last sum is over a set of representatives for the the elliptic points P of Γ; e_{P} is the order of the stabilizer of P in the image Γ̄ of Γ in Γ(1)/{±I}); [k(1 − 1/e_{P})] is the integer part of k(1 − 1/e_{P}).

We prove the result by applying the Riemann-Roch theorem to the compact Riemann surface Γ\ℍ^{∗}, but first we need to examine the relation between the zeros and poles of a Γ-invariant k -fold differential form on ℍ^{∗} and the zeros and poles of the corresponding modular form on Γ\ℍ^{∗}. It will be helpful to consider first a simple example.

Example 4.10. Let D be the unit disk, and consider the map w : D → D, z ↦ z^{e}. Let Q ↦ P. If Q ≠ 0, then the map is a local isomorphism, and so there is no difficulty. Thus we suppose that P and Q are both zero.

First suppose that f is a function on D (the target disk), and let f^{∗} = f ∘ w. If f has a zero of order m (regarded as function of w), then f^{∗} has a zero of order em, for if f(w) = aw^{m} + terms of higher degree, then f(z^{e}) = az^{em} + terms of higher degree. Thus

ord_{Q}(f^{∗}) = e · ord_{P} (f).

Now consider a k-fold differential form ω on D, and let ω^{∗} = w^{∗}(ω). Then ω = f(z) · (dz)^{k} for some f(z), and

ω^{∗} = f(z^{e}) · (dz^{e})^{k} = f(z^{e}) · (ez^{e−1} · dz)^{k} = e^{k} · f(z^{e}) · z^{k(e−1)} · (dz)^{k}.

Thus

ord_{Q}(ω^{∗}) = e ord_{P} (ω) + k(e − 1).

Lemma 4.11. Let f be a (meromorphic) modular form of weight 2k, and let ω be the corresponding k-fold differential form on Γ\ℍ^{∗}. Let Q ∈ ℍ^{∗} map to P ∈ Γ\ℍ^{∗}.

(a) If Q is an elliptic point with multiplicity e, then

ord_{Q}(f) = e ord_{P} (ω) + k(e − 1).

(b) If Q is a cusp, then

ord_{Q}(f) = ord_{P} (ω) + k.

(c) For the remaining points,

ord_{Q}(f) = ord_{P} (ω).

Proof. Let p be the quotient map ℍ → Γ\ℍ.

(a) We defined the complex structure near P so that, for appropriate neighbourhoods V of Q and U of P, there is a commutative diagram:

V ⟶_{≈}^{Q↦0} D
↓^{p}  ↓^{z↦z^{e}}
U ⟶_{≈}^{P ↦0} D

Thus this case is isomorphic to that considered in the example.

(b) Consider the map q : ℍ → (punctured disk), q(z) = exp(2πiz/h), and let ω^{∗} = g(q) · (dq)^{k} be a k-fold differential form on the punctured disk. Then dq = (2πi/h) · q · dz, and so the inverse image of ω^{∗} on ℍ is

ω = (cnst) · g(q(z)) · q(z)^{k} · (dz)^{k},

and so ω^{∗} corresponds to the modular form f(z) = (cnst) · g(q(z)) · q(z)^{k}. Thus f^{∗}(q) = g(q) · q^{k}, which gives our formula.

(iii) In this case, p is a local isomorphism near Q and P, and so there is nothing to prove.

We now prove the theorem. Let f ∈ M_{k}(Γ), and let ω be the corresponding k-fold differential on Γ\ℍ^{∗}. Because f is holomorphic, we must have

e ord_{P} (ω) + k(e − 1) = ord_{Q}(f) ≥ 0 at the image of an elliptic point;
ord_{P} (ω) + k ≥ ord_{Q}(f) ≥ 0 at the image of a cusp;
ord_{P} (ω) = ord_{Q}(f) ≥ 0 at the remaining cusps.

Fix a k-fold differential ω_{0}, and write ω = h · ω_{0}. Then

ord_{P} (h) + ord_{P} (ω_{0}) + k(1 − 1/e) ≥ 0 at the image of an elliptic point;
ord_{P} (h) + ord_{P} (ω_{0}) + k ≥ 0 at the image of a cusp;
ord_{P} (h) + ord_{P} (ω_{0}) ≥ 0 at the remaining points.

On combining these inequalities, we find that

div(h) + D ≥ 0,

where

D = div(ω_{0}) + ∑ k · P_{i} + ∑[k(1 − 1/e_{i})] · P_{i}

(the first sum is over the images of the cusps, and the second sum is over the images of the elliptic points). As we noted in Corollary 1.21, the degree of the divisor of a 1-fold differential form is 2g − 2; hence that of a k-fold differential form is k(2g − 2). Thus the degree of D is

k(2g − 2) + ν_{∞} · k + ∑_{P}[k(1 − 1/e_{P})].

Now the Riemann-Roch Theorem (1.22) tells us that the space of h’s has dimension

1 − g + k(2g − 2) + ν_{∞} · k + ∑_{P}[k(1 − 1/e_{P})]

for k ≥ 1. As the h’s are in one-to-one correspondence with the holomorphic modular forms of weight 2k, this proves the theorem in this case. For k = 0, we have already noted that modular forms are constant, and for k < 0 it is easy to see that there can be no modular forms.

Zeros of modular forms. Lemma 4.11 allows us to count the number of zero and poles of a meromorphic differential form.

Proposition 4.12. Let f be a (meromorphic) modular form of weight 2k; then

∑(ord_{Q}(f)/e_{Q} − k(1 − 1/e_{Q})) = k(2g − 2) + k · ν_{∞}

where the sum is over a set of representatives for the points in Γ\ℍ^{∗}, ν_{∞} is the number of inequivalent cusps, and e_{Q} is the ramification index of Q over p(Q) if Q ∈ ℍ and it is 1 if Q is a cusp.

Proof. Let ω be the associated k-fold differential form on Γ\ℍ^{∗}. We showed above that: ord_{Q}(f)/e_{Q} = ord_{P} (ω) + k(1 − 1/e_{Q}) for Q an elliptic point for Γ; ord_{Q}(f) = ord_{P} (ω) − k for Q a cusp; ord_{Q}(f) = ord_{P} (ω) at the remaining points. On summing these equations, we find that

∑ ord_{Q}(f)/e_{Q} − k(1 − 1/e_{Q}) = deg(div(ω)) + k · ν_{∞},

and we noted above that deg(div(ω)) = k(2g − 2).

Example 4.13. When Γ = Γ(1), this becomes

ord_{i∞}(f) + \frac{1}{2} ord_{i}(f) + \frac{1}{3} ord_{ρ}(f) + ∑ ord_{Q}(f) = −2k + k + \frac{1}{2}k + \frac{2}{3}k = \frac{k}{6}.

Here i∞, i, ρ are points in ℍ^{∗}, and the sum ∑ is over the remaining points in a fundamental domain.

Modular forms for Γ(1). We now describe all the modular forms for Γ(1).

Example 4.14. On applying Theorem 4.9 to the full modular group Γ(1), we obtain the following result: M_{k} = 0 for k < 0, dim M_{0} = 1, and

dim M_{k} = 1 − k + [\frac{k}{2}] + [\frac{2k}{3} ], k > 1.

Thus

\begin{aligned}k & = & 1 & 2 & 3 & 4 & 5 & 6 & 7 & …; \\ dim M_{k} & = & 0 & 1 & 1 & 1 & 1 & 2 & 1 & …. \end{aligned}

In fact, when k is increased by 6, the dimension increases by 1. Thus we have

(a) M_{k} = 0 for k < 0;

(b) dim(M_{k}) = [k/6] if k ≡ 1 mod 6; [k/6] + 1 otherwise; k ≥ 0.

Example 4.15. On applying (4.13) to the Eisenstein series G_{k}, k > 1, one obtains the following result:

k = 2: G_{2} has a simple zero at z = ρ, and no other zeros.

k = 3: G_{3} has a simple zero at z = i, and no other zeros.

k = 6: because Δ has no zeros in ℍ, it has a simple zero at ∞.

There is a geometric explanation for these statements. Let Λ = Λ(i, 1); then the torus ℂ/Λ has complex multiplication by i, and so the elliptic curve

Y^{2} = 4X^{3} − g_{2}(Λ)X − g_{3}(Λ)

has complex multiplication by i; clearly the curve

Y^{2} = X^{3} + X

has complex multiplication by i (and up to isogeny, it is the only such curve); this suggests that g_{3}(Λ) = 0. Similarly, G_{2}(Λ) = 0 “because” Y^{2} = X^{3} + 1 has complex multiplication by \sqrt[3]{1}. Finally, if Δ had no zero at ∞, the family of elliptic curves

Y^{2} = 4X^{3} − g_{2}(Λ)X − g_{3}(Λ)

over Γ(1)\ℍ would extend to a smooth family over Γ(1)\ℍ^{∗}, and this is not possible for topological reasons (its cohomology groups would give a nonconstant local system on Γ(1)\ℍ^{∗}, but the Riemann sphere is simply connected, and so admits no such system).

Proposition 4.16. (a) For k < 0, and k = 1, M_{k} = 0.

(b) For k = 0, 2, 3, 4, 5, M_{k} is a space of dimension 1, admitting as basis 1, G_{2}, G_{3}, G_{4}, G_{5} respectively; moreover S_{k}(Γ) = 0 for 0 ≤ k ≤ 5.

(c) Multiplication by Δ defines an isomorphism of M_{k−6} onto S_{k}.

(d) The graded k-algebra ⊕M_{k} = ℂ[G_{2}, G_{3}] with G_{2} and G_{3} of weights 2 and 3 respectively.

Proof. (a) See (4.14).

(b) Since the spaces are one-dimensional, and no G_{k} is identically zero, this is obvious.

(c) Certainly f ↦ fΔ is a homomorphism M_{k−6} → S_{k}. But if f ∈ S_{k}, then f/Δ ∈ M_{k−6} because Δ has only a simple zero at i∞ and f has a zero there. Now f ↦ f/Δ is inverse to f ↦ fΔ.

(d) We have to show that {G_{2}^{m} · G_{3}^{n} | 2m + 3n = k, m ∈ ℕ, n ∈ ℕ} forms a basis for M_{k}. We first show, by induction on k, that this set generates M_{k}. For k ≤ 3, we have already noted it. Choose a pair m ≥ 0 and n ≥ 0 such that 2m + 3n = k (this is always possible for k ≥ 2). The modular form g = G_{2}^{m} · G_{3}^{n} is not zero at infinity. If f ∈ M_{k}, then f − \frac{f(∞)}{g(∞)}g is zero at infinity, and so is a cusp form. Therefore, it can be written Δ · h with h ∈ M_{k−6}, and we can apply the induction hypothesis.

Thus ℂ[G_{2}, G_{3}] → ⊕M_{k} is surjective, and we want to show that it is injective. If not, the modular function G_{2}^{3}/G_{3}^{2} satisfies an algebraic equation over ℂ, and so is constant. But G_{2}(ρ) = 0 ≠ G_{3}(ρ) whereas G_{2}(i) ≠ 0 = G_{3}(i).

Remark 4.17. We have verified all the assertions in (4.3).

The Fourier coefficients of the Eisenstein series for Γ(1). For future use, we compute the coefficients in the expansion G_{k}(z) = ∑ a_{n}q^{n}.

The Bernoulli numbers B_{k}. They are defined by the formal power series expansion:

\frac{x}{e^{x} − 1} = 1 − \frac{x}{2} + ∑_{k=1}^{∞}(−1)^{k+1}B_{k} \frac{x^{2k}}{(2k)!}.

Thus B_{1} = 1/6; B_{2} = 1/30; ... ; B_{14} = 23749461029/870; ... Note that they are all rational numbers.

Proposition 4.18. For any integers k ≥ 1,

ζ(2k) = \frac{2^{2k−1}}{(2k)!} B_{k}π^{2k}.

Proof. Recall that (by definition)

cos(z) = \frac{e^{iz} + e^{−iz}}{2} , sin(z) = \frac{e^{iz} − e^{−iz}}{2i} .

Therefore,

cot(z) = i\frac{e^{iz} + e^{−iz}}{e^{iz} − e^{−iz}} = i\frac{e^{2iz} + 1}{e^{2iz} − 1} = i + \frac{2i}{e^{2iz} − 1}.

On replacing x with 2iz in the definition of the Bernoulli numbers, we find that

z cot(z) = 1 − ∑_{k=1}^{∞} B_{k} \frac{2^{2k}z^{2k}}{(2k)!} (∗).

There is a standard formula

sin(z) = z ∏_{n=1}^{∞}(1 − \frac{z^{2}}{n^{2}π^{2}} )

(see Cartan 1963, V.3.3). On forming the logarithmic derivative of this (i.e., forming d log(f) = f′/f) and multiplying by z, we find that

z cot z = 1 − ∑_{n=1}^{∞} \frac{2z^{2}/n^{2}π^{2}}{1 − z^{2}/n^{2}π^{2}}

= 1 + 2 ∑_{n=1}^{∞} \frac{1}{1 − n^{2}π^{2}/z^{2}}

= 1 + 2 ∑_{n=1}^{∞} ∑_{k=1}^{∞} \frac{z^{2k}}{n^{2k}π^{2k}}

= 1 + 2 ∑_{k=1}^{∞} (∑_{n=1}^{∞} n^{−2k}) \frac{z^{2k}}{π^{2k}} .

On comparing this formula with (∗), we obtain the result.

For example, ζ(2) = \frac{π^{2}}{2×3}, ζ(4) = \frac{π^{4}}{2×3^{2}×5}, ζ(6) = \frac{π^{6}}{3^{3}×5×7}, ....

Remark 4.19. Until a few years ago, when Apery showed that ζ(3) is irrational, nothing was known about the values of ζ at the odd positive integers.

The Fourier coefficients of G_{k}. For any integer n and number k, we write

σ_{k}(n) = ∑_{d|n} d^{k}.

Proposition 4.20. For any integer k ≥ 2,

G_{k}(z) = 2ζ(2k) + 2\frac{(2πi)^{2k}}{(2k − 1)!} ∑_{n=1}^{∞} σ_{2k−1}(n)q^{n}.

Proof. In the above proof, we showed above that

z cot(z) = 1 + 2 ∑_{n=1}^{∞} \frac{z^{2}}{z^{2} − n^{2}π^{2}} ,

and so (replace z with πz and divide by z)

π cot(πz) = \frac{1}{z} + 2 ∑_{n=1}^{∞} \frac{z}{z^{2} − n^{2}} = \frac{1}{z} + ∑_{n=1}^{∞} (\frac{1}{z + n} + \frac{1}{z − n}) .

Moreover, we showed that

cot(z) = i + \frac{2i}{e^{2iz} − 1},

and so

π cot(πz) = πi − \frac{2πi}{1 − q}
= πi − 2πi ∑_{n=1}^{∞} q^{n}

where q = e^{2πiz}. Therefore,

\frac{1}{z} + ∑_{n=1}^{∞} (\frac{1}{z + n} + \frac{1}{z − n}) = πi − 2πi ∑_{n=1}^{∞} q^{n}.

The (k − 1)^{th} derivative of this (k ≥ 2) is

∑_{n∈ℤ} \frac{1}{(n + z)^{k}} = \frac{1}{(k − 1)!}(−2πi)^{k} ∑_{n=1}^{∞} n^{k−1}q^{n}.

Now

G_{k}(z) =^{df} ∑_{(n,m)≠(0,0)} \frac{1}{(nz + m)^{2k}}

= 2ζ(2k) + 2 ∑_{n=1}^{∞} ∑_{m∈ℤ} \frac{1}{(nz + m)^{2k}}

= 2ζ(2k) + \frac{2(−2πi)^{2k}}{(2k − 1)!} ∑_{n=1}^{∞} ∑_{a=1}^{∞} a^{2k−1} · q^{an}

= 2ζ(2k) + \frac{2(2πi)^{2k}}{(2k − 1)!} ∑_{n=1}^{∞} σ_{2k−1}(a) · q^{n}.

The expansion of Δ and j. Recall that

Δ =^{df} g_{2}^{3} − 27g_{3}^{2}.

From the above expansions of G_{2} and G_{3}, one finds that

Δ = (2π)^{12} · (q − 24q^{2} + 252q^{3} − 1472q^{4} + ⋯)

Theorem 4.21 (Jacobi). Δ = (2π)^{12}q ∏_{n=1}^{∞}(1 − q^{n})^{24}, q = e^{2πiz}.

Proof. Let f(q) = q ∏_{n=1}^{∞}(1 − q^{n})^{24}. The space of cusp forms of weight 12 has dimension 1. Therefore, if we show that f(−1/z) = z^{12}f(z), then f will be a multiple of Δ. It is possible to prove by an elementary argument (due to Hurwitz), that f(−1/z) and z^{12}f(z) have the same logarithmic derivative; therefore

f(−1/z) = Cz^{12} · f(z),

some C. Put z = 1 to see C = 1. See Serre 1970, VII.4.4, for the details.

Write q ∏(1 − q^{n})^{24} = ∑_{n=1}^{∞} τ(n) · q^{n}. The function n ↦ τ(n) was studied by Ramanujan, and is called the Ramanujan τ-function. We have

τ(1) = 1, τ(2) = −24, ..., τ(12) = −370944,....

Evidently each τ(n) ∈ ℤ. Ramanujan made a number of interesting conjectures about τ(n), some of which, as we shall see, have been proved.

Recall that j(z) = \frac{1728g_{2}^{3}}{Δ} , Δ = g_{2}^{3} − 27g_{3}^{2}, g_{2} = 60G_{2}, g_{3} = 140G_{3}.

Theorem 4.22. The function

j(z) = \frac{1}{q} + 744 + 196 884q + 21 493 760q^{2} + c(3)q^{3} + c(4)q^{4} + ⋯, q = e^{2πiz},

where c(n) ∈ ℤ for all n.

Proof. Immediate consequence of the definition and the above calculations.

The size of the coefficients of a cusp form. Let f(z) = ∑ a_{n}q^{n} be a cusp form of weight 2k ≥ 2 for some congruence subgroup of SL_{2}(ℤ). For various reasons, for example, in order to obtain estimates of the number of times an integer can be represented by a quadratic form, one is interested in |a_{n}|.

Hecke showed that a_{n} = O(n^{k})—the proof is quite easy (see Serre 1970, VII.4.3, for the case of Γ(1)). Various authors improved on this—for example, Selberg showed in 1965 that a_{n} = O(n^{k−1/4+ε}) for all ε > 0. It was conjectured that a_{n} = O(n^{k−1/2} · σ_{0}(n)) (for the τ-function, this goes back to Ramanujan). The usual story with such conjectures is that they prompt an infinite sequence of papers proving results converging to the conjecture, but (happily) in this case Deligne proved in 1969 that the conjecture follows from the Weil conjectures for varieties over finite fields, and he proved the Weil conjectures in 1973. I hope to return to this question.

Modular forms as sections of line bundles. Let X be a topological manifold. A line bundle on X is a map of topological spaces π : L → X such that, for some open covering X = ⋃ U_{i} of X, π^{−1}(U_{i}) ≈ U_{i} × ℝ. Similarly, a line bundle on a Riemann surface is a map of complex manifolds π : L → X such that locally L is isomorphic to U × ℂ, and a line bundle on an algebraic variety is a map of algebraic varieties π : L → X such that locally for the Zariski topology on X, L ≈ U × 𝔸^{1}.

If L is a line bundle on X (say a Riemann surface), then for any open subset U of X, Γ(U, L) denotes the group of sections of L over U, i.e., the set of holomorphic maps f : U → L such that π ∘ f = identity map. Note that if L = U × ℂ, then Γ(U, L) can be identified with the set of holomorphic functions on U. (The Γ in Γ(U, L) should not be confused with a congruence group Γ.)

Now consider the following situation: Γ is a group acting freely and properly discontinuously on a Riemann surface H, and X = Γ\H. Write p for the quotient map H → X. Let π : L → X be a line bundle on X; then

p^{∗}(L) =^{df} {(h, l) ⊂ H × L | p(h) = π(l)}

is a line bundle on H (for example, p^{∗}(X ×L) = H ×L), and Γ acts on p^{∗}(L) through its action on H. Suppose we are given an isomorphism i : H × ℂ → p^{∗}(L). Then we can transfer the action of Γ on p^{∗}(L) to an action of Γ on H × ℂ over H. For γ ∈ Γ and (τ, z) ∈ H × ℂ, write

γ(τ, z) = (γτ, j_{γ}(τ)z), j_{γ}(τ) ∈ ℂ^{×}.

Then

γγ′(τ, z) = γ(γ′τ, j_{γ′}(τ)z) = (γγ′τ, j_{γ}(γ′τ) · j_{γ′}(z) · z).

Hence:

j_{γγ′}(τ) = j_{γ}(γ′τ) · j_{γ′}(τ).

Definition 4.23. An automorphy factor is a map j : Γ × ℍ → ℂ^{×} such that

(a) for each γ ∈ Γ, τ ↦ j_{γ}(τ) is a holomorphic function on ℍ;

(b) j_{γγ′}(τ) = j_{γ}(γ′τ) · j_{γ′}(τ).

Condition (b) should be thought of as a cocycle condition (in fact, that’s what it is). Note that if j is an automorphy factor, so also is j^{k} for any integer k.

Example 4.24. For any open subset H of ℂ with a group Γ acting on it, there is canonical automorphy factor j_{γ}(τ), namely,

Γ × H → ℂ, (γ, τ) ↦ (dγ)_{τ}.

By (dγ)_{τ} I mean the following: each γ defines a map H → H, and (dγ)_{τ} is the map on the tangent space at τ defined by γ. As H ⊂ ℂ, the tangent spaces at τ and at γτ are canonically isomorphic to ℂ, and so (dγ)_{τ} can be regarded as a complex number.

Suppose we have maps

M ⟶^{α} N ⟶^{β} P

of (complex) manifolds, then for any point m ∈ M, (d(β ∘ α))_{m} = (dβ)_{α(n)} ∘ (dα)_{m} (maps on tangent spaces). Therefore,

j_{γγ′}(τ) =_{df} (dγγ′)_{τ} = (dγ)_{γ′τ} · (dγ′)_{τ} = j_{γ}(γ′τ) · j_{γ′}(τ).

Thus j_{γ}(τ) =_{df} (dγ)_{τ} is an automorphy factor.

For example, consider Γ(1) acting on ℍ. If γ = (z ↦ \frac{az+b}{cz+d}), then

dγ = \frac{1}{(cz + d)^{2}} dz,

and so j_{γ}(τ) = (cz + d)^{−2}, and j_{γ}(τ)^{k} = (cz + d)^{−2k}.

Proposition 4.25. There is a one-to-one correspondence between the set of pairs (L, i) where L is a line bundle on Γ\H and i is an isomorphism H × ℂ ≈ p^{∗}(L) and the set of automorphy factors.

Proof. We have seen how to go (L, i) ↦ j_{γ}(τ). For the converse, use i and j to define an action of Γ on H × ℂ, and define L to be Γ\H × ℂ.

Remark 4.26. Every line bundle on ℍ is trivial (i.e., isomorphic to ℍ × ℂ), and so Proposition 4.25 gives us a classification of the line bundles on Γ\ℍ.

Let L be a line bundle on X. Then

Γ(X, L) = {F ∈ Γ(H, p^{∗}L) | F commutes with the actions of Γ}.

Suppose we are given an isomorphism p^{∗}L ≈ H × ℂ. We use it to identify the two line bundles on H. Then Γ acts on H × ℂ by the rule:

γ(τ, z) = (γτ, j_{γ}(τ)z).

A holomorphic section F : H → H × ℂ can be written F (τ) = (τ, f(τ)) with f(τ) a holomorphic map H → ℂ. What does it mean for F to commute with the action of Γ? We must have

F (γτ) = γF (τ), i.e., (γτ, f(γτ)) = (γτ, j_{γ}(τ)f(τ)).

Hence

f(γτ) = j_{γ}(τ) · f(τ).

Thus, if L_{k} is the line bundle on Γ\ℍ corresponding to j_{γ}(τ)^{−k}, where j_{γ}(τ) is the canonical automorphy factor (4.24), then the condition becomes

f(γτ) = (cz + d)^{2k} · f(τ),

i.e., condition (4.5a). Therefore the sections of L_{k} are in natural one-to-one correspondence with the functions on ℍ satisfying (4.5a,b). The line bundle L_{k} extends to a line bundle L_{k}^{∗} on the compactification Γ\ℍ^{∗}, and the sections of L_{k}^{∗} are in natural one-to-one correspondence with the modular forms of weight 2k.

Poincaré series. We want to construct modular forms for subgroups Γ of finite index in Γ(1). Throughout, we write Γ′ for the image of Γ in Γ(1)/{±I}.

Recall the standard way of constructing invariant functions: if h is a function on ℍ, then

f(z) =^{df} ∑_{γ∈Γ′} h(γz)

is invariant under Γ, provided the series converges absolutely (which it rarely will). Poincaré found a similar argument for constructing modular forms.

Let

Γ × ℍ → ℂ, (γ, z) ↦ j_{γ}(z)

be an automorphy factor for Γ; thus

j_{γγ′}(z) = j_{γ}(γ′z) · j_{γ′}(z).

Of course, we will be particularly interested in the case

j_{γ}(z) = (cz + d)^{2k}, γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

We wish to construct a function f such that f(γz) = j_{γ}(z) · f(z).

Try

f(z) = ∑_{γ∈Γ′} \frac{h(γz)}{j_{γ}(z)}.

If this series converges absolutely uniformly on compact sets, then

f(γ′z) = ∑_{γ∈Γ′} \frac{h(γγ′z)}{j_{γ}(γ′z)} = ∑_{γ∈Γ′} \frac{h(γγ′z)}{j_{γγ′}(z)}j_{γ′}(z) = j_{γ′}(z) · f(z)

as wished.

Unfortunately, there is little hope of convergence, for the following (main) reason: there may be infinitely many γ’s for which j_{γ}(z) = 1 identically, and so the sum contains infinitely many redundant terms. Let

Γ_{0} = {γ ∈ Γ′ | j_{γ}(z) = 1 identically}.

For example, if j_{γ}(z) = (cz + d)^{−2k}, then

Γ_{0} = \{± ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ| c = 0, d = 1\} = \{± ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) ∈ Γ\} =< ± ( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} )>

where h is the smallest positive integer such that ( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} ) ∈ Γ (thus h is the width

of the cusp i∞ for Γ). In particular, Γ_{0} is an infinite cyclic group.

If γ, γ′ ∈ Γ_{0}, then

j_{γγ′}(z) = j_{γ}(γ′z) · j_{γ′}(z) = 1 (all z),

and so Γ_{0} is closed under multiplication—in fact, it is a subgroup of Γ′.

Let h be a holomorphic function on ℍ invariant under Γ_{0}, i.e., such that h(γ_{0}z) = h(z) for all γ_{0} ∈ Γ_{0}. Let γ ∈ Γ′ and γ_{0} ∈ Γ_{0}; then

\frac{h(γ_{0}γz)}{j_{γ_{0}γ}(z)} = \frac{h(γz)}{j_{γ_{0}}(γz) · j_{γ}(z)} = \frac{h(γz)}{j_{γ}(z)},

i.e., h(γz)/j_{γ}(z) is constant on the coset Γ_{0}γ. Thus we can consider the series

f(z) = ∑_{Γ_{0}\Γ′} \frac{h(γz)}{j_{γ}(z)}

If the series converges absolutely uniformly on compact sets, then the previous argument shows that we obtain a holomorphic function f such that f(γz) = j_{γ}(z) · f(z).

Apply this with j_{γ}(z) = (cz + d)^{2k}, γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ), and Γ a subgroup of finite

index in Γ(1). As we noted above, Γ_{0} is generated by z ↦ z + h for some h, and a typical function invariant under z ↦ z + h is exp(2πinz/h), n = 0, 1, 2, …

Definition 4.27. The Poincaré series of weight 2k and character n for Γ is the series

φ_{n}(z) = ∑_{Γ_{0}\Γ′} \frac{exp(\frac{2πin·γ(z)}{h})}{(cz + d)^{2k}}

where Γ′ is the image of Γ in Γ(1)/{±I}.

We need a set of representatives for Γ_{0}\Γ′. Note that

( \begin{matrix}1 & m \\ 0 & 1 \end{matrix} ) · ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}a + mc & b + md \\ c & d \end{matrix} ) .

Using this, it is easily checked that ( \begin{matrix}a & b \\ c & d \end{matrix} ) and ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) are in the same coset

of Γ_{0} if and only if (c, d) = ±(c′, d′) and (a, b) ≡ ±(a′, b′) mod h. Thus a set of representatives for Γ_{0}\Γ′ can be obtained by taking one element of Γ′ for each pair (c, d), c > 0, which is the second row of a matrix in Γ′.

Theorem 4.28. The Poincaré series φ_{n}(z) for 2k ≥ 2, n ≥ 0, converges absolutely uniformly on compact subsets of ℍ; it converges absolutely uniformly on every fundamental domain D for Γ, and hence is a modular form of weight 2k for Γ. Moreover,

(a) φ_{0}(z) is zero at all finite cusps, and φ_{0}(i∞) = 1;

(b) for all n ≥ 1, φ_{n}(z) is a cusp form.

Proof. To see convergence, compare the Poincaré series with

∑_{m,n∈ℤ,(m,n)≠(0,0)} \frac{1}{|mz + n|^{2k}}

which converges uniformly on compact subsets of ℍ when 2k > 2. For the details of the proof, which is not difficult, see Gunning 1962, III.9.

Theorem 4.29. The Poincaré series φ_{n}(z), n ≥ 1, of weight 2k span M_{k}(Γ).

Before we can prove this, we shall need some preliminaries.

The geometry of ℍ. As Poincaré pointed out, ℍ can serve as a model for non-Euclidean hyperbolic plane geometry.

Recall that the axioms for hyperbolic geometry are the same as for Euclidean geometry, except that the axiom of parallels is replaced with the following axiom: suppose we are given a straight line and a point in the plane; if the line does not contain the point, then there exist at least two lines passing through the point and not intersecting the line.

The points of our non-Euclidean plane are the points of ℍ. A non-Euclidean “line” is a half-circle in ℍ orthogonal to the real axis, or a vertical half-line. The angle between two lines is the usual angle. To obtain the distance δ(z_{1}, z_{2}) between two points, draw the non-Euclidean line through z_{1} and z_{2}, let ∞_{1} and ∞_{2} be the points on the real axis (or i∞) on the “line” labeled in such a way that ∞_{1}, z_{1}, z_{2}, ∞_{2} follow one another cyclically around the circle, and define

δ(z_{1}, z_{2}) = log D(z_{1}, z_{2}, ∞_{1}, ∞_{2})

where D(z_{1}, z_{2}, z_{3}, z_{4}) is the cross-ratio.

The group PSL_{2}(ℝ) =^{df} SL_{2}(ℝ)/±I plays the same role as the group of orientation preserving affine transformations in the Euclidean plane, namely, it is the group of transformations preserving distance and orientation.

The measure μ(U) = ∫∫_{U} \frac{dxdy}{y^{2}} plays the same role as the usual measure dxdy on ℝ^{2}—it is invariant under translation by elements of PSL_{2}(ℝ). This follows from the invariance of the differential y^{−2}dxdy. (We prove something more general below.)

Thus we can consider ∫∫_{D} \frac{dxdy}{y^{2}} for any fundamental domain D of Γ—the invariance of the differential shows that this doesn’t depend on the choice of D. One shows that the integral does converge, and in fact that

∫_{D} dx · dy/y^{2} = 2π(2g − 2 + ν_{∞} + ∑(1 − 1/e_{P})).

See Shimura, 2.5. (There is a detailed discussion of the geometry of ℍ— equivalently, the open unit disk—in C. Siegel, Topics in Complex Functions II, Wiley, 1971, Chapter 3.)

Petersson inner product. Let f and g be two modular forms of weight 2k > 0 for a subgroup Γ of finite index in Γ(1).

Lemma 4.30. The differential f(z)·\overline{g(z)}·y^{2k−2}dxdy is invariant under the action of SL_{2}(ℝ)^{+}. (Here z = x + iy, so the notation is mixed.)

Proof. Let γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ). Then

f(γz) = (cz + d)^{2k} · f(z) (definition of a modular form)
\overline{g(γz)} = \overline{(cz + d)}^{2k} · \overline{g(z)} (the conjugate of the definition)
ℑ(γz) = \frac{ℑ(z)}{|cz + d|^{2}} (see the Introduction)
γ^{∗}(dx · dy) = \frac{dxdy}{|cz + d|^{4}} .

The last equation follows from the next lemma and the fact (4.8) that dγ/dz = 1/(cz + d)^{2}. On raising the third equation to the (2k − 2)th power, and multiplying, we obtain the result.

Lemma 4.31. For any holomorphic function w(z), the map z ↦ w(z) multiplies areas by |\frac{dw}{dz}|^{2}.

Proof. Write w(z) = u(x, y) + iv(x, y), z = x + iy. Thus, z ↦ w(z) is the map

(x, y) ↦ (u(x, y), v(x, y)),

and the Jacobian is

| \begin{matrix}u_{x} & v_{x} \\ u_{y} & v_{y} \end{matrix} | = u_{x}v_{y} − v_{x}u_{y}.

According to the Cauchy-Riemann equations, w′(z) = u_{x} + iv_{x}, u_{x} = v_{y}, u_{y} = −v_{x}, and so

|w′(z)|^{2} = u_{x}^{2} + v_{x}^{2} = u_{x}v_{y} − v_{x}u_{y}.

Lemma 4.32. Let D be a fundamental domain for Γ. If f or g is a cusp form, then the integral

∫∫ _{D} f(z) · \overline{g(z)} · y^{2k−2}dxdy

converges.

Proof. Clearly the integral converges if we exclude a neighbourhood of each of the cusps. Near the cusp i∞, f(z)·\overline{g(z)} = O(e^{−cy}) for some c > 0, and so the integral is dominated by ∫_{y_{1}}^{∞} e^{−cy}y^{k−2}dy < ∞. The other cusps can be handled similarly.

Let f and g be modular forms of weight 2k for some group Γ ⊂ Γ(1), and assume that one at least is a cusp form. The Petersson inner product of f and g is defined to be

<f, g>= ∫∫ _{D} f(z) · \overline{g(z)} · y^{2k−2}dxdy.

Lemma 4.30 shows that it is independent of the choice of D. It has the following properties:

• it is linear in the first variable, and semi-linear in the second;

• <f, g>= \overline{<g, f>};

• < f, f >> 0 for all f ≠ 0.

It is therefore a positive-definite Hermitian form on S_{k}(Γ), and so S_{k}(Γ) together with <, > is a finite-dimensional Hilbert space.

Completeness of the Poincaré series. Again let Γ be a subgroup of finite index in Γ(1).

Theorem 4.33. Let f be a cusp form of weight 2k ≥ 2 for Γ, and let φ_{n} be the Poincaré series of weight 2k and character n ≥ 1 for Γ. Then

<f, φ_{n}>= \frac{h^{2k}(2k − 2)!}{(4π)^{2k−1}} · n^{1−2k} · a_{n}

where h is the width of i∞ as a cusp for Γ and a_{n} is the n^{th} coefficient in the Fourier expansion of f :

f = ∑ a_{n}e^{\frac{2πinz}{h}} .

Proof. Write φ_{n} as a sum, and interchange the order of the integral and the sum. Look at a typical term. Write it as an integral over a fundamental domain for Γ_{0} in ℍ,

< f, φ_{n} >= ∫_{x=0}^{h} ∫_{y=0}^{∞} f(z) · exp(−2πinz/h) · y^{2(2k−1)} · dxdy.

Now write f(z) as a sum, and interchange the order of integration and summation. Evaluate. See Gunning, III.11, for the details.

Corollary 4.34. Every cusp form is a linear combination of Poincaré series φ_{n}(z), n ≥ 1.

Proof. If f is orthogonal to the subspace generated by the Poincaré series, then all the coefficients of its Fourier expansion are zero.

Eisenstein series for Γ(N). The Poincaré series of weight 2k > 2 and character 0 for Γ(N) is

ϕ_{0}(z) = ∑ \frac{1}{(cz + d)^{2k}} (sum over (c, d) ≡ (0, 1) mod N, gcd(c, d) = 1).

Recall (4.28) that this is a modular form of weight 2k for Γ(N) which takes the value 1 at i∞ and vanishes at all the other cusps.

For every complex-valued function ν on the (finite) set of inequivalent cusps for Γ(N), we want to construct a modular function f of weight 2k such that f|{cusps} = ν. Moreover, we would like to choose the f’s to be orthogonal (for the Petersson inner product) to the space of cusp forms. To do this, we shall construct a function (restricted Eisenstein series) which takes the value 1 at a particular cusp, takes the value 0 at the remaining cusps, and is orthogonal to cusp forms.

Write j_{γ}(z) = 1/(cz + d)^{2}, so that j_{γ}(z) is an automorphy factor:

j_{γγ′}(z) = j_{γ}(γ′z) · j_{γ′}(z).

Let P be a cusp for Γ(N), P ≠ i∞, and let σ ∈ Γ(1) be such that σ(P) = i∞. Define

φ(z) = j_{σ}(z)^{k} · φ_{0}(σz).

Lemma 4.35. The function φ(z) is a modular form of weight 2k for Γ(N); moreover φ takes the value 1 at P, and it is zero at every other cusp.

Proof. Let γ ∈ Γ(N). For the first statement, we have to show that φ(γz) = j_{γ}(z)^{−k}φ(z). From the definition of φ, we find that

φ(γz) = j_{σ}(γz)^{k} · φ_{0}(σγz).

As Γ(N) is normal, σγσ^{−1} ∈ Γ(N), and so

φ_{0}(σγz) = φ_{0}(σγσ^{−1} · σz) = j_{σγσ^{−1}}(σz)^{−k} · φ_{0}(σz).

On comparing this formula for φ(γz) with

j_{γ}(z)^{−k} · φ(z) = j_{γ}(z)^{−k} · j_{σ}(z)^{k} · φ_{0}(σz),

we see that it suffices to prove that

j_{σ}(γz) · j_{σγσ^{−1}}(σz)^{−1} = j_{γ}(z)^{−1} · j_{σ}(z),

or that

j_{σ}(γz) · j_{γ}(z) = j_{σγσ^{−1}}(σz) · j_{σ}(z).

But, because of the defining property of automorphy factors, this is just the obvious equality

j_{σγ}(z) = j_{σγσ^{−1}σ}(z).

The second statement is a consequence of the definition of φ and the properties of φ_{0}.

We now compute φ(z). Let T be a set of coset representatives for Γ_{0} in Γ(N). Then

φ(z) =^{df} j_{σ}(z)^{k} · φ_{0}(σz)
= j_{σ}(z)^{k} · ∑_{τ∈T} j_{τ}(σz)^{k}
= ∑_{τ∈T} j_{τσ}(z)^{k}
= ∑_{γ∈T σ} j_{γ}(z)^{k}.

Let σ = ( \begin{matrix}a_{0} & b_{0} \\ c_{0} & d_{0} \end{matrix} ), so that σ^{−1} = ( \begin{matrix}d_{0} & −b_{0} \\ −c_{0} & a_{0} \end{matrix} ), and P = σ^{−1} ( \begin{matrix}1 \\ 0 \end{matrix} ) = −d_{0}/c_{0}.

Note that

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ(N) ⟹ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}a_{0} & b_{0} \\ c_{0} & d_{0} \end{matrix} ) ≡ ( \begin{matrix}∗ & ∗ \\ c_{0} & d_{0} \end{matrix} ) mod N.

From this, we can deduce that T σ contains exactly one element of Γ(N)′ for each pair (c, d) with gcd(c, d) = 1 and (c, d) ≡ (c_{0}, d_{0}).

Definition 4.36. (a) A restricted Eisenstein series of weight 2k > 2 for Γ(N) is a series

G(z; c_{0}, d_{0}; N) = ∑(cz + d)^{−2k}

(sum over (c, d) ≡ (c_{0}, d_{0}) mod N, gcd(c, d) = 1). Here (c_{0}, d_{0}) is a pair such that gcd(c_{0}, d_{0}, N) = 1.

(b) A general Eisenstein series of weight 2k > 2 for Γ(N) is a series

G(z; c_{0}, d_{0}; N) = ∑(cz + d)^{−2k}

(sum over (c, d) ≡ (c_{0}, d_{0}) mod N, (c, d) ≠ (0, 0)). Here it is not required that gcd(c_{0}, d_{0}, N) = 1.

Consider the restricted Eisenstein series. Clearly,

G(z; c_{0}, d_{0}; N) = G(z; c_{1}, d_{1}; N)

if (c_{0}, d_{0}) ≡ ±(c_{1}, d_{1}) mod N. On the other hand, we get a restricted Eisenstein series for each cusp, and these Eisenstein series are linearly independent. On counting, we see that there is exactly one restricted Eisenstein series for each cusp, and so the distinct restricted Eisenstein series are linearly independent.

Proposition 4.37. The general Eisenstein series are the linear combinations of the restricted Eisenstein series.

Proof. Omitted.

Remark 4.38. (a) Sometimes Eisenstein series are defined to be the linear

combinations of restricted Eisenstein series.

(b) The Petersson inner product <f, g> is defined provided at least one of f or g is a cusp form. One finds that <f, g>= 0 (e.g., φ_{0} gives the 0^{th} coefficient) for the restricted Eisenstein series, and hence <f, g >= 0 for all cusp forms f and all Eisenstein series g : the space of Eisenstein series is the orthogonal complement of S_{k}(Γ) in M_{k}(Γ).

For more details on Eisenstein series for Γ(N), see Gunning 1962, IV.13.

Aside 4.39. In the one-dimensional case, compactifying Γ\ℍ presents no problem, and the Riemann-Roch theorem tells us there are many modular forms. The Poincaré series allow us to write down a set of modular forms that spans S_{k}(Γ). In the higher dimensional case (see 2.26), it is much more difficult to embed the quotient Γ\D of a bounded symmetric domain in a compact analytic space. Here the Poincaré series play a much more crucial role. In their famous 1964 paper, Baily and Borel showed that the Poincaré series can be used to give an embedding of the complex manifold Γ\D into projective space, and that the closure of the image is a projective algebraic variety. It follows that Γ\D has a canonical structure of an algebraic variety.

In the higher-dimensional case, the boundary of Γ\D, i.e., the complement of Γ\D in its compactification, is more complicated than in the one-dimensional case. It is a union of varieties of the form Γ′\D′ with D′ a bounded symmetric domain of lower dimension than that of D. The Eisenstein series then attaches to a cusp form on D′

MODULAR FUNCTIONS AND MODULAR FORMS 61

a modular form on D. (In our case, a cusp form on the zero-dimensional boundary is just a complex number.)

5. Hecke Operators

Hecke operators play a fundamental role in the theory of modular forms. After describing the problem they were first introduced to solve, we develop the theory of Hecke operators for the full modular group, and then for a congruence subgroup of the modular group.

Introduction. Recall that the cusp forms of weight 12 for Γ(1) form a onedimensional vector space over ℂ, generated by Δ = g_{2}^{3} − 27g_{3}^{2}, where g_{2} = 60G_{2} and g_{3} = 140G_{3}. In more geometric terms, Δ(z) is the discriminant of the elliptic curve ℂ/ℤz + ℤ. Jacobi showed that

Δ(z) = (2π)^{12} · q · ∏_{n=1}^{∞}(1 − q^{n})^{24}, q = e^{2πiz}.

Write f(z) = q · ∏_{n=1}^{∞}(1 − q)^{24} = ∑ τ(n) · q^{n}. Then n ↦ τ(n) is the Ramanujan τ-function. Ramanujan conjectured that it had the following properties:

(a) |τ(p)| ≤ 2 · p^{11/2},

(b) \{ \begin{matrix}τ(mn) & = & τ(m)τ(n) if gcd(m, n) = 1 \\ τ(p)τ(p^{n}) & = & τ(p^{n+1}) + p^{11}τ(p^{n−1}) if p is prime and n ≥ 1. \end{matrix}

Property (b) was proved by Mordell in 1917 in a paper in which he introduced the first examples of Hecke operators. To Δ we can attach a Dirichlet series

L(Δ, s) = ∑ τ(n)n^{−s}.

Proposition 5.1. The Dirichlet series L(Δ, s) has an Euler product expansion of the form

L(Δ, s) = ∏_{p prime} \frac{1}{(1 − τ(p)p^{−s} + p^{11−2s})}

if and only if (b) holds.

Proof. For a prime p, define

L_{p}(s) = ∑_{m≥0} τ(p^{m}) · p^{−ms} = 1 + τ(p) · p^{−s} + τ(p^{2}) · (p^{−s})^{2} + · · · .

If n ∈ ℕ has the factorization n = ∏ p_{i}^{r_{i}}, then the coefficient of (p^{−s})^{n} in ∏ L_{p}(s) is ∏ τ(p_{i}^{r_{i}}), which the first equation in (b) implies is equal to τ(n). Thus

L(Δ, s) = ∏ L_{p}(s).

Now consider

(1 − τ(p)p^{−s} + p^{11−2s}) · L_{p}.

By inspection, we find that the coefficient of (p^{−s})^{n} in this product is 1 for n = 0; 0 for n = 1; · · · · · · · · ·· τ(p^{n+1}) − τ(p)τ(p^{n}) + p^{11}τ(p^{n−1}) for n + 1.

Thus the second equation in (b) implies that (1 − τ(p)p^{−s} + p^{11−2s}) · L_{p} = 1, and hence that

L(Δ, s) = ∏_{p}(1 − τ(p)p^{−s} + p^{11−2s})^{−1}.

The argument can be run in reverse.

Proposition 5.2. Write

1 − τ(p)X + p^{11}X^{2} = (1 − aX)(1 − a′X);

Then the following conditions are equivalent:

(a) |τ(p)| ≤ 2 · p^{11/2};

(b) |a| = p^{11/2} = |a′|;

(c) a and a′ are conjugate complex numbers, i.e., a′ = ā.

Proof. First note that τ(p) is real (in fact, it is an integer).

(b) ⟹ (a): We have τ(p) = a+a′, and so (a) follows from the triangle inequality.

(c) ⟹ (b): We have that |a|^{2} = aā = aa′ = p^{11}.

(a) ⟹ (c): The discriminant of 1 − τ(p)X + p^{11}X^{2} is τ(p)^{2} − 4p^{11}, which (a) implies is < 0.

For each n ≥ 1, we shall define an operator:

T (n) : M_{k}(Γ(1)) → M_{k}(Γ(1)).

These operators will have the following properties:

T (m) ∘ T (n) = T (mn) if gcd(m, n) = 1;

T (p) ∘ T (p^{n}) = T (p^{n+1}) + p^{2k−1}T (p^{n−1}), p prime;

T (n) preserves the space of cusp forms, and is a Hermitian (self-adjoint) operator on S_{k}(Γ) :

<T (n)f, g>=<f, T (n)g>, f, g cusp forms.

Lemma 5.3. Let V be a finite-dimensional vector space over ℂ with a positive definite Hermitian form <, > .

(a) Let α : V → V be a linear map which is Hermitian (i.e., such that <αv, v′>=< v, αv′>); then V has a basis consisting of eigenvectors for α (thus α is diagonalizable).

(b) Let α_{1}, α_{2}, ... be a sequence of commuting Hermitian operators; then V has a basis consisting of vectors that are eigenvectors for all α_{i} (thus the α_{i} are simultaneously diagonalizable).

Proof. (a) Because ℂ is algebraically closed, α has an eigenvector e_{1}. Let V_{1} = (ℂ · e_{1})^{⊥}. Because α is Hermitian, V_{1} is stable under α, and so it has an eigenvector e_{2}. Let V_{2} = (ℂe_{1} + ℂe_{2})^{⊥}, and continue in this manner.

(b) From (a) we know that V = ⊕V (λ_{i}) where the λ_{i} are the distinct eigenvalues for α_{1} and V (λ_{i}) is the eigenspace for λ_{i}; thus α_{1} acts as multiplication by λ_{i} on V (λ_{i}). Because α_{2} commutes with α_{1} , it preserves each V (λ_{i}), and we can decompose each V (λ_{i}) further into a sum of eigenspaces for α_{2}. Continuing in this fashion, we arrive at a decomposition V = ⊕V_{j} such that each α_{i} acts as a scalar on each V_{j}. Now choose a basis for each V_{j} and take the union.

Remark 5.4. The pair (V, <, >) is a finite-dimensional Hilbert space. There is an analogous statement to the lemma for infinite-dimensional Hilbert spaces (it’s called the spectral theorem).

Proposition 5.5. Let f(z) = ∑ c(n)q^{n} be a modular form of weight 2k, k > 0, f ≠ 0. If f is an eigenfunction for all T (n), then c(1) ≠ 0, and when we normalize f so that c(1) = 1, then

T (n)f = c(n) · f.

Proof. See later (5.18).

Corollary 5.6. If f(z) is a normalized eigenform for all T (n), then c(n) is real.

Proof. The eigenvalues of a Hermitian operator are real, because

<αv, v>=<λv, v>= λ <v, v>, =<v, αv>=<v, λv>= λ̄ <v, v>

for any eigenvector v.

We deduce from these statements that if f is a normalized eigenform for all the T (n), then

c(m)c(n) = c(mn) if gcd(m, n) = 1;

c(p)c(p^{n}) = c(p^{n+1}) + p^{2k−1}c(p^{n−1}) if p is prime n ≥ 1.

Just as in the case of Δ, this implies that

L(f, s) =^{df} ∑ c(n) · n^{−s} = ∏_{p prime} \frac{1}{(1 − c(p)p^{−s} + p^{2k−1−2s})}.

Write 1 − c(p)X + p^{2k−1−2s} = (1 − aX)(1 − a′X). As before, the following statements are equivalent:

|c(p)| ≤ 2 · p^{\frac{k−1}{2}} ;

|a| = p^{\frac{k−1}{2}} = |a′|;

a and a′ are complex conjugates.

These statements are also referred to as the Ramanujan conjecture. As we mentioned in Section 3, they have been proved by Deligne.

Example 5.7. Because the space of cusp forms of weight 12 is one-dimensional, Δ is a simultaneous eigenform for the Hecke operators, and so Ramanujan’s Conjecture (b) for τ(n) does follow from the existence of Hecke operators with the above properties.

Note the similarity of L(f, s) to the L-function of an elliptic curve E/ℚ, which is defined to be

L(E, s) = ∏_{p good} \frac{1}{1 − a(p)p^{−s} + p^{1−2s}} .

Here 1 − a(p) + p = #E(𝔽_{p}). The Riemann hypothesis for E/𝔽_{p} is that |a(p)| ≤ 2\sqrt{p}. The number a(p) can also be realized as the trace of the Frobenius map on V_{ℓ}E. Since τ(p) is the trace of T (p) acting on an eigenspace, this suggests that there should be a relation of the form

“T (p) = Π_{p} + Π̄_{p}”

where Π_{p} is the Frobenius operator at p. We shall see that there do exist relations of this form, and that this is the key to Deligne’s proof that the Weil conjectures imply the (generalized) Ramanujan conjecture.

Conjecture 5.8 (Taniyama). Let E be an elliptic curve over ℚ. Then L(E, s) = L(f, s) for some normalized eigenform of weight 2 for Γ_{0}(N), where N is the conductor of E.

This conjecture is very important. A vague statement of this form was suggested by Taniyama in the 50’s, was promoted by Shimura in the 60’s, and then in 1967 Weil provided some rather compelling evidence for it. We shall discuss Weil’s work in Section 6. Since it is possible to list the normalized eigenforms of weight 2 for Γ_{0}(N) for a fixed N, the conjecture predicts how many elliptic curves with conductor N there are over ℚ. Computer searches have confirmed the number for small N. [The conjecture has been proved for most elliptic curves by Wiles, Taylor, and Diamond.]

It is known that Conjecture 5.8 implies Fermat’s last theorem (which for me, is the most compelling evidence for Fermat’s last theorem).

The conjecture is now subsumed by the Langlands program which (roughly speaking) predicts that all Dirichlet series arising from algebraic varieties (more generally, motives) occur among those arising from automorphic forms (better, automorphic representations) for reductive algebraic groups.

Abstract Hecke operators. Let L be the set of full lattices in ℂ. Recall (4.6) that modular forms are related to functions on L. We first define operators on L, which define operators on functions on L, and then operators on modular forms.

Let D be the free abelian group generated by the elements of L; thus an element of D is a finite sum

∑ n_{i}[Λ_{i}], n_{i} ∈ ℤ, Λ_{i} ∈ L.

For n = 1, 2, ... we define a ℤ-linear operator T (n) : D → D by setting

T (n)[Λ] = ∑[Λ′] (sum over all sublattices Λ′of Λ of index n).

The sum is obviously finite because any such sublattice Λ′ contains nΛ, and Λ/nΛ is finite. Write R(n) for the operator

R(n)[Λ] = [nΛ].

Proposition 5.9. (a) If m and n are relatively prime, then

T (m) ∘ T (n) = T (mn).

(b) If p is prime and n ≥ 1, then

T (p^{n}) ∘ T (p) = T (p^{n+1}) + pR(p) ∘ T (p^{n−1}).

Proof. (a) Note that

T (mn)[Λ] = ∑[Λ′′] (sum over Λ′′ with (Λ : Λ′′) = mn);

T (m)∘T (n)[Λ] = ∑[Λ′′] (sum over pairs (Λ′, Λ′′) with (Λ : Λ′) = n, (Λ′ : Λ′′) = m). But, if Λ′′ is a sublattice of Λ of index mn, then there is a unique chain

Λ ⊃ Λ′ ⊃ Λ

with Λ′ of index n in Λ, because Λ/mnΛ is the direct sum of a group of order m and a group of order n.

(b) Let Λ be a lattice. Note that

T (p^{n})∘T (p)[Λ] = ∑[Λ′′] (sum over pairs (Λ′, Λ′′) with (Λ : Λ′) = p, (Λ′ : Λ′′) = p^{n}); T (p^{n+1})[Λ] = ∑[Λ′′] (sum over Λ with (Λ : Λ′′) = p^{n+1}); pR(p) ∘ T (p^{n−1})[Λ] = p · ∑ R(p)[Λ′] (sum over Λ′ ⊂ Λ with (Λ : Λ′) = p^{n−1}).

Hence pR(p) ∘ T (p^{n−1})[Λ] = p · ∑[Λ′′] (sum over Λ′′ ⊂ pΛ with (pΛ : Λ′′) = p^{n−1}).

Each of these is a sum of sublattices Λ′′ of index p^{n+1} in Λ. Fix such a lattice, and let a be the number of times it occurs in the first sum, and b the number of times it occurs in the last sum. It occurs exactly once in the second sum, and so we have to prove:

a = 1 + pb.

There are two cases to consider.

The lattice Λ′′ is not contained in pΛ. Then b = 0, and a is the number of lattices Λ′ containing Λ′′ and of index p in Λ. Such a lattice contains pΛ, and its image in Λ/pΛ is of order p and contains the image of Λ′′, which is also of order p. Since the subgroups of Λ of index p are in one-to-one correspondence with the subgroups of Λ/pΛ of index p, this shows that there is exactly one lattice Λ′, namely Λ + pΛ′′, and so a = 1.

The lattice Λ′′ ⊂ pΛ. Here b = 1. Any lattice Λ′ of index p contains pΛ, and a fortiori Λ. We have to count the number of subgroups of Λ/pΛ of index p, and this is the number of lines through the origin in the 𝔽_{p}-plane, which is (p^{2} − 1)/(p − 1) = p + 1.

Corollary 5.10. For any m and n,

T (m) · T (n) = ∑_{d| gcd(m,n),d>0} d · R(d) ∘ T (mn/d^{2})

Proof. Prove by induction on s that

T (p^{r}) ∘ T (p^{s}) = ∑_{i≤r,s} p^{i} · R(p^{i}) ∘ T (p^{r+s−2i}),

and then apply (a) of the theorem.

Corollary 5.11. Let H be the ℤ-subalgebra of End(D) generated by the T (p) and R(p) for p prime; then H is commutative, and it contains T (n) for all n.

Proof. Obvious from 5.10.

Let F be a function L → ℂ. We can extend F by linearity to a function F : D → ℂ,

F (∑ n_{i}[Λ_{i}]) = ∑ n_{i}F (Λ_{i}).

For any operator T on D, we define T · F to be the function L → ℂ such that

(T · F)([Λ]) = F (T [Λ]).

For example,

(T (n) · F)([Λ]) = ∑ F ([Λ′]) (sum over sublattices Λ′ of Λ of index n)

and if F has weight 2k, so that F (λΛ) = λ^{−2k}F (Λ), then

R(n) · F = n^{−2k} · F.

Proposition 5.12. Let F : L → ℂ be a homogeneous function of weight 2k. Then T (n) · F is again of weight 2k, and for any m and n,

T (m) · T (n) · F = ∑_{d| gcd(m,n), d>0} d^{1−2k} · T (mn/d^{2}) · F.

In particular, if m and n are relatively prime, then

T (m) · T (n) · F = T (mn) · F,

and if p is prime and n ≥ 1, then

T (p) · T (p^{n}) · F = T (p^{n+1}) · F + p^{1−2k} · T (p^{n−1}) · F.

Proof. Immediate from Corollary 5.10 and the definitions.

Lemmas on 2 × 2 matrices. Before defining the action of Hecke operators on modular forms, we review some elementary results concerning 2 × 2 matrices with integer coefficients. Write M_{2}(ℤ) for the ring of 2 × 2 matrices with coefficients in ℤ.

Lemma 5.13. Let A be a 2 × 2 matrix with coefficients in ℤ and determinant n.

Then there is an invertible matrix U in M_{2}(ℤ) such that U · A = ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) with

ad = n, a ≥ 1, 0 ≤ b < d. (∗)

Moreover, the integers a, b, d are uniquely determined.

Proof. Apply row operations to A that are invertible in the ring M_{2}(ℤ) to get A into upper triangular form—see Math 676, 2.43 for the details. For the uniqueness, note that a is the gcd of the elements in the first column of A, d is the unique positive element such that ad = n, and b is obviously uniquely determined modulo d.

Remark 5.14. Let M(n) be the set of 2 × 2 matrices with coefficients in ℤ and determinant n. The group SL_{2}(ℤ) acts on M(n) by left multiplication, and the lemma provides us with a canonical set of representatives for the orbits:

M(n) = ⋃ SL_{2}(ℤ) · ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) (disjoint union over a, b, d as in the lemma).

Now let Λ be a lattice in ℂ. Choose a basis ω_{1}, ω_{2} for Λ, so that Λ = Λ(ω_{1}, ω_{2}).

For any α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M(n), define αΛ = Λ(aω_{1} + bω_{2}, cω_{1} + dω_{2}). Then αΛ is a

sublattice of Λ of index n, and every such lattice is of this form for some α ∈ M(n). Clearly αΛ = βΛ if and only if β = uα for u ∈ SL_{2}(ℤ). Thus we see that the sublattices of Λ of index n are precisely the lattices

Λ(aω_{1} + bω_{2}, dω_{2}), a, b, d ∈ ℤ, ad = n, a ≥ 1, 0 ≤ b < d − 1.

For example, consider the case n = p. Then the sublattices of Λ are in one-to-one correspondence with the lines through the origin in the 2-dimensional 𝔽_{p}-vector space Λ/pΛ. Write Λ/pΛ = 𝔽_{p}e_{1} ⊕ 𝔽_{p}e_{2} with e_{i} = ω_{i} (mod p) . The lines through the origin are determined by their intersections (if any) with the vertical line through (1, 0). Therefore there are p + 1 lines through the origin, namely,

𝔽_{p} · e_{1}, 𝔽_{p} · (e_{1} + e_{2}), 𝔽_{p} · (e_{1} + (p − 1)e_{2}), 𝔽_{p}(e_{2}).

Hence there are exactly p + 1 sublattices of Λ = ℤω_{1} + ℤω_{2} of index p, namely,

Λ(ω_{1}, pω_{2}), Λ(ω_{1} + ω_{2}, pω_{2}), …, Λ(pω_{1}, ω_{2}),

in agreement with the general result.

Remark 5.15. Let α ∈ M(n), and let Λ′ = αΛ. According to a standard theorem, we can choose bases ω_{1}, ω_{2} for Λ and ω′_{1}, ω′_{2} for Λ′ such that

ω′_{1} = aω_{1}, ω′_{2} = dω_{2}, a, d ∈ ℤ, ad = n, a|d, a ≥ 1

and a, d are uniquely determined. In terms of matrices, this says that

M(n) = ⋃ SL_{2}(ℤ) · ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) · SL_{2}(ℤ)

—disjoint union over a, d ∈ ℤ, ad = n, a|d, a ≥ 1. This decomposition of M(n) into a union of double cosets can also be proved directly by applying both row and column

operations, invertible in M_{2}(ℤ), to the matrix ( \begin{matrix}a & b \\ c & d \end{matrix} ).

Hecke operators for Γ(1). Recall 4.6 that we have a one-to-one correspondence between functions F on L of weight 2k and functions f on ℍ that are weakly modular of weight 2k, under which

F (Λ(ω_{1}, ω_{2})) = ω_{2}^{−2k} · f(ω_{1}/ω_{2}); f(z) = F (Λ(z, 1)).

Let f(z) be a modular form of weight 2k, and let F be the associated function of weight 2k on L. We define T (n) · f(z) to be the function on ℍ associated with n^{2k−1} · T (n) · F. The factor n^{2k−1} is inserted so that some formulas have integer coefficients rather than rational coefficients. Thus

T (n) · f(z) = n^{2k−1} · (T (n) · F)(Λ(z, 1)).

More explicitly,

T (n) · f(z) = n^{2k−1} · ∑ d^{−2k} f(\frac{az + b}{d})

where the sum is over the triples a, b, d satisfying (5.13(*)).

Proposition 5.16. (a) If f is a weakly modular form of weight 2k for Γ(1),

then T (n) · f is also weakly modular of weight 2k, and T (m) · T (n) · f = T (mn) · f if m and n are relatively prime; T (p) · T (p^{n}) · f = T (p^{n+1}) · f + p^{2k−1} · T (p^{n−1}) · f if p is prime and n ≥ 1.

(b) Let f be a modular form of weight 2k for Γ(1), with the Fourier expansion f = ∑_{m≥0} c(m)q^{m}, q = e^{2πiz}. Then T (n) · f is also a modular form, and

T (n) · f(z) = ∑_{m≥0} γ(m) · q^{m}

with

γ(m) = ∑_{a| gcd(m,n),a≥1} a^{2k−1} · c(\frac{mn}{a^{2}}).

Proof. (a) We know that

T (p) · T (p^{n}) · F (Λ(z, 1)) = T (p^{n+1}) · F (Λ(z, 1)) + p^{1−2k} · T (p^{n−1}) · F (Λ(z, 1)).

On multiplying through by (p^{n+1})^{2k−1} we obtain the second equation. The first is obvious.

(b) We know that

T (n) · f(z) = n^{2k−1} ∑_{a,b,d} d^{−2k} f(\frac{az + b}{d})

where the sum over a, b, d satisfying 5.13(*), i.e.,

ad = n, a ≥ 1, 0 ≤ b < d.

Therefore T (n) · f(z) is holomorphic on ℍ because f is. Moreover

T (n) · f(z) = n^{2k−1} ∑_{a,b,d} d^{−2k} ∑_{m≥0} c(m)q^{2πi \frac{az+b}{d} m}.

But

∑_{0≤b<d} e^{2πi \frac{bm}{d}} = \{ \begin{matrix}d & if d|m \\ 0 & otherwise. \end{matrix}

Set m/d = m′; then

T (n) · f(z) = n^{2k−1} ∑_{a,d,m′} d^{−2k+1} c(m′d)q^{am′}

where the sum is over the integers a, d, m′ such that ad = n and a ≥ 1. The coefficient of q^{t} in this is

∑_{a| gcd(n,t),a≥1} a^{2k−1} · c(\frac{t}{a} \frac{n}{a}).

When we substitute m for t in this formula, we obtain the required formula. Because γ(m) = 0 for m < 0, T (n) · f is holomorphic at i∞.

Corollary 5.17. Retain the notations of the proposition.

(a) The coefficients γ(0) = σ_{2k−1}(n) · c(0), γ(1) = c(m).

(b) If n = p is prime, then γ(m) = c(pm) if p does not divide m; γ(m) = c(pm) + p^{2k−1}c(m/p) if p|m.

(c) If f is a cusp form, then so also is T (n) · f.

Proof. These are all obvious consequences of the proposition.

Thus the T (n)’s act on the vector spaces M_{k}(Γ(1)) and S_{k}(Γ(1)), and satisfy the identities

T (m) ∘ T (n) = T (mn) if m and n relatively prime; T (p) ∘ T (p^{n}) = T (p^{n+1}) + p^{2k−1} · T (p^{n−1}) if p is prime and n ≥ 1.

Proposition 5.18. Let f = ∑ c(n)q^{n} be a nonzero modular form of weight 2k. Assume f is a simultaneous eigenform for all the T (n), say,

T (n) · f = λ(n) · f, λ(n) ∈ ℂ.

Then c(1) ≠ 0, and if f is normalized so that c(1) = 1, then

c(n) = λ(n)

for all n ≥ 1.

Proof. We have seen that the coefficient of q in T (n) · f is c(n). But, it is also λ(n) · c(1), and so c(n) = λ(n) · c(1). If c(1) were zero, then all c(n) would be zero, and f would be constant, which is impossible.

Corollary 5.19. Two normalized eigenforms of the same weight with the same eigenvalues are equal.

Proof. The proposition implies that the coefficients of their Fourier expansions are equal.

Corollary 5.20. If f = ∑ c(n)q^{n} is a normalized eigenform for the T (n), then c(m) · c(n) = c(mn) if m and n are relatively prime, c(p) · c(p^{n}) = c(p^{n+1}) + p^{2k−1}c(p^{n−1}) if p is prime and n ≥ 1.

Proof. We know that these relations hold for the eigenvalues.

With a modular form f, we can associate a Dirichlet series

L(f, s) = ∑_{n≥1} c(n) · n^{−s}.

The series ∑ n^{−s} converges for ℜ(s) > 1. The bounds on the values |c(n)| (see Section 4) show that L(f, s) converges to the right of some vertical line (if one accepts Deligne’s theorem and f is a cusp form of weight 2k, it converges for ℜ(s−k + \frac{1}{2}) > 1, i.e., for s > k + \frac{1}{2}).

Proposition 5.21. For any normalized eigenform f,

L(f, s) = ∏_{p} \frac{1}{1 − c(p)p^{−s} + p^{2k−1−2s}} .

Proof. This follows from 5.20, as in the proof of (5.1).

MODULAR FUNCTIONS AND MODULAR FORMS 71

The Hecke operators for Γ(1) are Hermitian. Before proving this, we make a small excursion.

Write GL_{2}(ℝ)^{+} for the group of real 2×2 matrices with positive determinant. Let

α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℝ)^{+},

and let f be a function on ℍ; we define

f|_{k}α = (det α)^{k} · (cz + d)^{−2k} · f(\frac{az + b}{cz + d}).

For example, if α = ( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ), then f|_{k}α = a^{2k} · a^{−2k} · f(z) = f(z); i.e., the centre of GL_{2}(ℝ)^{+} acts trivially. Note that f is weakly modular of weight 2k for Γ ⊂ Γ(1) if and only if f|_{k}α = f for all α ∈ Γ.

Recall that

T (n) · f(z) = n^{2k−1} · ∑ d^{−2k} · f(\frac{az + b}{d})

—sum over a, b, d, ad = n, a ≥ 1, 0 ≤ b < d. We can restate this as T (n) · f = ∑ n^{k−1} · f|_{k}α

where the α’s run through a particular set of representatives for the orbits Γ(1)\M(n). It is clear from the above remarks, that the right hand side is independent of the choice of the set of representatives.

Recall, that the Petersson inner product of two cusp forms f and g for Γ(1) is

<f, g>= ∫∫ _{D} f · ḡ · y^{2k−2} · dxdy

where z = x + iy and D is any fundamental domain for Γ(1).

Lemma 5.22. For any α ∈ GL_{2}(ℝ)^{+}, <f|_{k}α, g|_{k}α>=<f, g> .

Proof. Write ω(f, g) = f(z)ḡ(z)y^{k−2}dxdy, where z = x + iy. I claim that

ω(f|_{k}α, g|_{k}α) = α^{∗}ω(f, g),

and so

∫∫ _{D} ω(f|_{k}α, g|_{k}α) = ∫∫ _{D} α^{∗}ω(f, g) = ∫∫ _{αD} ω(f, g),

which equals <f, g> because αD is also a fundamental domain for Γ(1).

Since multiplying α by a scalar changes neither ω(f|_{k}α, g|_{k}α) nor α^{∗}ω(f, g), we can assume (in proving the claim) that det α = 1. Then

f|_{k}α = (cz + d)^{−2k} · f(αz)

ḡ|_{k}α = (cz̄ + d)^{−2k} · \overline{g(αz)}

and so

ω(f|_{k}α, g|_{k}α) = |cz + d|^{−4k} · f(αz) · \overline{g(αz)} · dx · dy.

On the other hand (see the proof of 4.30)

72 J. S. MILNE

ℑ(αz) = ℑ(z)/|cz + d|^{2} α^{∗}(dx · dy) = dx · dy/|cz + d|^{4}, and so α^{∗}(ω(f, g)) = f(αz)·\overline{g(αz)}·|cz+d|^{4−4k}·y^{2k−2}·|cz+d|^{−4}·dx·dy = ω(f|_{k}α, g|_{k}α).

Note that the lemma implies that <f|_{k}α, g>=<f, g|_{k}α^{−1}>, all α ∈ GL_{2}(ℝ)^{+}.

Theorem 5.23. For cusp forms f, g of weight 2k <T (n)f, g>=<f, T (n)g>, all n.

Because of (5.10), it suffices to prove the theorem for T (p), p prime. Recall that M(n) is the set of integer matrices with determinant n.

Lemma 5.24. There exists a common set of representatives {α_{i}} for the set of left orbits Γ(1)\M(p) and for the set of right orbits M(p)/(1).

Proof. Let α, β ∈ M(p); then (see 5.15)

Γ(1) · α · Γ(1) = Γ(1) · ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) · Γ(1) = Γ(1) · β · Γ(1).

Hence there exist elements u, v, u′, v′ ∈ Γ(1) such that

uαv = u′βv′

and so u′^{−1}uα = βv′v^{−1}, = γ say. Then Γ(1) · α = Γ(1) · γ and β · Γ(1) = γ · Γ(1).

For α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M(p), set α′ = ( \begin{matrix}d & −b \\ −c & a \end{matrix} ) = p · α^{−1} ∈ M(p). Let α_{i} be a

set of common representatives for Γ(1)\M(p) and M(p)/Γ(1), so that

M(p) = ⋃_{i} Γ(1) · α_{i} = ⋃_{i} α_{i} · Γ(1) (disjoint unions).

Then

M(p) = p · M(p)^{−1} = ⋃ p · Γ(1) · α_{i}^{−1} = ⋃ Γ(1) · α′_{i}.

Therefore,

<T (p)f, g>= p^{k−1} ∑_{i} <f|_{k}α_{i}, g>= p^{k−1} ∑_{i} <f, g|_{k}α_{i}^{−1}>= p^{k−1} ∑_{i} <f, g|_{k}α′_{i}>=<f, T (p)g> .

The ℤ-structure on the space of modular forms for Γ(1). Recall (4.20) that the Eisenstein series

G_{k}(z) =^{df} ∑_{(m,n)≠(0,0)} \frac{1}{(mz + n)^{2k}} = 2ζ(2k) + 2\frac{(2πi)^{2k}}{(2k − 1)!} ∑_{n=1}^{∞} σ_{2k−1}(n)q^{n}, q = e^{2πiz}.

For k ≥ 1, define the normalized Eisenstein series E_{k}(z) = G_{k}(z)/2ζ(2k).

Then, using that ζ(2k) = \frac{2^{2k−1}}{(2k)!} B_{k}π^{2k}, one finds that

E_{k}(z) = 1 + γ_{k} ∑_{n=1}^{∞} σ_{2k−1}(n)q^{n}, γ_{k} = (−1)^{k} · \frac{4k}{B_{k}} ∈ ℚ.

For example,

E_{2}(z) = 1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n},

E_{3}(z) = 1 − 504 ∑_{n=1}^{∞} σ_{5}(n)q^{n},
⋯
E_{6}(z) = 1 + \frac{54600}{691} ∑_{n=1}^{∞} σ_{11}(n)q^{n}.

Note that E_{2}(z) and E_{3}(z) have integer coefficients.

Lemma 5.25. The Eisenstein series G_{k}, k ≥ 2, is an eigenform of the T (n), with eigenvalue is σ_{2k−1}(n). The normalized eigenform is γ_{k}^{−1} · E_{k}. The corresponding Dirichlet series is

ζ(s) · ζ(s − 2k + 1).

Proof. The short proof that G_{k} is an eigenform, is to observe that M_{k} = S_{k}⊕ < G_{k}>, and that T (n) · G_{k} is orthogonal to S_{k} (because G_{k} is, T (n) is Hermitian, and T (n) preserves S_{k}). Therefore T (n) · G_{k} is a multiple of G_{k}.

The computational proof starts from the definition

G_{k}(Λ) = ∑_{λ∈Λ,λ≠0} \frac{1}{λ^{2k}} .

Therefore

T (p) · G_{k}(Λ) = ∑_{Λ′} ∑_{λ∈Λ′,λ≠0} \frac{1}{λ^{2k}}

where the outer sum is over the lattices Λ′ of index p in Λ. If λ ∈ pΛ, it lies in all Λ′, and so contributes (p + 1)/λ^{2k} to the sum. Otherwise, it lies in only one lattice Λ′, namely pΛ + ℤλ, and so it contributes 1/λ^{2k}. Hence

(T (p) · G_{k})(Λ) = G_{k}(Λ) + p ∑_{λ∈pΛ,λ≠0} \frac{1}{λ^{2k}} = G_{k}(Λ) + p^{1−2k}G_{k}(Λ) = (1 + p^{1−2k})G_{k}(Λ).

Therefore G_{k}(Λ), as a function on L, is an eigenform of T (p), with eigenvalue 1+p^{1−2k}. As a function on ℍ it is an eigenform with eigenvalue p^{2k−1}(1 + p^{1−2k}) = p^{2k−1} + 1 = σ_{2k−1}(p).

The normalized eigenform is

γ_{k}^{−1} + ∑_{n=1}^{∞} σ_{2k−1}(n)q^{n}, γ_{k} = (−1)^{k} · \frac{4k}{B_{k}} ,

and the associated Dirichlet series is

∑_{n=1}^{∞} \frac{σ_{2k−1}(n)}{n^{s}} = ∑_{a,d≥1} \frac{a^{2k−1}}{a^{s}d^{s}} = (∑_{d≥1} \frac{1}{d^{s}})(∑_{a≥1} \frac{1}{a^{s+1−2k}}) = ζ(s) · ζ(s − 2k + 1).

Let V be a vector space over ℂ. By a ℤ-structure on V, I mean a ℤ-module V_{0} ⊂ V which is free of rank equal to the dimension of V. Equivalently, it is a ℤ-submodule that is freely generated by a ℂ-basis for V, or a ℤ-submodule such that the natural map V_{0} ⊗_{ℤ} ℂ → V is an isomorphism (or a full lattice in V).

Let M_{k}(ℤ) be the ℤ-submodule of M_{k}(Γ(1)) consisting of modular forms f = ∑_{n=0}^{∞} a_{n}q^{n} with the a_{n} ∈ ℤ.

Proposition 5.26. The module M_{k}(ℤ) is a ℤ-structure on M_{k}(Γ(1)).

Proof. Recall that ⊕M_{k}(ℂ) = ℂ[G_{2}, G_{3}] = ℂ[𝔼_{2}, 𝔼_{3}]. It suffices to show that

⊕_{k} M_{k}(ℤ) = ℤ[E_{2}, E_{3}].

Note that E_{2}(z), E_{3}(z), and Δ′ = q ∏(1 − q^{n})^{24} all have integer coefficients. We prove by induction on k that M_{k}(ℤ) is the 2k^{th}-graded piece of ℤ[E_{2}, E_{3}] (here E_{2} has degree 4 and E_{3} has degree 6). Given f(z) = ∑ a_{n}q^{n}, a_{n} ∈ ℤ, write

f = a_{0}E_{2}^{a} · E_{3}^{b} + Δ · g

with 4a + 6b = 2k, and g ∈ M_{k−12}. Then a_{0} ∈ ℤ, and one checks by explicit calculation that g ∈ M_{k−12}(ℤ).

Proposition 5.27. The eigenvalues of the Hecke operators are algebraic integers.

Proof. Let M_{k}(ℤ) be the ℤ-module of modular forms with integer Fourier coefficients. It is stabilized by T (n), because.

T (n) · f(z) = ∑_{m≥0} γ(m) · q^{m}

with

γ(m) = ∑ a^{2k−1} · c(\frac{mn}{a^{2}}) (sum over a|m,a ≥ 1).

The matrix of T (n) with respect to a basis for M_{k}(ℤ) integer coefficients, and this shows that the eigenvalues of T (n) are algebraic integers.

Aside 5.28. For Siegel modular forms of all levels, the analogous result was only proved fairly recently (by Chai and Faltings), using difficult algebraic geometry. See Section 7.

Geometric interpretation of Hecke operators. Before discussing Hecke operators for a general group, we explain the geometric significance of Hecke operators. Fix a subgroup Γ of finite index in Γ(1).

Let α ∈ GL_{2}(ℝ)^{+}. Then α defines a map x ↦ αx : ℍ → ℍ, and we would like to define a map α : Γ\ℍ → Γ\ℍ, Γz ↦ “αΓz”. Unfortunately, far from being normal in GL_{2}(ℝ)^{+}. If we try defining α(Γz) = Γαz we run into the problem that the orbit Γαz depends on the choice of z (because α^{−1}Γα ≠ Γ in general, even if α has integer coefficients and Γ = Γ(N)).

In fact, αΓz is not even a Γ-orbit. Instead, we need to consider the union of the orbits meeting αΓz, i.e., we need to look at ΓαΓz. Any coset (right or left) of Γ in GL_{2}(ℝ)^{+} that meets ΓαΓ is contained in it, and so we can we can write

ΓαΓ = ⋃ Γα_{i} (disjoint union),

and then ΓαΓz = ⋃ Γα_{i}z (disjoint union). Thus α, or better, the double coset Γα, defines a “many-valued map”

Γ\ℍ → Γ\ℍ, Γz ↦ {Γα_{i}z}.

Since “many-valued maps” don’t exist in my lexicon, we shall have to see how to write this in terms of honest maps. First we give a condition on α that ensures that the “map” is at least finitely-valued.

Lemma 5.29. Let α ∈ GL_{2}(ℝ)^{+}. Then ΓαΓ is a finite union of right (and of left) cosets if and only if α is a scalar multiple of a matrix with integer coefficients.

Proof. Omit. [Note that the next lemma shows that this is equivalent to α^{−1}Γα being commensurable with Γ.]

Lemma 5.30. Let α ∈ GL_{2}(ℝ)^{+}. Write

Γ = ⋃(Γ ∩ α^{−1}Γα)β_{i} (disjoint union);

then

Γα = ⋃ Γα_{i} (disjoint union)

with α_{i} = α · β_{i}.

Proof. We are given that Γ = ⋃(Γ ∩ α^{−1}Γα)β_{i}. Therefore

α^{−1}ΓαΓ = ⋃_{i} α^{−1}Γα · (Γ ∩ α^{−1}Γα) · β_{i} = ⋃_{i}(α^{−1}ΓαΓ ∩ α^{−1}Γα) · β_{i}.

But α^{−1}ΓαΓ ⊃ α^{−1}Γα, and so we can drop it from the right hand term. Therefore

α^{−1}ΓαΓ = ⋃_{i} α^{−1}Γαβ_{i}.

On multiplying by α, we find that ΓαΓ = ⋃_{i} Γαβ_{i}, as claimed.

If Γαβ_{i} = Γαβ_{j}, then β_{i}β_{j}^{−1} ∈ α^{−1}Γα; since it also lies in Γ, this implies that i = j.

Now let Γ_{α} = Γ ∩ α^{−1}Γα, and write Γ = ⋃ Γ_{α} · β_{i} (disjoint union). Consider

Γ_{α}\ℍ
↙  ↘ α
Γ\ℍ  Γ\ℍ.

The map α sends an orbit Γ_{α} ·x to Γ·αx—this is now well-defined—and the left hand arrow sends an orbit Γ_{α} · x to Γ · x.

Let f be a modular function, regarded as a function on Γ\ℍ. Then f ∘ α is a function on Γ_{α}\ℍ, and its “trace” ∑ f ∘ α ∘ β_{i} is invariant under Γ, and is therefore a function on Γ\ℍ. This function is ∑ f ∘ α_{i} = T (p) · f. Similarly, a (meromorphic) modular form can be thought of as a k-fold differential form on Γ\ℍ, and T (p) can be interpreted as the pull-back followed by the trace in the above diagram.

Remark 5.31. In general a diagram of finite-to-one maps

Y
↙  ↘
X  Z.

is called a correspondence on X × Z. The simplest example is obtained by taking Y to be the graph of a map φ : X → Z; then the projection Y → X is a bijection. A correspondence is a is a “many-valued mapping”, correctly interpreted: an element x ∈ X is “mapped” to the images in Z of its inverse images in Y. The above observation shows the Hecke operator on modular functions and forms is defined by a correspondence, which we call the Hecke correspondence.

The Hecke algebra. The above discussion suggests that we should define an action of double cosets ΓαΓ on modular forms. It is convenient first to define an abstract algebra, H(Γ, Δ), called the Hecke algebra.

Let Γ be a subgroup of Γ(1) of finite index, and let Δ be a set of real matrices with positive determinant, closed under multiplication, and such that for α ∈ Δ, the double coset ΓαΓ contains only finitely many left and right cosets for Γ. Define H(Γ, Δ) to be the free ℤ-module generated by the double cosets ΓαΓ, α ∈ Δ. Thus an element of H(Γ, Δ) is a finite sum,

∑ n_{α}ΓαΓ, α ∈ Δ, n_{α} ∈ ℤ.

Write [α] for ΓαΓ when it is regarded as an element of H(Γ, Δ).

We define a multiplication on H(Γ, Δ) as follows. Note that if ΓαΓ meets a right coset Γα′, then it contains it. Therefore, we can write ΓαΓ = ∪Γα_{i}, ΓβΓ = ∪Γβ_{i} (finite disjoint unions). Then ΓαΓ · ΓβΓ = ΓαΓβΓ = ∪ΓαΓβ_{j} = ∪_{i,j}Γα_{i}β_{j}; therefore ΓαΓβΓ is a finite union of double cosets. Define

[α] · [β] = ∑ c_{α,β}^{γ} · [γ]

where the union is over the γ ∈ Δ such that ΓγΓ ⊂ ΓαΓβΓ, and c_{α,β}^{γ} is the number of pairs (i, j) with Γα_{i}β_{j} = Γγ.

Example 5.32. Let Γ = Γ(1), and let Δ be the set of matrices with integer coefficients and positive determinant. Then H(Γ, Δ) is the free abelian group on the generators

Γ(1) ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) Γ(1), a|d, ad > 0, a ≥ 1, a, d ∈ ℤ.

Write T (a, d) for the element Γ(1) ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) Γ(1) of H(Γ, Δ). Thus H(Γ, Δ) has a

quite explicit set of free generators, and it is possible to write down (complicated) formulas for the multiplication.

For a prime p, we define T (p) to be the element T (1, p) of H(Γ, Δ). We would like to define

T (n) = M(n) =^{df} {matrices with integer coefficients and determinant n}.

We can’t do this because M(n) is not a double coset, but it is a finite union of double cosets (see 5.15), namely,

M(n) = ⋃ Γ(1) · ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) · Γ(1), a|d, ad = n, a ≥ 1, a, d ∈ ℤ.

This suggests defining

T (n) = ∑ T (a, d), a|d, ad = n, a ≥ 1, a, d ∈ ℤ.

As before, we let D be the free abelian group on the set of lattices L in ℂ. A double coset [α] acts on D according to the rule:

[α] · Λ = αΛ.

(To compute αΛ, choose a basis ( \begin{matrix}ω_{1} \\ ω_{2} \end{matrix} ) for Λ, and let αΛ be the lattice with

basis α · ( \begin{matrix}ω_{1} \\ ω_{2} \end{matrix} ); this is independent of the choice of the basis, and of the choice of

a representative for the double coset ΓαΓ.) We extend this by linearity to an action of H(Γ, Δ) on D. It is immediate from the various definitions that T (n) (element of H(Γ, Δ)) acts on D as the T (n) on defined at the start of this section. The relation in (5.10) implies that the following relation holds in the ring H(Γ, Δ):

T (n)T (m) = ∑_{d| gcd(m,n)} d · T (d, d) · T (nm/d^{2}) (∗)

In particular, for relatively prime integers m and n,

T (n)T (m) = T (nm),

and for a prime p,

T (p) · T (p^{n}) = T (p^{n+1}) + p · T (p, p) · T (p^{n−1}).

The ring H(Γ, Δ) acts on the set of functions on L:

[α] · F = ∑ F (α_{i}Λ) if Γα = ∪Γα_{i}.

The relation (∗) implies that

T (n) · T (m) · F = ∑_{d| gcd(m,n)} d^{1−2k} · T (mn/d^{2}) · F

for F a function on L of weight 2k.

Finally, we make H(Γ, Δ) act on M_{k}(Γ(1)) by

[α] · f = det(α)^{k−1} · ∑ f|_{k}α_{i} (∗∗)

if ΓαΓ = ∪Γ(1) · α_{i}. Recall that f|_{k}α = (det α)^{k} · (cz + d)^{k} · f(\frac{az+b}{cz+d}), if α = ( \begin{matrix}a & b \\ c & d \end{matrix} ).

The element T (n) ∈ H(Γ, Δ) acts on M_{k}(Γ(1)) as in the old definition, and (*) implies that

T (n) · T (m) · f = ∑_{d| gcd(m,n)} d^{2k−1} · T (\frac{mn}{d^{2}}) · f.

We now define a Hecke algebra for Γ(N). For this we take Δ(N) to be the set of integer

matrices α such that n =^{df} det(α) is positive and prime to N, and α ≡ ( \begin{matrix}1 & 0 \\ 0 & n \end{matrix} ) mod

N.

Lemma 5.33. Let Δ′(N) be the set of integer matrices with positive determinant prime to N. Then the map

Γ(N) · α · Γ(N) ↦ Γ(1) · α · Γ(1) : H(Γ(N), Δ(N)) → H(Γ(1), Δ′(N))

is an isomorphism.

Proof. Elementary. (See Ogg 1969, pIV-10.)

Let T^{N}(a, d) and T^{N}(n) be the elements of H(Γ(N), Δ(N)) corresponding to T (a, d) and T (n) in H(Γ(1), Δ′(N)) under the isomorphism in the lemma. Note that H(Γ(1), Δ′(N)) is a subring of H(Γ(1), Δ). From the identity (∗), we obtain the identity

T^{N}(n)T^{N}(m) = ∑_{d| gcd(n,m)} d · T^{N}(d, d) · T^{N}(mn/d^{2}) (***)

for (mn, N) = 1.

When we let H(Γ(N), Δ(N)) act on M_{k}(Γ(N)) by the rule (∗∗), the identity (∗) translates into a slightly different identity for operators on M_{k}(Γ(N)). (The key point

is that ( \begin{matrix}d & 0 \\ 0 & d \end{matrix} )∈ Δ′(N) if gcd(d, N) = 1 but not Δ(N)—see Ogg 1969, pIV-12).

For f ∈ M_{k}(Γ(N)), we have the identity is

T^{N}(n) · T^{N}(m) · f = ∑_{d|m,n} d^{2k−1} · R_{d} · T^{N}(mn/d^{2}) · f

for (mn, N) = 1. Here R_{d} is a matrix in Γ(1) such that R_{d} ≡ ( \begin{matrix}d^{−1} & 0 \\ 0 & d \end{matrix} ) mod N.

The term R_{d} causes problems. Let V = M_{k}(Γ(N)). If d ≡ 1 mod N, then R_{d} ∈ Γ(N), and so it acts as the identity map on V. Therefore d ↦ R_{d} defines an action of (ℤ/Nℤ)^{×} on V, and so we can decompose V into a direct sum,

V = ⊕V (ε),

over the characters ε of (ℤ/Nℤ)^{×}, where

V (ε) = {f ∈ V | f|R_{d} = ε(d) · f}.

Lemma 5.34. The operators R_{n} and T^{N}(m) on V commute for (nm, N) = 1 . Hence V (ε) is invariant under T^{N}(m).

Proof. See Ogg 1969, pIV-13.

Let M_{k}(Γ(N), ε) = V (ε). Then T^{N}(n) acts on M_{k}(Γ(N), ε) with the basic identity:

T^{N}(n) · T^{N}(m) = ∑_{d| gcd(n,m)} d^{2k−1} · ε(d) · T^{N}(nm/d^{2}),

for (nm, N) = 1.

Proposition 5.35. Let f ∈ M_{k}(Γ(N), ε) have the Fourier expansion f = ∑ a_{n}q^{n}. Assume that f is an eigenform for all T^{N}(n), and normalize it so that a_{1} = 1. Then

L_{N}(f, s) =^{df} ∑_{gcd(n,N)=1} a_{n}n^{−s} = ∏_{gcd(p,N)=1} \frac{1}{(1 − a_{p}p^{−s} + ε(p)p^{2k−1−2s})}.

Proof. Essentially the same as the proof of Proposition 5.1.

Let U = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) (it would be too confusing to continue denoting it as T). Then

U^{N} ∈ Γ(N), and so

f ↦ f|_{k}U^{m} = f(z + m)

defines an action of ℤ/Nℤ on V =^{df} M_{k}(Γ(N)). We can decompose V into a direct sum over the characters of ℤ/Nℤ. But the characters of ℤ/Nℤ are parametrized by the N^{th} roots of one in ℂ—the character corresponding to ζ is m mod N ↦ ζ^{m}. Thus

V = ⊕V (ζ), ζ an N^{th} root of 1,

where V (ζ) = {f ∈ V | U^{m} · f = ζ^{m}f}. Alas V (ζ) is not invariant under T^{N}(n). To remedy this, we have to consider, for each t|N,

V (t) = ⊕V (ζ), ζ a primitive (N/t)^{th} root of 1.

Let m be an integer divisible only by the primes dividing N; we define

T^{t}(m) = m^{k−1} · ∑_{0≤b<m} f|_{k} ( \begin{matrix}1 & bN/t \\ 0 & m \end{matrix} ) .

For a general n > 1, we write n = mn_{0} with gcd(n_{0}, N) = 1, and set

T^{t}(n) = T (n_{0}) · T^{t}(m).

We then have the relation:

T^{t}(n) · T^{t}(m) · f = ∑_{d| gcd(n,m)} ε(d)d^{2k−1}T^{t}(nm/d^{2}) · f

for f ∈ V (, t) =^{df} V (ε) ∩ V (t).

Theorem 5.36. Let f ∈ V (t, ε) have the Fourier expansion f(z) = ∑ a_{n}q^{n}. If a_{1} = 1 and f is an eigenform for all the T^{t}(n) with gcd(n, N) = 1, then the associated Dirichlet series has the Euler product expansion

∑ a_{n}n^{−s} = ∏_{p} \frac{1}{1 − a_{p}p^{−s} + ε(p)p^{2k−1−2s}} .

Proof. See Ogg 1969, pIV-10.

In the statement of the theorem, we have extended ε from (ℤ/Nℤ)^{×} to ℤ/Nℤ by setting ε(p) = 0 for p|N. Thus ε(p) = 0 if p|N, and a_{p} = 0 if p|\frac{N}{t}. This should be compared with the L-series of an elliptic curve E with conductor N, where the p-factor of the L-series is (1±p^{−s})^{−1} if p|N but p^{2} does not divide N, and is 1 if p^{2}|N.

Proposition 5.37. Let f and g be cusp forms for Γ(N) of weight 2k and character ε. Then

<T (n) · f, g>= ε(n) <f, T (n) · g>

Proof. See Ogg 1969, pIV-24.

Unlike the case of forms for Γ(1), this does not imply that the eigenvalues are real. It does imply that M_{k}(Γ(N), ε, t) has a basis of eigenforms for the T (n) with gcd(n, N) = 1 (but not for all T (n)’s).

For a summary of the theory of Hecke operators for Γ_{0}(N), see Math 679, especially Section 26.

Part II: Applications to Arithmetic Geometry

In this part we apply the preceding theory, first to obtain elliptic modular curves defined over number fields, and then to study the zeta functions of modular curves and of elliptic curves. There is considerable overlap between this part and Math 679.

6. The Modular Equation for Γ_{0}(N)

For any congruence subgroup Γ of Γ(1), the algebraic curve Γ\ℍ^{∗} is defined over a specific number field. As a first step toward proving this general statement, we find in this section a canonical polynomial F (X, Y) with coefficients in ℚ such that the curve F (X, Y) = 0 is birationally equivalent to X_{0}(N) =^{df} Γ_{0}(N)\ℍ^{∗}.

Recall that

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) | c ≡ 0 mod N\} .

If γ = ( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} ), then

( \begin{matrix}N^{−1} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}a & N^{−1}b \\ Nc & d \end{matrix} ) for ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ(1),

and so

Γ_{0}(N) = Γ(1) ∩ γ^{−1}Γ(1)γ.

Note that −I ∈ Γ_{0}(N). In the map

SL_{2}(ℤ) → SL_{2}(ℤ/Nℤ)

the image of Γ_{0}(N) is the group of all matrices of the form ( \begin{matrix}a & b \\ 0 & a^{−1} \end{matrix} ) in SL_{2}(ℤ/Nℤ).

This group obviously has order N · φ(N), and so (cf. 2.23),

μ =^{df} (Γ̄(1) : Γ̄_{0}(N)) = (Γ(1) : Γ_{0}(N)) = N · ∏_{p|N}(1 + \frac{1}{p}).

(Henceforth, Γ̄ denotes the image of Γ in SL_{2}(ℤ)/{±I}.) Consider the set of pairs (c, d) of positive integers satisfying:

gcd(c, d) = 1, d|N, 0 ≤ c < N/d. (∗)

For each such pair, we choose a pair (a, b) of integers such that ad − bc = 1. Then the

matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) form a set of representatives for Γ_{0}(N)\Γ(1). (Check that they

are not equivalent under left multiplication by elements of Γ_{0}(N), and that there is the correct number.)

If 4|N then Γ̄_{0}(N) contains no elliptic elements of order 2, and if 9|N then it contains no elliptic elements of order 3. The cusps for Γ_{0}(N) are represented by the pairs (c, d) satisfying (*), modulo the equivalence relation:

(c, d) ∼ (c′, d′) if d = d′ and c′ = c + m, some m ∈ ℤ.

For each d, there are exactly φ(gcd(d, N/d)) inequivalent pairs, and so the number of cusps is

∑_{d|N,d>0} φ(gcd(d, \frac{N}{d})).

It is now possible to use Theorem 2.22 to compute the genus of X_{0}(N). (See Shimura 1971, p25, for more details on the above material.)

Theorem 6.1. The field ℂ(X_{0}(N)) of modular functions for Γ_{0}(N) is generated (over ℂ) by j(z) and j(Nz). The minimum polynomial F (j, Y) ∈ ℂ(j)[Y] of j(Nz) over ℂ(j) has degree μ. Moreover, F (j, Y) is a polynomial in j and has coefficients in ℤ, i.e., F (X, Y) ∈ ℤ[X, Y]. When N > 1, F (X, Y) is symmetric in X and Y, and when N = p is prime,

F (X, Y) ≡ X^{p+1} + Y^{p+1} − X^{p}Y^{p} − XY mod p.

Proof. Let γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) be an element of Γ_{0}(N) with c = Nc′, c′ ∈ ℤ. Then

j(Nγz) = j (\frac{Naz + Nb}{cz + d}) = j (\frac{Naz + Nb}{Nc′z + d}) = j (\frac{a(Nz) + Nb}{c′(Nz) + d}) = j(Nz)

because ( \begin{matrix}a & Nb \\ c′ & d \end{matrix} ) ∈ Γ(1). Therefore ℂ(j(z), j(Nz)) is contained in the field of

modular functions for Γ_{0}(N).

The curve X_{0}(N) is a covering of X(1) of degree μ = (Γ(1) : Γ_{0}(N)). From Proposition 1.16 we know that the field of meromorphic functions ℂ(X_{0}(N)) on X_{0}(N) has degree μ over ℂ(X(1)) = ℂ(j), but we shall prove this again. Let {γ_{1} = 1, ..., γ_{μ}} be a set of representatives for the right cosets of Γ_{0}(N) in Γ(1), so that,

Γ(1) = ∪Γ_{0}(N)γ_{i} (disjoint union).

For any γ ∈ Γ(1), {γ_{1}γ, ..., γ_{μ}γ} is also a set of representatives for the right cosets of Γ_{0}(N) in Γ(1)—the set {Γ_{0}(N)γ_{i}γ} is just a permutation of the set {Γ_{0}(N)γ_{i}}.

If f(z) is a modular function for Γ_{0}(N), then f(γ_{i}z) depends only on the coset Γ_{0}(N)γ_{i}. Hence the functions {f(γ_{i}γz)} are a permutation of the functions {f(γ_{i}z)}, and any symmetric polynomial in the f(γ_{i}z) is invariant under Γ(1); since such a polynomial obviously satisfies the other conditions, it is a modular function for Γ(1), and hence a rational function of j. We have shown that f(z) satisfies a polynomial of degree μ with coefficients in ℂ(j), namely, ∏(Y − f(γ_{i}z)). Since this holds for every f ∈ ℂ(X_{0}(N)), we see that ℂ(X_{0}(N)) has degree at most μ over ℂ(j).

Next I claim that all the f(γ_{i}z) are conjugate to f(z) over ℂ(j): for let F (j, Y) be the minimum polynomial of f(z) over ℂ(j); in particular, F (j, Y) is monic and irreducible when regarded as a polynomial in Y with coefficients in ℂ(j); on replacing z with γ_{i}z and remembering that j(γ_{i}z) = j(z), we find that F (j(z), f(γ_{i}z)) = 0, which proves the claim.

If we can show that the functions j(Nγ_{i}z) are distinct, then it will follow that the minimum polynomial of j(Nz) over ℂ(j) has degree μ; hence [ℂ(X_{0}(N)) : ℂ(j)] = μ and ℂ(X_{0}(N)) = ℂ(j(z))[j(Nz)].

Suppose j(Nγ_{i}z) = j(Nγ_{i′}z) for some i ≠ i′. Recall that j defines an isomorphism Γ(1)\ℍ^{∗} → (Riemann sphere), and so j(Nγ_{i}z) = j(Nγ_{i′}z) all z implies that there exists a γ ∈ Γ(1) such that Nγ_{i}z = γNγ_{i′}z all z, and this implies that

( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} ) γ_{i} = ±γ ( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} ) γ_{i′}.

Hence γ_{i}γ_{i′}^{−1} ∈ Γ(1) ∩ ( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} )^{−1} Γ(1) ( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} ) = Γ_{0}(N), and this contradicts the

fact that γ_{i} and γ_{i′} lie in different cosets.

The minimum polynomial of j(Nz) over ℂ(j) is F (j, Y) = ∏(Y − j(Nγ_{i}z)). The symmetric polynomials in the j(Nγ_{i}z) are holomorphic on ℍ. As they are rational functions of j(z), they must in fact be polynomials in j(z), and so F (X, Y) ∈ ℂ[X, Y] (rather than ℂ(X)[Y]).

But we know (4.22) that

j(z) = q^{−1} + ∑_{n=0}^{∞} c_{n}q^{n} (∗).

with the c_{n} ∈ ℤ. Consider j(Nγz) for some γ = ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ Γ(1). Then Nγz =

( \begin{matrix}Na′ & Nb′ \\ c′ & d′ \end{matrix} ) z, and j(Nγz) is unchanged when we act on the matrix on the left by

an element of Γ(1). Therefore (see 5.15)

j(Nγz) = j(\frac{az + b}{d})

for some integers a, b, d with ad = N. On substituting \frac{az+b}{d} for z in (*) and noting that e^{2πi(az+b)/d} = e^{2πib/d} · e^{2πiaz/d}, we find that j(Nγz) has a Fourier expansion in powers of q^{1/N} whose coefficients are in ℤ[e^{2πi/N}], and hence are algebraic integers. The same is then true of the symmetric polynomials in the j(Nγ_{i}z). We know that these symmetric polynomials lie in ℂ[j(z)], and I claim that in fact they are polynomials in j with coefficients that are algebraic integers.

Consider a polynomial P = ∑ c_{n}j^{n} ∈ ℂ[j] whose coefficients are not all algebraic integers. If c_{m} is the coefficient having the largest subscript among those that are not algebraic integers, then the coefficient of q^{−m} in the q-expansion of P is not an algebraic integer, and so P can not be equal to a symmetric polynomial in the j(Nγ_{i}z).

Thus F (X, Y) = ∑ c_{m,n}X^{m}Y^{n} with the c_{m,n} algebraic integers (and c_{0,μ} = 1). When we substitute (*) into the equation

F (j(z), j(Nz)) = 0,

and equate coefficients of powers of q, we obtain a set of linear equations for the c_{m,n} with rational coefficients. When we adjoin the equation

c_{0,μ} = 1,

then the equations determine the c_{m,n} uniquely (because there is only one monic minimum equation for j(Nz) over ℂ(j)). Because the system of linear equations has a solution in ℂ, it also has a solution in ℚ; because the solution is unique, the solution in ℂ must in fact lie in ℚ. Thus the c_{m,n} ∈ ℚ, but we know that they are algebraic integers, and so they lie in ℤ.

Now assume N > 1. On replacing z with −1/Nz in the equation F (j(z), j(Nz)) = 0, we obtain

F (j(−1/Nz), j(−1/z)) = 0,

which, because of the invariance of j, is just the equation

F (j(Nz), j(z)) = 0.

This shows that F (Y, X) is a multiple of F (X, Y) (recall that F (X, Y) is irreducible in ℂ(X)[Y], and hence in ℂ[X, Y]), say, F (Y, X) = cF (X, Y). On equating coefficients, one sees that c^{2} = 1, and so c = ±1. But c = −1 would imply that F (X, X) = 0, and so X − Y would be a factor of F (X, Y), which contradicts the irreducibility. Hence c = 1, and F (X, Y) is symmetric.

Finally, suppose N = p, a prime. The argument following (*) shows in this case that the functions j(pγ_{i}z) for i ≠ 1 are exactly the functions:

j (\frac{z + m}{p}) , m = 0, 1, 2, …, p − 1.

Let ζ_{p} = e^{2πi/p}, and let 𝔭 denote the prime ideal (1 − ζ_{p}) in ℤ[ζ_{p}]. Then 𝔭^{p−1} = (p). When we regard the functions j(\frac{z+m}{p}) as power series in q, then we see that they are all congruent modulo 𝔭 (meaning that their coefficients are congruent modulo 𝔭), and so

f(j(z), Y) =_{df} (Y − j(pz)) ∏_{m=0}^{p−1}(Y − j(\frac{z + m}{p})) ≡ (Y − j(pz))(Y − j(z/p))^{p}

≡ (Y − j(z)^{p})(Y^{p} − j(z)) (mod 𝔭).

This implies the last equation in the theorem.

Example 6.2. For N = 2, the equation is

X^{3} + Y^{3} − X^{2}Y^{2} + 1488XY (X + Y) − 162000(X^{2} + Y^{2}) + 40773375XY + 8748000000(X + Y) − 157464000000000 = 0.

Rather a lot of effort (for over a century) has been put into computing F (X, Y) for small values of N. For a discussion of how to do it (complete with dirty tricks), see Birch’s article in Modular Functions of One Variable, Vol I, SLN 320 (Ed. W. Kuyk).

The modular equation F_{N}(X, Y) = 0 was introduced by Kronecker, and used by Kronecker and Weber in the theory of complex multiplication. For N = 3, it was computed by Smith in 1878; for N = 5 it was computed by Berwick in 1916; for N = 7 it was computed by Herrmann in 1974; for N = 11 it was computed by MACSYMA in 1984. This last computation took 20 hours on a VAX-780; the result is a polynomial of degree 21 with coefficients up 10^{60} which takes 5 pages to write out. See Kaltofen and Yui, On the modular equation of order 11, Proc. of the Third MACSYMA’s user’s Conference, 1984, pp472-485.

Clearly one gets nowhere with brute force methods in this subject.

7. The Canonical Model of X_{0}(N) over ℚ

After reviewing some algebraic geometry, we define the canonical model of X_{0}(N) over ℚ.

Review of some algebraic geometry. (This is essentially Section 13 of my notes on Algebraic Geometry (Math 631).) Theorem 6.1 will allow us to define a model of X_{0}(N) over ℚ, but before explaining this I need to review some of the terminology from algebraic geometry.

First we need a slightly more abstract notion of sheaf than that on p11.

Definition 7.1. A presheaf F on a topological space X is a map assigning to each open subset U of X a set F (U) and to each inclusion U ⊃ U′ a “restriction” map

a ↦ a|U′ : F (U) → F (U′).

The restriction map corresponding to U ⊃ U is required to be the identity map, and if U ⊃ U′ ⊃ U′′, then the restriction map F (U) → F (U′′) is required to be the composite of the restriction maps F (U) → F (U′) and F (U′) → F (U′′).

If the sets F (U) are abelian groups and the restriction maps are homomorphisms, then F is called a presheaf of abelian groups (similarly for a sheaf of rings, modules, etc.). A presheaf F is a sheaf if for every open covering {U_{i}} of U ⊂ X and family of elements a_{i} ∈ F (U_{i}) agreeing on overlaps (that is, such that a_{i}|U_{i} ∩ U_{j} = a_{j}|U_{i} ∩ U_{j} for all i, j), there is a unique element a ∈ F (U) such that a_{i} = a|U_{i} for all i. A ringed space is a pair (X, O_{X}) where X is a topological space and O_{X} is a sheaf of rings on X. With the obvious notion of morphism, the ringed spaces form a category.

Let k_{0} be a field, and let k be an algebraic closure of k_{0}. An affine k_{0}-algebra A is a finitely generated k_{0}-algebra A such that A ⊗_{k_{0}} k is an integral domain.

This is stronger than saying that A itself is an integral domain—in fact, A can be an integral domain without A ⊗_{k_{0}} k being reduced. Consider for example the algebra

A = k_{0}[X, Y]/(X^{p} + Y^{p} + a)

where p = char(k_{0}) and a ∉ k_{0}^{p}; then A is an integral domain because X^{p} + Y^{p} + a is irreducible, but obviously

A ⊗_{k_{0}} k = k[X, Y]/(X^{p} + Y^{p} + a) = k[X, Y]/((X + Y + α)^{p}), α^{p} = a,

is not reduced. This problem arises only because of inseparability: if k_{0} is perfect, then A ⊗_{k_{0}} k is reduced whenever A is finitely generated k_{0}-algebra that is an integral domain. However, even then A ⊗_{k_{0}} k need not be an integral domain—consider for example A = k[X]/(f(X)). We have the following criterion: a finitely generated algebra A over a perfect field k_{0} is an affine k-algebra if and only if A is an integral domain and k_{0} is algebraically closed in A (i.e., an element of A algebraic over k_{0} is already in k_{0}).

Example 7.2. An algebra k_{0}[X, Y]/(f(X, Y)) is an affine k_{0}-algebra if and only if f(X, Y) is absolutely irreducible, i.e., it is irreducible in k[X, Y].

Let A be a finitely generated k_{0}-algebra. We can write

A = k_{0}[x_{1}, ..., x_{n}] = k_{0}[X_{1}, ..., X_{n}]/(f_{1}, ..., f_{m}),

and then

A ⊗_{k_{0}} k = k[X_{1}, ..., X_{n}]/(f_{1}, ..., f_{m}).

Thus A is an affine algebra if and only if the elements f_{1}, ..., f_{m} of k_{0}[X_{1}, ..., X_{n}] generate a prime ideal when regarded as elements of k[X_{1}, ..., X_{n}].

Let A be an affine k_{0}-algebra. Define specm(A) to be the set of maximal ideals in A, and endow it with the topology having as basis the sets D(f), f ∈ A, where D(f) = {𝔪 | f ∉ 𝔪}. There is a unique sheaf of k_{0}-algebras O on specm(A) such that O(D(f)) = A_{f} =^{df} A[f^{−1}] for all f. Here O is a sheaf in the abstract sense—the elements of O(U) are not functions on U with values in k_{0}, although we may wish to think of them as if they were. If f ∈ A and 𝔪_{v} ∈ specm A, then we define f(v) to be the image of f in the κ(v) =^{df} A/𝔪_{v}. Then v ↦ f(v) is not a function on specm(A) in the conventional sense because (unless k_{0} = k) the fields κ(v) are varying with v, but it does make sense to speak of the set V (f) of zeros of f in X, and this zero set is the complement of D(f).

The ringed space

Specm(A) =^{df} (specm(A), O),

as well as any ringed space isomorphic to such a space, is called an affine variety over k_{0}. A ringed space (X, O_{X}) is a prevariety over k_{0} if there is a finite covering (U_{i}) of X by open subsets such that (U_{i}, O_{X}|U_{i}) is an affine variety over k_{0} for all i. A morphism of prevarieties over k_{0} is a morphism of ringed spaces; in more detail, a morphism of prevarieties (X, O_{X}) → (Y, O_{Y}) is a continuous map φ : X → Y and, for every open subset U of Y, a map ψ : O_{Y} (U) → O_{X}(φ^{−1}(U)) satisfying certain natural conditions. A prevariety X over k is separated if for all pairs of morphisms φ, ψ : Z → X, the set where φ and ψ agree is closed in Z. A variety is a separated prevariety.

When V = Specm B and W = Specm A, there is a one-to-one correspondence between the set of morphisms of varieties W → V and the set of homomorphisms of k_{0}- algebras A → B. If A = k_{0}[X_{1}, ..., X_{m}]/𝔞 and B = k_{0}[Y_{1}, ..., Y_{n}]/𝔟, a homomorphism A → B is determined by a family of polynomials, P_{i}(Y_{1}, ..., Y_{n}), i = 1, ..., m; the morphism W → V sends (y_{1}, …, y_{n}) to (…, P_{i}(y_{1}, ..., y_{n}), …); in order to define a homomorphism, the P_{i} must be such that F ∈ 𝔞 ⇒ F (P_{1}, ..., P_{n}) ∈ 𝔟; two families P_{1}, ..., P_{m} and Q_{1}, ..., Q_{m} determine the same map if and only if P_{i} ≡ Q_{i} mod 𝔟 for all i.

There is a canonical way of associating a variety X over k with a variety X_{0} over k; for example, if X_{0} = Specm(A), then X = Specm(A ⊗_{k_{0}} k). We then call X_{0} a model for X over k_{0}. When X ⊂ 𝔸^{n}, to give a model for X over k_{0} is the same as to give an ideal 𝔞_{0} ⊂ k_{0}[X_{1}, ..., X_{n}] such that 𝔞_{0} generates the ideal of X,

I(X) =^{df} {f ∈ k[X_{1}, …, X_{n}] | f = 0 on X}.

Of course, X need not have a model over k_{0}—for example, an elliptic curve E over k will have a model over k_{0} ⊂ k if and only if its j-invariant j(E) lies in k_{0}. Moreover, when X has a model over k_{0}, it will usually have a large number of them, no two of which are isomorphic over k_{0}. For example, let X be a nondegenerate quadric surface in ℙ^{3} over ℚ^{al} (the algebraic closure of ℚ); thus X is isomorphic to the surface

X^{2} + Y^{2} + Z^{2} + W^{2} = 0.

The models over X over ℚ are defined by equations

aX^{2} + bY^{2} + cZ^{2} + dW^{2} = 0, a, b, c, d ∈ ℚ.

Thus classifying the models of X over ℚ is equivalent to classifying quadratic forms over ℚ in 4 variables; this has been done, but it is quite complicated—there are an infinite number.

Let X be a variety over k_{0}. A point of X with coordinates in k_{0}, or a point of X rational over k_{0}, is a morphism Specm k_{0} → X. For example, if X is affine, say X = Specm A, then a point of X with coordinates in k_{0} is a k_{0}-homomorphism A → k_{0}. If A = k[X_{1}, ..., X_{n}]/(f_{1}, ..., f_{m}), then to give a k_{0}-homomorphism A → k_{0} is the same as to give an n-tuple (a_{1}, ..., a_{n}) such that

f_{i}(a_{1}, ..., a_{n}) = 0 i = 1, ..., m;

thus a point of X with coordinates in k_{0} is exactly what you expect it to be. Similar remarks apply to projective varieties. We write X(k_{0}) for the points of X with coordinates in k_{0}.

It is possible to define the notion of a point of X with coordinates in any k_{0}-algebra R, and we write X(R) for the set of such points. For example, when X = Specm A,

X(R) = Hom_{k−alg}(A, R).

When k = k_{0}, X(k_{0}) = X. What is the relation of the sets X(k_{0}) and X when k ≠ k_{0}? Let v ∈ X. Then v corresponds to a maximal ideal 𝔪_{v} (actually, it is a maximal ideal), and we write κ(v) for the residue field O_{v}/𝔪_{v}. It is a finite extension of k_{0}, and we call the degree of κ(v) over k_{0} the degree of v. Then X(k_{0}) can be identified with the points v of X of degree 1. (Suppose for example that X is affine, say X = Specm A. Then a point of X is a maximal ideal 𝔪_{v} in A. Obviously, 𝔪_{v} is the kernel of a k_{0}-homomorphism A → k_{0} if and only if κ(v) =^{df} A/𝔪_{v} = k_{0}, in which case it is the kernel of exactly one such homomorphism.)

The set X(k) can be identified with the set of points on X_{k}, where X_{k} is the variety over k associated with X. When k_{0} is perfect, there is an action of Gal(k/k_{0}) on X(k), and one can show that there is a natural one-to-one correspondence between the orbits of the action and the points of X. (Again suppose X = Specm A, and let v ∈ X; associate with v the set of k_{0}-homomorphisms A → k with kernel 𝔪_{v}.)

Assume k_{0} is perfect. As we just noted, if X_{0} is a variety over k_{0}, then there is an action of Gal(k/k_{0}) on X_{0}(k). The variety X =^{df} X_{0,k} and the action of Gal(k/k_{0}) on X(k) then determines X_{0}: for example, if X = Specm A, then the action of Gal(k/k_{0}) on X(k) determines an action of Gal(k/k_{0}) on A and X_{0} = Specm A^{Gal(k/k_{0})}.

All of the usual theory of algebraic varieties over algebraically closed fields carries over mutatis mutandis to varieties over a nonalgebraically closed field.

Curves and Riemann surfaces. Fix a field k_{0}, and let X be an algebraic variety over k_{0}. The function field k_{0}(X) of X is the field of fractions of O_{X}(U) for any open affine subset U of X; for example, if X = Specm A, then k_{0}(X) is the field of fractions of A. The dimension of X is defined to be the transcendence degree of k_{0}(X) over k_{0}. An algebraic curve is an algebraic variety of dimension 1.

To each point v of X there is attached a local ring O_{v}. For example, if X = Specm A, then a point v of X is a maximal ideal 𝔪 in A, and the local ring attached to v is A_{𝔪}. An algebraic variety is said to be regular if all the local rings A_{𝔪} are regular (“regular” is a weaker condition than “nonsingular”; nonsingular implies regular, and the two are equivalent when the ground field k_{0} is algebraically closed).

Consider an algebraic curve X. Then X is regular if and only if the local rings attached to it are discrete valuation rings. For example, Specm A is a regular curve if and only if A is a Dedekind domain. A regular curve X defines a set of discrete valuation rings in k_{0}(X), each of which contains k_{0}, and X is complete if and only if this set includes all the discrete valuation rings in k_{0}(X) having k_{0}(X) as field of fractions and containing k_{0}.

A field K containing k_{0} is said to be a function field in n variables over k_{0} if it is finitely generated and has transcendence degree n over k_{0}. The field of constants of K is the algebraic closure of k_{0} in K. Thus the function field of an algebraic variety over k_{0} of dimension n is a function field in n variables over k_{0} having k_{0} as its field of constants (whence the terminology).

Theorem 7.3. The map X ↦ k_{0}(X) defines an equivalence from the category of complete regular algebraic curves over k_{0} to the category of function fields in one variable over k_{0} having k_{0} as field of constants.

Proof. The curve corresponding to the field K can be constructed as follows: take X to be the set of discrete valuation rings in K containing k_{0} and having K as their field of fractions; define a subset U of X to be open if it omits only finitely many elements of X; for such a U, define O_{X}(U) to be the intersection of the discrete valuation rings in U.

Corollary 7.4. A regular curve U can be embedded into a complete regular curve Ū; the map U ↪ Ū is universal among maps from U into complete regular curves.

Proof. Take Ū to be the complete regular algebraic curve attached to k_{0}(U). There is an obvious identification of U with an open subset of Ū.

Example 7.5. Let F (X, Y) be an absolutely irreducible polynomial in k_{0}[X, Y], and let A = k_{0}[X, Y]/(F (X, Y)). Thus A is an affine k_{0}-algebra, and C =^{df} Specm A is the curve: F (X, Y) = 0. Let C^{ns} be the complement in C of the set of maximal ideals of A containing the ideal (∂F/∂X, ∂F/∂X) mod F (X, Y). Then C^{ns} is a nonsingular curve, and hence can be embedded into a complete regular curve C̄.

There is a geometric way of constructing C̄, at least in the case that k_{0} = k is algebraically closed. First consider the plane projective curve C′ defined by the homogeneous equation

Z^{deg(F)}F (X/Z, Y/Z) = 0.

This is a projective (hence complete) algebraic curve which, in general, will have singular points. It is possible to resolve these singularities geometrically, and so obtain a nonsingular projective curve (see W. Fulton 1969, p179).

Theorem 7.6. Every compact Riemann surface X has a unique structure of a complete nonsingular algebraic curve.

Proof. We explain only how to construct the associated algebraic curve. The underlying set is the same; the topology is that for which the open sets are those with finite complements; the regular functions on an open set U are the holomorphic functions on U which are meromorphic on the whole of X.

Remark 7.7. Theorems 7.3 and 7.6 depend crucially on the hypothesis that the variety has dimension 1.

In general, many different complete nonsingular algebraic varieties can have the same function field. A nonsingular variety U over a field of characteristic zero can be embedded in a complete nonsingular variety Ū, but this is a very difficult theorem (proved by Hironaka in 1970), and Ū is very definately not unique. For a variety of dimension > 3 over a field of characteristic p > 0, even the existence of Ū is not known.

For a curve, “complete” is equivalent to “projective”; for smooth surfaces they are also equivalent, but in higher dimensions there are many complete nonprojective varieties (although Chow’s lemma says that a complete variety is not too far away from a projective variety).

Many compact complex manifolds of dimension > 1 have no algebraic structure.

The curve X_{0}(N) over ℚ. According to Theorem 7.6, there is a unique structure of an complete nonsingular curve on X_{0}(N) compatible with its structure as a Riemann surface. We write X_{0}(N)_{ℂ} for X_{0}(N) regarded as an algebraic curve over ℂ. Note that X_{0}(N)_{ℂ} is the unique complete nonsingular curve over ℂ having the field ℂ(j(z), j(Nz) of modular functions for Γ_{0}(N) as its field of rational functions.

Now write F_{N}(X, Y) for the polynomial constructed in Theorem 6.1, and let C be the curve over ℚ defined by the equation:

F_{N}(X, Y) = 0.

As is explained above, we can remove the singular points of C to obtain a nonsingular curve C^{ns} over ℚ, and then we can embed C^{ns} into a complete regular curve C̄. The coordinate functions x and y are rational functions on C̄, they generate the field of rational functions on C̄, and they satisfy the relation F_{N}(x, y) = 0; these statements characterize C̄ and the pair of functions x, y on it.

Let C̄_{ℂ} be the curve defined by C̄ over ℂ. It can also be obtained in the same way as C̄ starting from the curve F_{N}(X, Y) = 0, now thought of as a curve over ℂ. There is a unique isomorphism C̄_{ℂ} → X_{0}(N)_{ℂ} making the rational functions x and y on C̄_{ℂ} correspond to the functions j(z) and j(Nz) on X_{0}(N). We can use this isomorphism to identify the two curves, and so we can regard C̄ as being a model of X_{0}(N)_{ℂ} over ℚ. We write it X_{0}(N)_{ℚ}. (In fact, we often omit the subscripts from X_{0}(N)_{ℂ} and X_{0}(N)_{ℚ}.)

We can be a little more explicit: on an open subset, the isomorphism X_{0}(N) → C̄_{ℂ} is simply the map [z] ↦ (j(z), j(Nz)) (regarding this pair as a point on the affine curve F_{N}(X, Y) = 0).

The action of Aut(ℂ) on X_{0}(N) corresponding to the model X_{0}(N)_{ℚ} has the following description: for τ ∈ Aut(ℂ), τ[z] = [z′] if τj(z) = j(z′) and τj(Nz) = j(Nz′).

The curve X_{0}(N)_{ℚ} is called the canonical model of X_{0}(N) over ℚ. The canonical model X(1)_{ℚ} of X(1) is just the projective line ℙ^{1} over ℚ. If the field of rational functions on ℙ^{1} is ℚ(T), then the identification of ℙ^{1} with X(1) is made in such a way that T corresponds to j.

The quotient map X_{0}(N) → X(1) corresponds to the map of algebraic curves X_{0}(N)_{ℚ} → X(1)_{ℚ} defined by the inclusion of function fields ℚ(T) → ℚ(x, y), T ↦ x. On an open subset of X_{0}(N)_{ℚ}, it is the projection map (a, b) ↦ a.

8. Modular Curves as Moduli Varieties

For over 100 years algebraic geometers have worked with “moduli varietes” that classify isomorphism classes of certain objects but, as far as I know, a precise definition of a moduli variety was not given before Mumford’s work in the 1960’s. In this section I explain the general notion of a moduli variety, and then I explain how to realize the modular curves as moduli varieties for elliptic curves with additional structure.

The general notion of a moduli variety. Fix a field k which initially we assume to be algebraically closed. A moduli problem over k is a contravariant functor F from the category of algebraic varieties over k to the category of sets. Thus for each variety V over k we are given a set F(V), and for each regular map φ : W → V, we are given map φ^{∗} : F(V) → F(W). Typically, F(V) will be the set of isomorphism classes of certain objects over V.

A solution to the moduli problem is a variety V over k together with an identification V (k) = F(k) and certain additional data sufficient to determine V uniquely. More precisely:

Definition 8.1. A pair (V, α) consisting of a variety V over k together with a bijection α : F(k) → V (k) is called a solution to the moduli problem F if it satisfies the following conditions:

(a) Let T be a variety over k and let f ∈ F(T); a point t ∈ T (k) can be regarded as a map Specm k → V, and so (by the functoriality of F) f defines an element f_{t} of T (k); we therefore have a map t ↦ α(f_{t}) : T (k) → V (k), and this map is required to be regular (i.e., defined by a morphism of algebraic varieties);

(b) (Universality) Let Z be a variety over k and let β : F(k) → Z(k) be a map such that, for any pair (T, f) as in (a), the map t ↦ β(f_{t}) : T (k) → Z(k) is regular; then the map β ∘ α^{−1} : V (k) → Z(k) is regular.

A variety V that occurs as the solution of a moduli problem is called a moduli variety.

Proposition 8.2. Up to a unique isomorphism, there exists at most one solution to a moduli problem.

Proof. Suppose there are two solutions (V, α) and (V′, α′). Then because of the universality of (V, α), α′ ∘ α^{−1} : V → V′ is a regular map, and because of the universality of (V′, α′), its inverse is also a regular map.

Of course, in general there may exist no solution to a moduli problem, and when there does exist a solution, it may be very difficult to prove it. Mumford was given the Fields medal mainly because of his construction of the moduli varieties of curves and abelian varieties.

Remark 8.3. It is possible to modify the above definition for the case that the ground field k_{0} is not algebraically closed. For simplicity, we assume k_{0} to be perfect, and we let k be an algebraic closure of k_{0}. Now V is a variety over k_{0} and α is a family of maps α(k′) : F(k′) → V (k′) (one for each algebraic extension k′ of k_{0}) compatible with inclusions of fields, and (V_{k}, α(k)) is required to be a solution to the moduli problem over k. If (V, α) and (V′, α′) are two solutions to the same moduli problem, then α′ ∘ α^{−1} : V (k) → V′(k) and its inverse are both regular maps commuting with the action of Gal(k/k_{0}); they are both therefore defined over k_{0}. Consequently, up to a unique isomorphism, there again can be at most one solution to a moduli problem.

Note that we don’t require α(k′) to be a bijection when k′ is not algebraically closed. In particular, V need not represent the functor F. When V does represent the functor, V is called a fine moduli variety; otherwise it is a coarse moduli variety.

The moduli variety for elliptic curves. We show that 𝔸^{1} is the moduli variety for elliptic curves over a perfect field k_{0}.

An elliptic curve E over a field k′ is a curve given by an equation of the form,

Y^{2}Z + a_{1}XY Z + a_{3}Y Z^{2} = X^{3} + a_{2}X^{2}Z + a_{4}XZ^{2} + a_{6}Z^{3} (∗)

for which the discriminant Δ(a_{1}, a_{2}, a_{3}, a_{4}, a_{6}) ≠ 0. It has a distinguished point (0 : 1 : 0), and an isomorphism of elliptic curves over k′ is an isomorphism of varieties carrying the distinguished point on one curve to the distinguished point on the second. (There is a unique group law on E having the distinguished element as zero, and a morphism of elliptic curves is automatically a homomorphism of groups.)

Let V be a variety over a field k′. An elliptic curve (better, family of elliptic curves) over V is a map of algebraic varieties E → V where E is the subvariety of V × ℙ^{1} defined by an equation of the form (*) with the a_{i} regular functions on V; Δ(a_{1}, a_{2}, a_{3}, a_{4}, a_{6}) is now a regular function on V which is required to have no zeros.

For any variety V, let E(V) be the set of isomorphism classes of elliptic curves over V. Then E is a contravariant functor, and so can be regarded as a moduli problem over k_{0}.

For any field k′ containing k_{0}, the j-invariant defines a map

E ↦ j(E) : E(k′) → 𝔸^{1}(k′) = k′,

and the theory of elliptic curves (Math 679) shows that this map is an isomorphism if k′ is algebraically closed (but not in general otherwise).

Theorem 8.4. The pair (𝔸^{1}, j) is a solution to the moduli problem E.

Proof. For any k_{0}-homomorphism σ : k′ → k′′, j(σE) = σj(E), and so it remains to show that (𝔸^{1}, j) satisfies the conditions (a) and (b) over k.

Let E → T be a family of elliptic curves over T, where T is a variety over k. The map t ↦ j(E_{t}) : T (k) → 𝔸^{1}(k) is regular because j(E_{t}) = c_{4}^{3}/Δ where c_{4} is a polynomial in the a_{i}’s and Δ is a nowhere zero polynomial in the a_{i}’s.

Now let (Z, β) be a pair as in (b). We have to show that j ↦ β(E_{j}) : 𝔸^{1}(k) → Z(k), where E_{j} is an elliptic curve over k with j-invariant j, is regular. Let U be the open subset of 𝔸^{1} obtained by removing the points 0 and 1728. Then

E : Y^{2}Z + XY Z = X^{3} − \frac{36}{u − 1728}XZ^{2} − \frac{1}{u − 1728}Z^{3}, u ∈ U,

is an elliptic curve over U with the property that j(E_{u}) = u (Silverman 1986, p52). Because of the property possessed by (Z, β), E/U defines a regular map u ↦ β(E_{u}) : U → Z. But this is just the restriction of the map j ↦ β(E_{j}) to U(k), which is therefore regular, and it follows that j itself is regular.

The curve Y_{0}(N)_{ℚ} as a moduli variety. Let k be a perfect field, and let N be a positive integer not divisible by the characteristic of k (so there is no restriction on N when k has characteristic zero). Let E be an elliptic curve over k. When k is an algebraically closed field, a cyclic subgroup of E of order N is simply a cyclic subgroup of E(k) of order N in the sense of abstract groups. When k is not algebraically closed, a cyclic subgroup of E is a Zariski-closed subset S such that S(k^{al}) is cyclic subgroup of S(k^{al}) of order N. Thus S(k^{al}) is a cyclic subgroup of order N of E(k^{al}) that is stable (as a set—not elementwise) under the action of Gal(k^{al}/k), and every such group arises from a (unique) S.

An isomorphism from one pair (E, S) to a second (E′, S′) is an isomorphism E → E′ mapping S onto S′.

These definitions can be extended in a natural way to families of elliptic curves over varieties.

For any variety V over k, define E_{0,N}(V) to be the set of isomorphism classes of pairs (E, S) where E is an elliptic curve over V, and S is a cyclic subgroup of E of order N. Then E_{0,N} is a contravariant functor, and hence is a moduli problem.

Recall that Λ(ω_{1}, ω_{2}) is the lattice generated by a pair (ω_{1}, ω_{2}) with ℑ(ω_{1}/ω_{2}) > 0. Note that Λ(ω_{1}, N^{−1}ω_{2})/Λ(ω_{1}, ω_{2}) is a cyclic subgroup of order N of the elliptic curve ℂ/Λ(ω_{1}, ω_{2}).

Lemma 8.5. The map

ℍ → E_{0,N}(ℂ), z ↦ (ℂ/Λ(z, 1), Λ(z, N^{−1})/Λ(z, 1))

induces a bijection Γ_{0}(N)\ℍ → E_{0,N}(ℂ).

Proof. Easy—see Math 679, p124.

Let E′_{0,N}(k) denote the set of isomorphism classes of homomorphisms of elliptic curves α : E → E′ over k whose kernel is a cyclic subgroup of E of order N. The map

α ↦ (E, Ker(α)) : E′_{0,N}(k) → E_{0,N}(k)

is a bijection; its inverse is (E, S) ↦ (E → E/S). For example, the element (ℂ/Λ(z, 1), Λ(z, N^{−1})/Λ(z, 1)) of E_{0,N}(ℂ) corresponds to the element (ℂ/Λ(z, 1) →^{N} ℂ/Λ(Nz, 1)) of E′_{0,N}(ℂ).

Let F_{N}(X, Y) be the polynomial defined in Theorem 6.1 and let C be the (singular) curve F_{N}(X, Y) = 0 over ℚ. For any field k ⊃ ℚ, consider the map

E′_{0,N}(k) → 𝔸^{2}(k), (E, E′) ↦ (j(E), j(E′)).

When k = ℂ, the above discussion shows that the image of this map is contained in C(ℂ), and this implies that the same is true for any k.

Recall that Y_{0}(N) = Γ_{0}(N)\ℍ. There is an affine curve Y_{0}(N)_{ℚ} ⊂ X_{0}(N)_{ℚ} which is a model of Y_{0}(N) ⊂ X_{0}(N). (This just says that the set of cusps on X_{0}(N) is defined over ℚ.)

Theorem 8.6. Let k be a field, and let N be an integer not divisible by the characteristic of k. The moduli problem E_{0,N} has a solution (M, α) over k. When k = ℚ, M is canonically isomorphic to Y_{0}(N)_{ℚ}, and the map

E_{0,N}(k) ⟶^{α} M(k) ⟶^{≈} Y_{0}(N)_{ℚ}(k) ⟶^{(j,j_{N})} C(k)

is (E, S) ↦ (j(E), j(E/S)).

Proof. When k = ℚ, it is possible to prove that Y_{0}(N)_{ℚ} is a solution to the moduli problem in much the same way as for 𝔸^{1} above. If p ∤ N, then it is possible to show that Y_{0}(N)_{ℚ} has good reduction at p, and the curve Y_{0}(N)_{𝔽_{p}} over 𝔽_{p} it reduces to is a solution to the moduli problem over 𝔽_{p}.

The curve Y (N) as a moduli variety. Let N be a positive integer, and let ζ ∈ ℂ be a primitive Nth root of 1. A level-N structure on an elliptic curve E is a pair of points t = (t_{1}, t_{2}) in E(k) such that the map

(m, m′) ↦ (mt, mt′) : ℤ/Nℤ × ℤ/Nℤ → E(k)

is injective. This means that E(k)_{N} has order N^{2}, and t_{1} and t_{2} form a basis for E(k)_{N} as a ℤ/nℤ-module. For any variety V over a field k ⊃ ℚ[ζ], define E_{N}(V) to be the set of isomorphism classes of pairs (E, t) where E is an elliptic curve over V and t = (t_{1}, t_{2}) is a level-N structure on E such that e_{N}(t_{1}, t_{2}) = ζ (here e_{N} is the Weil pairing—see Silverman III.8). Then E_{N} is a contravariant functor, and hence is a moduli problem.

Lemma 8.7. The map

ℍ → E_{N}(ℂ)

z ↦ (ℂ//Λ(z, 1), (z, 1) mod Λ(z, 1)) induces a bijection Γ(N)\ℍ → E_{N}(ℂ).

Proof. Easy.

Theorem 8.8. Let k be a field containing ℚ[ζ], where ζ is a primitive Nth root of 1. The moduli problem E_{N} has a solution (M, α) over k. When k = ℂ, M is canonically isomorphic to Y (N)_{ℂ} (= X(N)_{ℂ} with the cusps removed). Let M be the solution to the moduli problem E_{N} over ℚ[ζ]; then M has good reduction at the prime ideals not dividing N.

Proof. Omit.

Example 8.9. For N = 2, the solution to the moduli problem is 𝔸^{1}. In this case, there is a universal elliptic curve with level-2 structure over 𝔸^{1}, namely, the curve

E : Y^{2}Z = X(X − Z)(X − λZ).

Here λ is the coordinate on 𝔸^{1}, and the map E → 𝔸^{1} is (x : y : z, λ) ↦ λ. The level-2 structure is the pair of points (0 : 0 : 1), (1 : 0 : 1). The curve E is universal in the following sense: for any family of elliptic curves E′ → V with level-2 structure over a variety V (with the same base field k), there is a unique morphism V → 𝔸^{1} such that E′ is the pull-back of E. In this case the map E(k) → 𝔸^{1}(k) is an isomorphism for all fields k ⊃ ℚ, and 𝔸^{1} is a fine moduli variety.

9. Modular Forms, Dirichlet Series, and Functional Equations

The most famous Dirichlet series, ζ(s) =^{df} ∑_{n=1}^{∞} n^{−s}, was shown by Riemann (in 1859) to have an analytic continuation to the whole complex plane except for a simple pole at s = 1, and to satisfy a functional equation

Z(s) = Z(1 − s)

where Z(s) = π^{−s/2}Γ(s/2)ζ(s). One now believes (Hasse-Weil conjecture) that all Dirichlet series arising as the zeta functions of algebraic varieties over number fields should have meromorphic continuations to the whole complex plane and satisfy functional equations. In this section we investigate the relation between Dirichlet series with functional equations and modular forms.

We saw in (2.12) that the modular group Γ(1) is generated by the matrices T =

( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). Therefore a modular function f(z) of weight 2k

satisfies the following two conditions:

f(z + 1) = f(z), f(−1/z) = (−z)^{2k}f(z).

The first condition implies that f(z) has a Fourier expansion f(z) = ∑ a_{n}q^{n}, and so defines a Dirichlet series φ(s) = ∑ a_{n}n^{−s}. Hecke showed that the second condition implies that the Dirichlet series satisfies a functional equation, and conversely every Dirichlet series satisfying a functional equation of the correct form (and certain holomorphicity conditions) arises from a modular form. Weil extended this result to the subgroup Γ_{0}(N) of Γ(1), which needs more than two generators (and so we need more than one functional equation for the Dirichlet series). In this section we explain Hecke’s and Weil’s results, and in later sections we explain the implications of Weil’s results for elliptic curves over ℚ.

The Mellin transform. Let a_{1}, a_{2}, … be a sequence of complex numbers such that a_{n} = O(n^{M}) for some M. This can be regarded as the sequence of coefficients of either the power series f(q) = ∑_{1}^{∞} a_{n}q^{n}, which is absolutely convergent for |q| < 1 at least, or for the Dirichlet series φ(s) = ∑_{1}^{∞} a_{n}n^{−s}, which is absolutely convergent for ℜ(s) > M + 1 at least. In this subsection, we give explicit formulae that realize the formal correspondence between f(y) and φ(s).

Recall that the gamma function Γ(s) is defined by the formula,

Γ(s) = ∫_{0}^{∞} e^{−x}x^{s−1}dx, ℜ(s) > 0.

It has the following properties: Γ(s + 1) = sΓ(s), Γ(1) = 1, and Γ(\frac{1}{2}) = \sqrt{π}; Γ(s) extends to a function that is holomorphic on the whole complex plane, except for simple poles at s = −n, where it has a residue \frac{(−1)^{n}}{n!} , n = 0, 1, 2, . . ..

Proposition 9.1 (Mellin Inversion Formula). For every real c > 0,

e^{−x} = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} Γ(s)x^{−s}ds, x > 0.

(The integral is taken upwards on a vertical line.)

Proof. Regard the integral as taking place on a vertical circumference on the Riemann sphere. The calculus of residues shows that the integral is equal to

2πi ∑_{n=0}^{∞} res_{s=−n}x^{−s}Γ(s) = 2πi ∑_{n=0}^{∞} \frac{(−x)^{n}}{n!} = 2πi · e^{−x}.

Theorem 9.2. Let a_{1}, a_{2}, … be a sequence of complex numbers such that a_{n} = O(n^{M}) for some M. Write f(x) = ∑_{1}^{∞} a_{n}e^{−nx} and ϕ(s) = ∑_{1}^{∞} a_{n}n^{−s}. Then

Γ(s)ϕ(s) = ∫_{0}^{∞} f(x)x^{s−1}dx for ℜ(s) > max(0, M + 1), (∗)

f(x) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} ϕ(s)Γ(s)x^{−s}ds for c > max(0, M + 1) and ℜ(x) > 0. (∗∗)

Proof. First consider (∗). Formally we have

∫_{0}^{∞} f(x)x^{s−1}dx = ∫_{0}^{∞} ∑_{1}^{∞} a_{n}e^{−nx}x^{s−1}dx

= ∑_{1}^{∞} ∫_{0}^{∞} a_{n}e^{−nx}x^{s−1}dx

= ∑_{1}^{∞} a_{n}Γ(s)n^{−s}
= Γ(s)ϕ(s)

on writing x for nx in the last integral and using the definition of Γ(s). The only problem is in justifying the interchange of the integral with the summation sign.

The equation (∗∗) follows from Proposition 9.1.

The functions f(x) and ϕ(s) are called the Mellin transforms of each other.

The equation (∗) provides a means of analytically continuing ϕ(s) provided f(x) tends to zero sufficiently rapidly at x = 0. In particular, if f(x) = O(x^{A}) for every A > 0 as x → 0 through real positive values, then Γ(s)ϕ(s) can be extended to a holomorphic function over the entire complex plane. Of course, this condition on f(x) implies that x = 0 is an essential singularity.

We say that a function φ(s) on the complex plane is bounded on vertical strips, if for all real numbers a < b, φ(s) is bounded on the strip a ≤ ℜ(s) ≤ b as ℑ(s) → ±∞.

Theorem 9.3 (Hecke 1936). Let a_{0}, a_{1}, a_{2}, … be a sequence of complex numbers such that a_{n} = O(n^{M}) for some M. Given λ > 0, k > 0, C = ±1, write

φ(s) = ∑ a_{n}n^{−s}; (φ(s) converges for ℜ(s) > M + 1)
Φ(s) = (\frac{2π}{λ})^{−s} Γ(s)φ(s);
f(z) = ∑_{n≥0} a_{n}e^{2πinz/λ}; (converges for ℑ(z) > 0).

Then the following conditions are equivalent:

(i) The function Φ(s) + \frac{a_{0}}{s} + \frac{Ca_{0}}{k−s} can be analytically continued to a holomorphic function on the entire complex plane which is bounded on vertical strips, and it satisfies the functional equation

Φ(k − s) = CΦ(s).

(ii) In the upper half plane, f satisfies the functional equation

f(−1/z) = C(z/i)^{k}f(z).

Proof. Given (ii), apply (∗) to obtain (i); given (i), apply (∗∗) to obtain (ii).

Remark 9.4. Let Γ′(λ) be the subgroup of Γ(1) generated by the maps z ↦ z + λ and z ↦ −1/z. A modular form of weight k and multiplier C for Γ′(λ) is a holomorphic function f(z) on ℍ such that

f(z + λ) = f(z), f(−1/z) = C(z/i)^{k}f(z),

and f is holomorphic at i∞. This is a slightly more general notion than in Section 4—if k is an even integer and C = 1 then it agrees with it.

The theorem says that there is a one-to-one correspondence between modular forms of weight k and multiplier C for Γ′(λ) whose Fourier coefficients satisfy a_{n} = O(n^{M}) for some M, and Dirichlet series satisfying (i). Note that Φ(s) is holomorphic if f is a cusp form.

For example ζ(s) corresponds to a modular form of weight 1/2 and multiplier 1 for Γ′(2).

Weil’s theorem. Given a sequence of complex numbers a_{1}, a_{2}, … such that a_{n} = O(n^{M}) for some M, write

L(s) = ∑_{n=1}^{∞} a_{n}n^{−s}, Λ(s) = (2π)^{−s}Γ(s)L(s), f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}. (1)

More generally, let m > 0 be an integer, and let χ be a primitive character on (ℤ/mℤ)^{×} (primitive means that it is not a character on (ℤ/dℤ)^{×} for any proper divisor d of n). As usual, we extend χ(s) to the whole of ℤ/mℤ by setting χ(n) = 0 if n is not relatively prime to m. We write

L_{χ}(s) = ∑_{n=1}^{∞} a_{n}χ(n)n^{−s}, Λ_{χ}(s) = (\frac{m}{2π})^{−s}Γ(s)L_{χ}(s), f_{χ}(z) = ∑_{n=1}^{∞} a_{n}χ(n)e^{2πinz}.

Note that L_{χ} and f_{χ} are the Mellin transforms of each other.

For any χ, the associated Gauss sum is

g(χ) = ∑_{n=1}^{m} χ(n)e^{−2πin/m}.

Obviously χ̄(a)g(χ) = ∑ χ(n)e^{−2πian/m}, and hence

χ(n) = m^{−1}g(χ) ∑ χ̄(a)e^{2πian/m}.

It follows from this last equation that

f_{χ} = m^{−1}g(χ) ∑_{1}^{m} χ̄(a)f|_{k} ( \begin{matrix}m & a \\ 0 & m \end{matrix} ) .

Theorem 9.5. Let f(z) be a modular form of weight 2k for Γ_{0}(N), and suppose

that f|_{k} ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ) = C(−1)^{k}f for some C = ±1. Define

C_{χ} = Cg(χ)χ(−N)/g(χ̄).

Then Λ_{χ}(s) satisfies the functional equation:

Λ_{χ}(s) = C_{χ}N^{k−s}Λχ̄(2k − s) whenever gcd(m, N) = 1.

Proof. Apply Theorem 9.3.

The most interesting result is the converse to this theorem.

Theorem 9.6 (Weil 1967). Fix a C = ±1, and suppose that for all but finitely many primes p not dividing N the following condition holds: for every primitive character χ of (ℤ/pℤ)^{×}, Λ(s) and Λ_{χ}(s) can be analytically continued to holomorphic functions in the entire complex plane and that each of them is bounded on vertical strips; suppose also that they satisfy the functional equations:

Λ(s) = CN^{k−s}Λ(2k − s)
Λ_{χ}(s) = C_{χ}N^{k−s}Λχ̄(2k − s)

where C_{χ} is defined above; suppose further that the Dirichlet series L(s) is absolutely convergent for s = k − ϵ for some ϵ > 0. Then f(z) is a cusp form of weight 2k for Γ_{0}(N).

Proof. Several pages of manipulation of 2 × 2 matrices.

Let E be an elliptic curve over ℚ, and let L(E, s) be the associated L-series. As we shall see shortly, it is generally conjectured that L(s) satisfies the hypotheses of the theorem, and hence is attached to a modular form f(z) of weight 2 for Γ_{0}(N). Granted this, one can show that there is nonconstant map α : X_{0}(N) → E (defined over ℚ) such that the pull-back of the canonical differential on E is the differential on X_{0}(N) attached to f(z).

Remark 9.7. Complete proofs of the statements in this section can be found in Ogg 1969, especially Chapter V. They are not particularly difficult—it would only add about 5 pages to the notes to include them.

10. Correspondences on Curves; the Theorem of Eichler-Shimura

In this section we sketch a proof of the key theorem of Eichler and Shimura relating the Hecke correspondence T_{p} to the Frobenius map. In the next section we explain how this enables us to realize certain zeta functions as the Mellin transforms of modular forms.

The ring of correspondences of a curve. Let X and X′ be projective nonsingular curves over a field k which, for simplicity, we take to be algebraically closed.

A correspondence T between X and X′ is a pair of finite surjective morphisms

X ⟵^{α} Y ⟶^{β} X′.

It can be thought of as a many-valued map X → X′ sending a point P ∈ X(k) to the set {β(Q_{i})} where the Q_{i} run through the elements of α^{−1}(P) (the Q_{i} need not be distinct). Better, define Div(X) to be the free abelian group on the set of points of X; thus an element of Div(X) is a finite formal sum

D = ∑ n_{P} P, n_{P} ∈ ℤ, P ∈ C.

A correspondence T then defines a map

Div(X) → Div(X′), P ↦ ∑ β(Q_{i}),

(notations as above). This map multiplies the degree of a divisor by deg(α). It therefore sends the divisors of degree zero on X into the divisors of degree zero on X′, and one can show that it sends principal divisors to principal divisors. It therefore defines a map T : J(X) → J(X′) where

J(X) =^{df} Div^{0}(X)/{principal divisors}.

We define the ring of correspondences A(X) on X to be the subring of End(J(X)) generated by the maps defined by correspondences.

If T is the correspondence

X ⟵^{β} Y ⟶^{α} X′.

then the transpose T′ of T is the correspondence

X ⟵^{α} Y ⟶^{β} X′.

A morphism α : X → X′ can be thought of as a correspondence

X ← Γ → X′

where Γ ⊂ X × X′ is the graph of α and the maps are the projections.

Aside 10.1. Attached to any complete nonsingular curve X there is an abelian variety Jac(X) whose set of points is J(X). The ring of correspondences is the endomorphism ring of Jac(X)—see the next section.

The Hecke correspondence. Let Γ be a subgroup of Γ(1) of finite index, and let α be a matrix with integer coefficients and determinant > 0. Write ΓαΓ = ∪Γα_{i} (disjoint union). Then we get a map

T (α) : J(X(Γ)) → J(X(Γ)), [z] ↦ ∑[α_{i}z].

As was explained on in Section 5, this is the map defined by the correspondence:

X(Γ) ← X(Γ_{α}) →^{α} X(Γ)

where Γ_{α} = Γ ∩ α^{−1}Γα. In this way, we get a homomorphism H → A from the ring of Hecke operators into the ring of correspondences.

Consider the case Γ = Γ_{0}(N) and T = T (p) the Hecke correspondence defined by

the double coset Γ_{0}(N) ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) Γ_{0}(N). Assume that p ∤ N. We give two further

descriptions of T (p).

First, identify a point of Y_{0}(N) (over ℂ) with an isomorphism class of homomorphisms E → E′ of elliptic curves with kernel a cyclic group of order N. The subgroup E_{p} of E of points of order dividing p is isomorphic to (ℤ/pℤ) × (ℤ/pℤ). Hence there are p + 1 cyclic subgroups of E_{p} of order p, say S_{0}, S_{1}, …, S_{p} (they correspond to the lines through the origin in 𝔽_{p}^{2}). Then (as a many-valued map), T (p) sends α : E → E′ to {E_{i} → E′_{i} | i = 0, 1, …, p} where E_{i} = E/S_{i} and E′_{i} = E′/α(S_{i}).

Second, regard Y_{0}(N) as the curve C defined by the polynomial F_{N}(X, Y) constructed in Theorem 6.1 (of course, this isn’t quite correct—there is a map Y_{0}(N) → C, [z] ↦ (j(z), j(Nz)), which is an isomorphism over the nonsingular part of C). Let (j, j′) be a point on C; then there are elliptic curves E and E′ (well-defined up to isomorphism) such that j = j(E) and j′ = j(E′). The condition F_{N}(j, j′) = 0 implies that there is a homomorphism α : E → E′ with kernel a cyclic subgroup of order N. Then T (p) maps (j, j′) to {(j_{i}, j′_{i}) | i = 0, …, p} where j_{i} = j(E/S_{i}) and j′_{i} = j(E′/αS_{i}).

These last two descriptions of the action of T (p) are valid over any field of characteristic 0.

The Frobenius map. Let C be a curve defined over a field k of characteristic p ≠ 0. Assume (for simplicity) that k is algebraically closed. If C is defined by equations ∑ c_{i_{0}i_{1}···}X_{0}^{i_{0}}X_{1}^{i_{1}} ⋯ = 0 and q is a power of p, then C^{(q)} is the curve defined by the equations ∑ c_{i_{0}i_{1}···}^{q}X_{0}^{i_{0}}X_{1}^{i_{1}} ⋯ = 0, and the Frobenius map Π_{q} : C → C^{(q)} sends the point (a_{0} : a_{1} : ⋯) to (a_{0}^{q} : a_{1}^{q} : ⋯). Note that if C is defined over 𝔽_{q}, so that the equations can be chosen to have coefficients c_{i_{0}i_{1}···} in 𝔽_{q}, then C = C^{(q)} and the Frobenius map is a map from C to itself.

Recall that a nonconstant morphism α : C → C′ of curves defines an inclusion α^{∗} : k(C′) ↪ k(C) of function fields, and that the degree of α is defined to be [k(C) : α^{∗}k(C′)]. The map α is said to be separable or purely inseparable according as k(C) is a separable of purely inseparable extension of α^{∗}k(C′). If the separable degree of k(C) over α^{∗}k(C′) is m, then the map C(k) → C′(k) is m : 1 except on a finite set (assuming k to be algebraically closed).

Proposition 10.2. The Frobenius map Π_{q} : C → C^{(q)} is purely inseparable of degree q, and any purely inseparable map φ : C → C′ of degree q (of complete nonsingular curves) factors as

C ⟶^{Π_{q}} C^{(q)} ⟶^{≈} C′

Proof. See Silverman 1986, II.2.12. [First check that

Π_{q}^{∗}k(C) = k(C^{(q)}) = k(C)^{q} =^{df} {a^{q} | a ∈ k(C)}

Then show that k(C) is purely inseparable of degree q over k(C)^{q}, and that this statement uniquely determines k(C)^{q}. The last sentence is obvious when k(C) = k(T) (field of rational functions in T), and the general case follows because k(C) is a separable extension of such a field k(T)].

Brief review of the points of order p on elliptic curves. Let E be an elliptic curve over an algebraically closed field k. The map p : E → E (multiplication by p) is of degree p^{2}. If k has characteristic zero, then the map is separable, which implies that its kernel has order p^{2}. If k has characteristic p, the map is never separable: either it is purely inseparable (and so E has no points of order p) or its separable and inseparable degrees are p (and so E has p points of order dividing p). In the first case, (10.2) tells us that multiplication by p factors as

E → E^{(p^{2})} →^{≈} E.

Hence this case occurs only when E ≈ E^{(p^{2})}, i.e., when j(E) = j(E^{(p^{2})}) = j(E)^{p^{2}}. Thus if E has no points of order p, then j(E) ∈ 𝔽_{p^{2}}.

The Eichler-Shimura theorem. The curve X_{0}(N) is defined over ℚ and the Hecke correspondence T (p) is defined over some number field K. For almost all primes p ∤ N, X_{0}(N) will reduce to a nonsingular curve X̃_{0}(N).^{4} For such a prime p, the correspondence T (p) defines a correspondence T̃(p) on X̃_{0}(N).

Theorem 10.3. For a prime p where X_{0}(N) has good reduction,

T̄_{p} = Π_{p} + Π′_{p}

(equality in the ring A(X̃_{0}(N)) of correspondences on X̃_{0}(N) over the algebraic closure 𝔽 of 𝔽_{p}; here Π′_{p} is the transpose of Π_{p}).

Proof. We show that they agree as many-valued maps on an open subset of X̃_{0}(N).

Over ℚ_{p}^{al} we have the following description of T_{p} (see above): a homomorphism of elliptic curves α : E → E′ with cyclic kernel of order N defines a point (j(E), j(E′)) on X_{0}(N); let S_{0}, …, S_{p} be the subgroups of order p in E; then

T_{p}(j(E), j(E′)) = {(j(E_{i}), j(E′_{i}))}

where E_{i} = E/S_{i} and E′_{i} = E′/α(S_{i}).

Consider a point P̃ on X̃_{0}(N) with coordinates in 𝔽. Ignoring a finite number of points of X̃_{0}(N), we can suppose P̃ ∈ Ỹ_{0}(N) and hence is of the form (j(Ẽ), j(Ẽ′)) for some map α̃ : Ẽ → Ẽ′. Moreover, we can suppose that Ẽ has p points of order dividing p.

4In fact, it is known that X_{0}(N) has good reduction for all primes p ∤ N, but this is hard to prove. It is easy to see that X_{0}(N) does not have good reduction at primes dividing N.

Let α : E → E′ be a lifting of α̃ to ℚ_{p}^{al}. The reduction map E_{p}(ℚ_{p}^{al}) → Ẽ_{p}(𝔽_{p}^{al}) has a kernel of order p. Number the subgroups of order p in E so that S_{0} is the kernel of this map. Then each S_{i}, i ≠ 0, maps to a subgroup of order p in Ẽ.

The map p : Ẽ → Ẽ factors as

Ẽ →^{φ} Ẽ/S_{i} →^{ψ} Ẽ.

When i = 0, φ is a purely inseparable map of degree p (it is the reduction of the map E → E/S_{0}—it therefore has degree p and has zero (visible) kernel), and so ψ must be separable of degree p (we are assuming Ẽ has p points of order dividing p). Proposition 10.2 shows that there is an isomorphism Ẽ^{(p)} → Ẽ/S_{0}. Similarly Ẽ′^{(p)} ≈ Ẽ′/S_{0}. Therefore

(j(Ẽ_{0}), j(Ẽ′_{0})) = (j(Ẽ^{(p)}), j(Ẽ^{′(p)})) = (j(Ẽ)^{p}, j(Ẽ′)^{p}) = Π_{p}(j(Ẽ), j(Ẽ′)).

When i ≠ 0, φ is separable (its kernel is the reduction of S_{i}), and so ψ is purely inseparable. Therefore Ẽ ≈ Ẽ_{i}^{(p)}, and similarly Ẽ′ ≈ Ẽ′_{i}^{(p)}. Therefore

(j(Ẽ_{i})^{(p)}, j(Ẽ′_{i})^{(p)}) = (j(Ẽ), j(Ẽ′)).

Hence

{(j(Ẽ_{i}), j(Ẽ′_{i})) | i = 1, 2, …, p}

is the inverse image of Π_{p}, i.e., it is Π′_{p}(j(Ẽ), j(Ẽ′)). This completes the proof of the theorem.

11. Curves and their Zeta Functions

We begin by reviewing the theory of the zeta functions of curves over ℚ; then we explain the relation between the various representations of the ring of correspondences; finally we explain the implications of the Eichler-Shimura theorem for the zeta functions of the curves X_{0}(N) and elliptic curves; in particular, we state the conjecture of Taniyama-Weil, and briefly indicate how it implies Fermat’s last theorem.

Two elementary results. We begin with two results from linear algebra that will be needed later.

Proposition 11.1. Let Λ be a free ℤ-module of finite rank, and let α : Λ → Λ be a ℤ-linear map with nonzero determinant. Then the kernel of the map

α̃ : (Λ ⊗ ℚ)/Λ → (Λ ⊗ ℚ)/Λ

defined by α has order | det(α)|.

Proof. Consider the commutative diagram:

0 ⟶ Λ ⟶ Λ ⊗ ℚ ⟶ (Λ ⊗ ℚ)/Λ ⟶ 0
↓^{α}  ↓^{α⊗1}  ↓α̃
0 ⟶ Λ ⟶ Λ ⊗ ℚ ⟶ (Λ ⊗ ℚ)/Λ ⟶ 0.

Because det(α) ≠ 0, the middle vertical map is an isomorphism. Therefore the snake lemma gives an isomorphism

Ker(α̃) → Coker(α),

and it is easy to see that Coker(α) is finite with order equal to det(α) (especially if the map is given by a diagonal matrix).

Let V be a real vector space. To give the structure of a complex vector space on V (compatible with its real structure), it suffices to give an ℝ-linear map J : V → V such that J^{2} = −1.

The map J extends by linearity to V ⊗_{ℝ} ℂ, and V ⊗_{ℝ} ℂ splits as a direct sum

V ⊗_{ℝ} ℂ = V^{+} ⊕ V^{−},

V^{±} the ±1 eigenspaces of J.

Proposition 11.2. (a) The map

V ⟶^{v↦v⊗1} V ⊗_{ℝ} ℂ ⟶^{project} V^{+}

is an isomorphism of complex vector spaces.

(b) Denote by w ↦ w̄ the map v ⊗ z ↦ v ⊗ z̄ : V ⊗_{ℝ} ℂ → V ⊗_{ℝ} ℂ; this is an ℝ-linear involution of V ⊗_{ℝ} ℂ interchanging V^{+} and V^{−}.

Proof. Easy exercise.

Corollary 11.3. Let α be an endomorphism of V that is ℂ-linear. Write A for the matrix of α regarded as an endomorphism of V, and A_{1} for the matrix of α as a ℂ-linear endomorphism of V. Then

A ∼ A_{1} ⊕ Ā_{1}.

(By this I mean that the matrix A is similar to the matrix ( \begin{matrix}A_{1} & 0 \\ 0 & Ā_{1} \end{matrix} )

Proof. Follows immediately from the above Proposition. [In the case that V has dimension 2, we can identify V (as a real or complex vector space) with ℂ; for the map “multiplication by α = a + ib” the statement becomes,

( \begin{matrix}a & −b \\ b & a \end{matrix} ) ∼ ( \begin{matrix}a + ib & 0 \\ 0 & a − ib \end{matrix} ) ,

which is obviously true because the two matrices are semisimple and have the same trace and determinant.]

The zeta function of a curve over a finite field. The next theorem summarizes what is known.

Theorem 11.4. Let C be a complete nonsingular curve of genus g over 𝔽_{q}. Let N_{n} be the number of points of C with coordinates in 𝔽_{q^{n}}. Then there exist algebraic integers α_{1}, α_{2}, …, α_{2g} (independent of n) such that

N_{n} = 1 + q^{n} − ∑_{i=1}^{2g} α_{i}^{n}; (∗)

moreover, the numbers q/α_{i}^{−1} are a permutation of the α_{i}, and for each i, |α_{i}| = q^{1/2}.

All but the last of these assertions follow in a straightforward way from the Riemann-Roch theorem (see M. Eichler, Introduction to the Theory of Algebraic Numbers and Function, Academic Press, 1966, V.5.1). The last is the famous “Riemann hypothesis” for curves, proved in this case by Weil in the 1940’s.

Define Z(C, t) to be the power series with rational coefficients such that

log Z(C, t) = ∑_{n=1}^{∞} N_{n}t^{n}/n.

Then (*) is equivalent to the formula

Z(C, t) = \frac{(1 − α_{1}t) ⋯ (1 − α_{n}t)}{(1 − t)(1 − qt)}

(because − log(1 − at) = ∑ a^{n}t^{n}/n).

Define ζ(C, s) = Z(C, q^{−s}). Then the “Riemann hypothesis” is equivalent to ζ(C, s) having all its zeros on the line ℜ(s) = 1/2, whence its name. One can show that ζ(C, s) = ∏_{x∈C} \frac{1}{(1−ℕx^{−s})}, where ℕx is the number of elements in the residue field at x, and so the definition of ζ(C, s) is quite similar to that of ζ(ℚ, s).

The zeta function of a curve over ℚ. Let C be a complete nonsingular curve over ℚ. For all but finitely many primes p, the reduction C(p) of C modulo p will be a complete nonsingular curve over 𝔽_{p}. We call the primes for which this is true the “good primes” for C and the remainder the “bad primes”. We set

ζ(C, s) = ∏_{p} ζ_{p}(C, s)

where ζ_{p}(C, s) is the zeta function of C(p) when p is a good prime and is as defined in (Serre, Seminaire DPP 1969/70; Oeuvres, Vol II, pp 581–592) when p is a bad prime.

On comparing the expansion of ζ(C, s) as a Dirichlet series with ∑ n^{−s} and using the Riemann hypothesis, one finds that ζ(C, s) converges for ℜ(s) > 3/2. It is conjectured that it can be analytically continued to the entire complex plane except for simple poles at the negative integers, and that it satisfies a functional equation relating ζ(s) to ζ(2 − s). Note that we can write

ζ(C, s) = \frac{ζ(s)ζ(s − 1)}{L(C, s)}

where

L(C, s) = ∏_{p} \frac{1}{(1 − α_{1}(p)p^{−s}) ⋯ (1 − α_{2g}(p)p^{−s})}.

For an elliptic curve E over ℚ, there is a pleasant geometric definition of the factors of L(E, s) at the bad primes. Choose a Weierstrass minimal model for E, and reduce it mod p. If E(p) has a node at which each of the two tangents are rational over 𝔽_{p}, then the factor is (1 − p^{−s})^{−1}; if E(p) has a node at which the tangents are not separately rational over 𝔽_{p} (this means that the tangent cone is a homogeneous polynomial of degree two variables with coefficients in 𝔽_{p} that does not factor over 𝔽_{p}), then the factor is (1 + p^{−s})^{−1}; if E(p) has a cusp, then the factor is 1.

The geometric conductor of E is defined to be

N = ∏_{p} p^{f_{p}}

where f_{p} = 0 if E has good reduction at p, f_{p} = 1 if E(p) has a node as its only singularity, and f_{p} ≥ 2 if E(p) has a cusp (with equality unless p = 2, 3). Write

Λ(s) = (2π)^{−s}Γ(s)L(E, s).

Then it is conjectured that Z(s) can be analytically continued to the entire complex plane as a holomorphic function, and satisfies the functional equation:

Λ(s) = ±N^{1−s}Λ(2 − s).

More generally, let m > 0 be a prime not dividing N and let χ be primitive character of (ℤ/mℤ)^{×}. If

L(E, s) = ∑ c_{n}n^{−s},

we define

L_{χ}(E, s) = ∑ c_{n}χ(n)n^{−s},

and

Λ_{χ}(E, s) = (m/2π)^{s}Γ(s)L_{χ}(E, s).

It is conjectured that Λ_{χ}(E, s) can be analytically continued to the whole complex plane as a holomorphic function, and that it satisfies the functional equation

Λ_{χ}(E, s) = ±(g(χ)χ(−N)/g(χ̄))N^{1−s}Λχ̄(E, 2 − s)

where

g(χ) = ∑_{n=1}^{m} χ(n)e^{2πin/m}.

Review of elliptic curves. (See also Math 679 or, for complete details, Silverman 1986.) Let E be an elliptic curve over an algebraically closed field k, and let A = End(E). Then A ⊗_{ℤ} ℚ is ℚ, an imaginary quadratic field, or a quaternion algebra over ℚ (the last case only occurs when k has characteristic p ≠ 0, and then only for supersingular elliptic curves).

Because E has genus 1, the map ∑ n_{i}[P_{i}] ↦ ∑ n_{i}P_{i} : Div^{0}(E) → E(k) defines an isomorphism J(k) → E(k).

Here A is the full ring of correspondences of E. Certainly, any element of A can be regarded as a correspondence on E. Conversely a correspondence

E ← Y → E

defines a map E(k) → E(k), and it is easy to see that this is regular.

There are three natural representations of A.

First, let W = Tgt_{0}(E). This is a one-dimensional vector space over k. Since every element α of A fixes 0, α defines an endomorphism dα of W. We therefore obtain a homomorphism ρ : A → End(W).

Next, for any prime ℓ ≠ char(k), the Tate module T_{ℓ}E of E is a free ℤ_{ℓ}-module of rank 2. We obtain a homomorphism ρ_{ℓ} : A → End(T_{ℓ}E).

Finally, when k = ℂ, H_{1}(E, ℤ) is a free ℤ-module of rank 2, and we obtain a homomorphism ρ_{B} : A → End(H_{1}(E, ℤ)).

Proposition 11.5. When k = ℂ,

ρ_{B} ⊗ ℤ_{ℓ} ∼ ρ_{ℓ}, ρ_{B} ⊗ ℂ ∼ ρ ⊕ ρ̄.

(By this I mean that they are isomorphic as representations; from a more down-to-earth point of view, this means that if we choose bases for the various modules, then

the matrix (ρ_{B}(α)) is similar to (ρ_{ℓ}(α)) and to ( \begin{matrix}ρ(α) & 0 \\ 0 & ρ̄(α) \end{matrix} ) for all α ∈ A.)

Proof. Write E = ℂ/Λ. Then ℂ is the universal covering space of E and Λ is the group of covering transformations. Therefore Λ = π_{1}(E, 0). From algebraic topology, we know that H_{1} is the maximal abelian quotient of π_{1}, and so (in this case), H_{1}(E, ℤ) ≅ π_{1}(E, 0) ≅ Λ (canonical isomorphisms).

The map ℂ → E defines an isomorphism ℂ → Tgt_{0}(E). But Λ is a lattice in ℂ (regarded as a real vector space), which means that the canonical map Λ ⊗_{ℤ} ℝ → ℂ is an isomorphism. Now the relation ρ_{B} ∼ ρ ⊕ ρ̄ follows from (11.3).

Next, note that the group of points of order ℓ^{N} on E, E_{ℓ^{N}}, is equal to ℓ^{−N}Λ/Λ. There are canonical isomorphisms

Λ ⊗_{ℤ} (ℤ/ℓ^{N}ℤ) = Λ/ℓ^{N}Λ ⟶^{ℓ^{−N}} ℓ^{−N}Λ/Λ = E_{ℓ^{N}} .

When we pass to the inverse limit, these isomorphisms give an isomorphism Λ ⊗ ℤ_{ℓ} ≅ T_{ℓ}E.

Remark 11.6. There is yet another representation of A. Let Ω^{1}(E) be the space of holomorphic differentials on E. It is a one-dimensional space over k. Moreover, there is a canonical pairing

Ω^{1}(E) × Tgt_{0}(E) → k.

This is nondegenerate. Therefore the representation of A on Ω^{1}(E) is the transpose of the representation on Tgt_{0}(E). Since both representations are one-dimensional, this means that they are equal.

Proposition 11.7. For any nonzero endomorphism α of E, the degree of α is equal to det(ρ_{ℓ}α).

Proof. Suppose first that k = ℂ, so that we can identify E(ℂ) with ℂ/Λ. Then E(ℂ)_{tors} = (Λ ⊗ ℚ)/Λ, and (11.1) shows that the kernel of the map E(ℂ)_{tors} → E(ℂ)_{tors} defined by α is finite and has order equal to det(ρ_{B}(α)). But the order of the kernel is deg(α) and (11.5) shows that det(ρ_{B}(α)) = det(ρ_{ℓ}(α)).

For the case of a general k, see Silverman 1986, V, Proposition 2.3.

Corollary 11.8. Let E be an elliptic curve over 𝔽_{p}; then the numbers α_{1} and α_{2} occurring in (11.4) are the eigenvalues of Π_{p} acting on T_{ℓ}E for any ℓ ≠ p.

Proof. The elements of E(𝔽_{q}) are exactly the elements of E(𝔽̄_{p}) that are fixed by Π_{q} =^{df} Π_{p}^{n}, i.e., E(𝔽_{q}) is the kernel of the endomorphism Π_{p}^{n} − 1. This endomorphism is separable (Π_{p} obviously acts as zero on the tangent space), and so

N_{n} = deg(Π_{p}^{n} − 1) = det(ρ_{ℓ}(Π_{p})) = (α_{1}^{n} − 1)(α_{2}^{n} − 1) = q − α_{1}^{n} − α_{2}^{n} + 1.

We need one last fact.

Proposition 11.9. Let α′ be the transpose of the endomorphism α of E; then ρ_{ℓ}(α′) is the transpose of ρ_{ℓ}(α).

The zeta function of X_{0}(N): case of genus 1. When N is one of the integers 11, 14, 15, 17, 19, 20, 21, 24, 17, 32, 36, or, 49, the curve X_{0}(N) has genus 1. Recall (discussion before Theorem 6.1) that the number of cusps^{5} of Γ_{0}(N) is ∑_{d|N} φ(d, N/d). If N is prime, then there are two cusps, 0 and i∞, and they are both rational over ℚ. If N is one of the above values, and we take i∞ to be the zero element of X_{0}(N), then it becomes an elliptic curve over ℚ.

Lemma 11.10. There is a natural one-to-one correspondence between the cusp forms of weight 2 for Γ_{0}(N) and the holomorphic differential forms X_{0}(N) (over ℂ).

5For a description of the cusps on X_{0}(N) and their fields of rationality, see Ogg, Rational points on certain elliptic modular curves, Proc. Symp. P. Math, 24, AMS, 1973, 221-231.

Proof. We know that f ↦ fdz gives a one-to-one correspondence between the meromorphic modular forms of weight 2 for Γ_{0}(N) and the meromorphic differentials on X_{0}(N), but Lemma 4.11 shows that the cusp forms correspond to the holomorphic differential forms.

Assume X_{0}(N) has genus one. Let ω be a holomorphic differential on X_{0}(N); when we pull it back to ℍ and write it f(z)dz, we obtain a cusp form f(z) for Γ_{0}(N) of weight 2. It is automatically an eigenform for T (p) all p ∤ N, and we assume that it is normalized so that f(z) = ∑ a_{n}q^{n} with a_{1} = 1. Then T (p) · f = a_{p}f. One can show that a_{p} is real.

Now consider X̃_{0}(N), the reduction of X_{0}(N) modulo p. Here we have endomorphisms Π_{p} and Π′_{p}, and Π_{p} ∘ Π′_{p} = deg(Π_{p}) = p. Therefore

(I_{2} − ρ_{ℓ}(Π_{p})T)(I_{2} − ρ_{ℓ}(Π′_{p})T) = I_{2} − (ρ_{ℓ}(Π_{p} + Π′_{p}))T + pT^{2}.

According to the Eichler-Shimura theorem, we can replace Π_{p}+Π′_{p} by T̃(p), and since the ℓ-adic representation doesn’t change when we reduce modulo p, we can replace T̃(p) by T (p). The right hand side becomes

I_{2} − ( \begin{matrix}a_{p} & 0 \\ 0 & a_{p} \end{matrix} ) T + pT^{2}.

Now take determinants, noting that Π_{p} and Π′_{p}, being transposes, have the same characteristic polynomial. We get that

(1 − a_{p}T + pT^{2})^{2} = det(1 − Π_{p}T)^{2}.

On taking square roots, we conclude that

(1 − a_{p}T + pT^{2}) = det(1 − Π_{p}T) = (1 − α_{p}T)(1 − ᾱ_{p}T).

On replacing T with p^{−s} in this equation, we obtain the equality of the p-factors of the Euler products for the Mellin transform of f(z) and of L(X_{0}(N), s). We have therefore proved the following theorem.

Theorem 11.11. The zeta function of X_{0}(N) (as a curve over ℚ) is, up to a finite number of factors, the Mellin transform of f(z).

Corollary 11.12. The strong Hasse-Weil conjecture (see below) is true for X_{0}(N).

Proof. Apply Theorem 9.5.

Review of the theory of curves. We repeat the above discussion with E replaced by a general (projective nonsingular) curve C. Proofs can be found (at least when the ground field is ℂ) in Griffiths 1989. Let C be a complete nonsingular curve over an algebraically closed field k. Attached to C there is an abelian variety J, called the Jacobian variety of C such that

J(k) = Div^{0}(C)/{principal divisors}.

In the case that C is an elliptic curve, J = C, i.e., an elliptic curve is its own Jacobian.

When k = ℂ it is easy to define J, at least as a complex torus. As we have already mentioned, the Riemann-Roch theorem shows that the holomorphic differentials Ω^{1}(C) on C form a vector space over k of dimension g = genus of C.

Now assume k = ℂ. The map

H_{1}(C, ℤ) → Ω^{1}(C)^{∨}, γ ↦ (ω ↦ ∫_{γ} ω),

identifies H_{1}(C, ℤ) with a lattice in Ω^{1}(C)^{∨} (linear dual to the vector space Ω^{1}(C)). Therefore we have a g-dimensional complex torus Ω^{1}(C)^{∨}/H_{1}(C, ℤ). One proves that there is a unique abelian variety J over ℂ such that J(ℂ) = Ω^{1}(C)^{∨}/H_{1}(C, ℤ). (Recall that not every compact complex manifold of dimension > 1 arises from an algebraic variety.)

We next recall two very famous theorems. Fix a point P ∈ C.

Abel’s Theorem: Let P_{1}, …, P_{r} and Q_{1}, …, Q_{r} be elements of C(ℂ); then there is a meromorphic function on C(ℂ) with its poles at the P_{i} and its zeros at the Q_{i} if and only if, for any path γ_{i} from P to P_{i} and path γ′_{i} from P to Q_{i}, there exists a γ in H_{1}(C(ℂ), ℤ) such that

∑_{i=1}^{r} ∫_{γ_{i}} ω − ∑_{i=1}^{r} ∫_{γ′_{i}} ω = ∫_{γ} ω all ω.

Jacobi Inversion Formula: For any linear mapping l : Ω^{1}(C) → ℂ, there exist g points P_{1}, …, P_{g} in C(ℂ) and paths γ_{1}, …, γ_{g} from P to P_{i} such that l(ω) = ∑ ∫_{γ_{i}} ω for all ω ∈ Ω^{1}(C).

These two statements combine to show that there is an isomorphism:

∑ n_{i}P_{i} ↦ (ω ↦ ∑ n_{i} ∫_{γ_{i}} ω) : Div^{0}(C)/{principal divisors} → J(ℂ).

(The γ_{i} are paths from P to P_{i}.) The construction of J is much more difficult over a general field k. (See my second article in: Arithmetic Geometry, eds. G. Cornell and Silverman, Springer, 1986.)

The ring of correspondences A of C can be identified with the endomorphism ring of J, i.e., with the ring of regular maps α : J → J such that α(0) = 0.

Again, there are three representations of A.

First, we have a representation ρ of A on Tgt_{0}(J) = Ω^{1}(C)^{∨}. This is a vector space of dimension g over the ground field k.

Second, for any ℓ ≠ char(k), we have a representation on the Tate module T_{ℓ}(J) = \underleftarrow{lim} J_{ℓ^{n}}(k). This is a free ℤ_{ℓ}-module of rank 2g.

Third, when k = ℂ, we have a representation on H_{1}(C, ℤ). This is a free ℤ-module of rank 2.

Proposition 11.13. When k = ℂ,

ρ_{B} ⊗ ℤ_{ℓ} ∼ ρ_{ℓ}, ρ_{B} ⊗ ℂ ∼ ρ ⊕ ρ̄.

Proof. This can be proved exactly as in the case of an elliptic curve.

The rest of the results for elliptic curves extend in an obvious way to a curve C of genus g and its Jacobian variety J(C).

The zeta function of X_{0}(N): general case. Exactly as in the case of genus 1, the Eichler-Shimura theorem implies the following result.

Theorem 11.14. Let f_{1}, f_{2}, ..., f_{g} be a basis for the cusp forms of degree 2 for Γ_{0}(N), chosen to be normalized eigenforms for the Hecke operators T (p) for p prime to N. Then, apart from the factors corresponding to a finite number of primes, the zeta function of X_{0}(N) is equal to the product of the Mellin transforms of the f_{i}.

Theorem 11.15. Let f be a cusp form of weight 2, which is a normalized eigenform for the Hecke operators, and write f = ∑ a_{n}q^{n}. Then for all primes p ∤ N, |a_{p}| ≤ 2p^{1/2}.

Proof. In the course of the proof of the theorem, one finds that a_{p} = α + ᾱ where α occurs in the zeta function of the reduction of X_{0}(N) at p. Thus this follows from the Riemann hypothesis.

Remark 11.16. As discussed in Section 4, Deligne has proved the analogue of Theorem 11.15 for all weights: let f be a cusp form of weight 2k for Γ_{0}(N) and assume f is an eigenform for all the T (p) with p a prime not dividing N and that f is “new” (see below); write f = ∑_{1}^{∞} a_{n}q^{n} with a_{1} = 1; then

|a_{p}| ≤ 2p^{2k−1/2},

for all p not dividing N. The proof identifies the eigenvalues of the Hecke operator with sums of eigenvalues of Frobenius endomorphisms acting on the étale cohomology of a power of the universal elliptic curve; thus the conjecture follows from the Riemann hypothesis for such varieties. See Deligne, Sém. Bourbaki, Fév. 1969. In fact, Deligne’s paper Weil II simplifies the proof (for a few hints concerning this, see E. Freitag and R. Kiehl, Etale Cohomology and the Weil Conjecture, p278).

The Conjecture of Taniyama and Weil. Let E be an elliptic curve over ℚ. Let N be its geometric conductor. It has an L-series

L(E, s) = ∑_{n=1}^{∞} a_{n}q^{n}.

For any prime m not dividing N, and primitive character χ : (ℤ/mℤ)^{×} → ℂ^{×}, let

Λ_{χ}(E, s) = N^{s/2} (\frac{m}{2π} )^{s} Γ(s) ∑_{n=1}^{∞} a_{n}χ(n)q^{n}.

Conjecture 11.17 (Strong Hasse-Weil conjecture). For all m prime to N, and all primitive Dirichlet characters χ, Λ_{χ}(E, s) has an analytic continuation of ℂ, bounded in vertical strips, satisfying the functional equation

Λ_{χ}(E, s) = ±(g(χ)χ(−N)/g(χ̄))N^{1−s}Λχ̄(E, 2 − s)

where

g(χ) = ∑_{n=1}^{m} χ(n)e^{2πin/m}.

An elliptic curve E over ℚ is said to be modular if there is a nonconstant map X_{0}(N) → E (defined over ℚ).

Remark 11.18. Let C be a complete nonsingular curve, and fix a rational point P on C (assumed to exist). Then there is a canonical map φ_{P} : C → J(C) sending P to 0, and the map is universal: for any abelian variety A and regular map φ : C → A sending P to 0, there is a unique map ψ : J(C) → A such that ψ ∘ φ_{P} = φ. Thus to say that E is modular means that there is a surjective homomorphism J_{0}(N) → E.

Theorem 11.19. An elliptic curve E over ℚ is modular if and only if it satisfies the strong Hasse-Weil conjecture (and in fact, there is a map X_{0}(N) → E with N equal to the geometric conductor of E).

Proof. Suppose E is modular, and let ω be the Néron differential on E. The pull-back of ω to X_{0}(N) can be written f(z)dz with f(z) a cusp form of weight 2 for Γ_{0}(N), and the Eichler-Shimura theorem shows that Λ(E, s) is the Mellin transform of f. (Actually, it is not quite this simple...)

Conversely, suppose E satisfies the strong Hasse-Weil conjecture. Then according to Weil’s theorem, Λ(E, s) is the Mellin transform of a cusp form f. The cusp form has rational Fourier coefficients, and the next proposition shows that there is a quotient E′ of J_{0}(N) whose L-series is the Mellin transform of f; thus we have found a modular elliptic curve having the same zeta function as E, and a theorem of Faltings then shows that there is an isogeny E′ → E.

Theorem 11.20 (Faltings 1983). Let E and E′ be elliptic curves over ℚ. If ζ(E, s) = ζ(E′, s) then E is isogenous to E′.

Proof. See his paper proving Mordell’s conjecture (Invent. Math. 1983).

Suppose M|N; then we have a map X_{0}(N) → X_{0}(M), and hence a map J_{0}(N) → J_{0}(M). The intersection of the kernels is the “new” part of J_{0}(N), J_{0}^{new}(N).

Similarly, it is possible to define a subspace S_{0}^{new}(N) of new cusp forms of weight 2.

Proposition 11.21. There is a one-to-one correspondence between the elliptic curves over ℚ that are images of X_{0}(N) but of no X_{0}(M) with M < N and newforms for Γ_{0}(N) that are eigenforms with rational eigenvalues.

Proof. Given a “new” form f(z) = ∑ a_{n}q^{n} as in the Proposition, we define an elliptic curve E equal to the intersection of the kernels of the endomorphisms T (p)−a_{p} acting on J(X_{0}(N)). Some quotient of E by a finite subgroup will be the modular elliptic curve sought.

Conjecture 11.22 (Taniyama). Let E be an elliptic curve over ℚ with geometric conductor N. Then there is a nonconstant map X_{0}(N) → E; in particular, every elliptic curve over ℚ is a modular elliptic curve.

We have proved the following.

Theorem 11.23. The strong Hasse-Weil conjecture for elliptic curves over ℚ is equivalent to the Taniyama-Weil conjecture.

Conjecture 6.22 was suggested (a little vaguely) by Taniyama ^{6} in 1955, and promoted by Shimura. Weil gave rather compelling evidence for it.

Notes. There is a vast literature on the above questions. The best introduction to it is: Elliptic curves and modular functions, H.P.F. Swinnerton-Dyer and B.J. Birch, in Modular Functions of One Variable IV, (eds. Birch and Kuyk), SLN 476, QA343.M72 v.4, pp 2–32. See also: Manin, Parabolic points and zeta-functions of modular curves, Math. USSR 6 (1972), 19–64.

Fermat’s last theorem.

Theorem 11.24. The Taniyama conjecture implies Fermat’s last theorem.

Idea: It is clear that the Taniyama conjecture restricts the number of elliptic curves over ℚ that there can be with small conductor. For example, X_{0}(N) has genus zero for N = 1, 2, 3, ..., 10, 12, 13, 16, 18, 25 and so for these values, the Taniyama-Weil conjecture implies that there can be no elliptic curve with this conductor. (Tate showed a long time ago that there is no elliptic curve over ℚ with conductor 1, that is, with good reduction at every prime.)

More precisely, the one proves the following:

Theorem 11.25. Let p be a prime > 2, and suppose that

a^{p} − b^{p} = c^{p}

with a, b, c all nonzero integers and gcd(a, b, c) = 1. Then the elliptic curve

E : Y^{2} = X(X − a^{p})(X + b^{p})

is not a modular elliptic curve.

Proof. We can assume that p > 163; moreover that 2|b and a ≡ 3 mod 3. An easy calculation shows that the curve has bad reduction exactly at the primes p dividing abc, and at each such prime the reduced curve has a node. Thus the geometric conductor is a product of the primes dividing abc.

Suppose that E is a Weil curve. There is a weight 2 cusp form for Γ_{0}(N) with integral q-expansion, and Ribet proves that there is a cusp form of weight 2 for Γ_{0}(2) such that f ≡ f′ modulo ℓ. But X_{0}(2) has genus zero, and so there are no cusp forms of weight 2.

Remark 11.26. Ribet’s proof is very intricate; it involves a delicate interplay between three primes ℓ, p, and q, which is one more than most of us can keep track of (Ribet, On modular representations of Gal(ℚ̄/ℚ) arising from modular forms, Invent. Math 100 (1990), 431–476). As far as I know, the idea of using the elliptic curve in (11.25) to attempt to prove Fermat’s last theorem is due to G. Frey. He has published many talks about it, see for example, Frey, Links between solutions of A − B = C and elliptic curves, in Number Theory, Ulm 1987, (ed. H. Schlickewei and Wirsing), SLN 1380.

6Taniyama was a very brilliant Japanese mathematician who was the main founder of the theory of complex multiplication of abelian varietes of dimension > 1. He killed himself in late 1958, shortly after his 31st birthday.

Application to the conjecture of Birch and Swinnerton-Dyer. Recall (Silverman 1986) that, for an elliptic curve E over ℚ, the conjecture of Birch and Swinnerton-Dyer predicts that

lim_{s→1}(s − 1)^{−r}L(E, s) = \frac{Ω ∏_{p} c_{p}[TS(E/ℚ)]R(E/ℚ)}{[E(ℚ)_{tors}]^{2}}

where r = rank(E(ℚ)), Ω = ∫_{E(ℝ)} |ω| where ω is the Néron differential on E, the product of the c_{p} is over the bad primes, TS is the Tate-Shafarevich group of E, and R(E/ℚ) is the discriminant of the height pairing.

Now suppose E is a modular elliptic curve. Put the equation for E in Weierstrass minimal form, and let ω = dx/(2y + a_{1}x + a_{3}) be the Néron differential. Assume α^{∗}ω = fdz, for f(z) a newform for Γ_{0}(N). Then L(E, s) is the Mellin transform of f(z). Write f(z) = c(q + a_{2}q^{2} + ...)q^{−1}dq, where c is a positive rational number. Conjecturally c = 1, and so I drop it.

Assume that i∞ maps to 0 ∈ E. Then q is real for z on the imaginary axis between 0 and i∞. Therefore j(z) and j(Nz) are real, and, as we explained (end of Section 8) this means that the image of the imaginary axis in X_{0}(N)(ℂ) is in X_{0}(N)(ℝ), i.e., the points in the image of the imaginary axis have real coordinates.

The Mellin transform formula (cf. 9.2) implies that

L(E, 1) = Γ(1)L(E, 1) = ∫_{0}^{i∞} f(z)dz.

Define M by the equation

∫_{0}^{i∞} f(z)dz = M · ∫_{E(ℝ)} ω.

Intuitively at least, M is the winding number of the map from the imaginary axis from 0 to i∞ onto E(ℝ). The image of the point 0 in X_{0}(N) is known to be a point of finite order, and this implies that the winding number is a rational number. Thus, for a modular curve (suitably normalized), the conjecture of Birch and Swinnerton-Dyer can be restated as follows.

Conjecture 11.27 (Birch and Swinnerton-Dyer). (a) E(ℚ) is infinite if and

only if M = 0.

(b) If M ≠ 0, then M[E(ℚ)]^{2} = [TS(E/ℚ)] ∏_{p} c_{p}.

Remark 11.28. Some remarkable results have been obtained in this context by Kolyvagin and others. (See: Rubin, The work of Kolyvagin on the arithmetic of elliptic curves, SLN 1399, MR 90h:14001), and the papers of Kolyvagin.)

More details can be found in the article of Birch and Swinnerton-Dyer mentioned above. Winding numbers and the mysterious c are discussed in Mazur and Swinnerton-Dyer, Inventiones math., 25, 1-61, 1974. See also the article of Manin mentioned above and my Math 679 notes.

12. Complex Multiplication for Elliptic Curves

The theory of complex multiplication is not only the most beautiful part of mathematics but also of the whole of science. D. Hilbert.

It was known to Gauss that ℚ[ζ_{n}] is an abelian extension of ℚ. Towards the end of the 1840’s Kronecker had the idea that cyclotomic fields, and their subfields, exhaust the abelian extensions of ℚ, and furthermore, that every abelian extension of a quadratic imaginary number field E = ℚ[\sqrt{−d}] is contained in the extension given by adjoining to E roots of 1 and certain special values of the modular function j. Many years later, he was to refer to this idea as the most cherished dream^{7} of his youth (mein liebster Jugendtraum) (Kronecker, Werke, V, p435).

Abelian extensions of ℚ. Let ℚ^{cyc} = ∪ℚ[ζ_{n}]; it is a subfield of the maximal abelian extension ℚ^{ab} of ℚ.

Theorem 12.1 (Kronecker-Weber). The field ℚ^{cyc} = ℚ^{ab}.

The proof has two steps.

Elementary part. Note that there is a homomorphism

χ : Gal(ℚ[ζ_{n}]/ℚ) → (ℤ/nℤ)^{×}, σζ = ζ^{χ(m)},

which is obviously injective. Proving that it is surjective is equivalent to proving that the cyclotomic polynomial

Φ_{n}(X) =^{df} ∏_{(m.n)=1}(X − ζ^{m})

is irreducible in ℚ[X], or that Gal(ℚ[ζ_{n}]/ℚ) acts transitively on the primitive n^{th} roots of 1. One way of doing this is to look modulo p, and exploit the Frobenius map (see Math 594f, 5.9).

Application of class field theory. For any abelian extension F of ℚ, class field theory provides us with a surjective homomorphism (the Artin map)

ϕ : 𝕀 → Gal(F/ℚ)

where 𝕀 is the group of idèles of ℚ (see Math 776). When we pass to the inverse limit over all F ’s, then we obtain an exact sequence

1 → (ℚ^{×} · ℝ^{+})^{−} → 𝕀 → Gal(ℚ^{ab}/ℚ) → 1

where ℝ^{+} = {r ∈ ℝ | r > 0}, and the bar denotes the closure.

Consider the homomorphisms

𝕀 → Gal(ℚ^{ab}/ℚ) → Gal(ℚ^{cyc}/ℚ) ⟶^{χ} \underleftarrow{lim}(ℤ/mℤ)^{×} = ℤ̂^{×}.

All maps are surjective. In order to show that the middle map is an isomorphism, we have to prove that the kernel of 𝕀 → ℤ̂^{×} is (ℚ^{×} · ℝ^{+})^{−}; it clearly contains (ℚ^{×} · ℝ^{+})^{−}.

Note that ℤ̂ = ∏ ℤ_{ℓ}, and that ℤ̂^{×} = ∏ ℤ_{ℓ}^{×}. There is therefore a canonical embedding i : ℤ̂ ↪ 𝕀, and to complete the proof of the theorem, it suffices to show:

7See a series of articles in preparation by N. Schappacher for a complete and very careful treatment of Kronecker’s Jugendtraum and the work that grew out of it.

(i) the composite ℤ̂^{×} ⟶^{i}𝕀 → ℤ̂^{×} is the identity map; (ii) (ℚ^{×} · ℝ^{+})^{−} · i(ℤ̂^{×}) = 𝕀.

Assume these statements, and let α ∈ 𝕀. Then (ii) says that α = a · i(z) with a ∈ (ℚ^{×} · ℝ^{+})^{−} and z ∈ ℤ^{×}, and (i) shows that φ(a · z) = z. Thus, if α ∈ Ker(φ), then z = 1, and α ∈ (ℚ^{×} · ℝ^{+})^{−}.

The proofs of (i) and (ii) are left as an exercise (see Math 776, V.5.9).

Alternative: Find the kernel of ϕ : (𝔸^{×}/ℚ^{×}) → Gal(ℚ[ζ_{n}]/ℚ), and show that every open subgroup of finite index contains such a subgroup.

Alternative: For a proof using only local (i.e., not global) class field theory, see Math 776, I.4.16.

Orders in K. Let K be a quadratic imaginary number field. An order of K is a subring R containing ℤ and free of rank 2 over ℤ. Clearly every element of R is integral over ℤ, and so R ⊂ O_{K} (ring of integers in K). Thus O_{K} is the unique maximal order.

Proposition 12.2. Let R be an order in K. Then there is a unique integer f > 0 such that R = ℤ + f · O_{K}. Conversely, for any integer f > 0, ℤ + f · O_{K} is an order in K.

Proof. Let {1, α} be a ℤ-basis for O_{K}, so that O_{K} = ℤ + ℤα. Then R ∩ ℤα is a subgroup of ℤα, and hence equals ℤαf for some positive integer f. Now ℤ + fO_{K} ⊂ ℤ + ℤαf ⊂ R. Conversely, if m + nα ∈ R, m, n ∈ ℤ, then nα ∈ R, and so n ∈ fℤ. Thus, m + nα ∈ ℤ + fαℤ ⊂ ℤ + fO_{K}.

The number f is called the conductor of R. We often write R_{f} for ℤ + f · O_{K}.

Proposition 12.3. Let R be an order in K. The following conditions on an R-submodule 𝔞 of K are equivalent:

(a) 𝔞 is a projective R-module;

(b) R = {a ∈ K | a · 𝔞 ⊂ 𝔞};

(c) 𝔞 = x · O_{K} for some x ∈ 𝕀 (this means that for all primes v of O_{K}, 𝔞 · O_{v} = x_{v} · O_{v}).

Proof. For (b) ⟹ (c), see Shimura 1971, (5.4.2), p 122.

Such an R-submodule of K is called a proper R-ideal. A proper R-ideal of the form αR, α ∈ K^{×}, is said to be principal. If 𝔞 and 𝔟 are two proper R-ideals, then

𝔞 · 𝔟 =^{df} {∑ a_{i}b_{i} | a_{i} ∈ 𝔞, b_{i} ∈ 𝔟}

is again a proper R-ideal.

Proposition 12.4. For any order R in K, the proper R-ideals form a group with respect to multiplication, with R as the identity element.

Proof. Shimura 1971, Proposition 4.11, p105.

The class group Cl(R) is defined to be the quotient of the group of proper R-ideals by the subgroup of principal ideals. When R is the full ring of integers in E, then Cl(R) is the usual class group.

Remark 12.5. The class number of R is

h(R) = h · f · (O_{K}^{×} : R^{×})^{−1} · ∏_{p|f} (1 − (\frac{K}{p} ) p^{−1})

where h is the class number of O_{K}, and (\frac{K}{p}) = 1, −1, 0 according as p splits in K, stays prime, or ramifies. (If we write {±1} for the Galois group of K over ℚ, then p ↦ (\frac{K}{p}) is the reciprocity map.) See Shimura 1971, Exercise 4.12.

Elliptic curves over ℂ. For any lattice Λ in ℂ, the Weierstrass ℘ and ℘′ functions realize ℂ/Λ as an elliptic curve E(Λ), and every elliptic curve over ℂ arises in this way. If Λ and Λ′ are two lattices, and α is an element of ℂ such that αΛ ⊂ Λ′, then [z] ↦ [αz] is a homomorphism E(Λ) → E(Λ′), and every homomorphism is of this form; thus

Hom(E(Λ), E(Λ′)) = {α ∈ ℂ | αΛ ⊂ Λ′}.

In particular, E(Λ) ≈ E(Λ′) if and only if Λ′ = αΛ for some α ∈ ℂ^{×}.

These statements reduce much of the theory of elliptic curves over ℂ to linear algebra. For example, End(E) is either ℤ or an order R in a quadratic imaginary field K. Consider E = E(Λ); if End(E) ≠ ℤ, then there is an α ∈ ℂ, α ∉ ℤ, such that αΛ ⊂ Λ, and

End(E) = {α ∈ ℂ | αΛ ⊂ Λ},

which is an order in ℚ[α] having Λ as a proper ideal.

When End(E) = R ≠ ℤ, we say E has complex multiplication by R.

Write E = E(Λ), so that E(ℂ) = ℂ/Λ. Clearly E_{n}(ℂ), the set of points of order dividing n on E, is equal to n^{−1}Λ/Λ, and so it is a free ℤ/nℤ-module of rank 2. The inverse limit, T_{ℓ}E =^{df} \underleftarrow{lim} E_{ℓ^{m}} = \underleftarrow{lim} ℓ^{−m}Λ/Λ = Λ ⊗ ℤ_{ℓ}, and so V_{ℓ}E = Λ ⊗ ℚ_{ℓ}.

Algebraicity of j. When R is an order in a quadratic imaginary field K ⊂ ℂ, we write Ell(R) for the set of isomorphism classes of elliptic curves over ℂ with complex multiplication by R.

Proposition 12.6. For each proper R-ideal 𝔞, E(𝔞) =^{df} ℂ/𝔞 is an elliptic curve with complex multiplication by R, and the map 𝔞 ↦ ℂ/𝔞 induces a bijection

Cl(R) → Ell(R).

Proof. If 𝔞 is a proper R-ideal, then

End(E(𝔞)) = {α ∈ ℂ | α𝔞 ⊂ 𝔞} (see above)
= {α ∈ K | α𝔞 ⊂ 𝔞} (easy)
= R (definition of proper R-ideal).

Since E(α ·𝔞) ≈ E(𝔟) we get a well-defined map Cl(R) → Ell(R). Similar arguments show that it is bijective.

Corollary 12.7. Up to isomorphism, there are only finitely many elliptic curves over ℂ with complex multiplication by R; in fact there are exactly h(R).

With an elliptic curve E over ℂ, we can associate its j-invariant j(E) ∈ ℂ, and E ≈ E′ if and only if j(E) = j(E′). For an automorphism σ of ℂ, we define σE to be the curve obtained by applying σ to the coefficients of the equation defining E. Clearly j(σE) = σj(E).

Theorem 12.8. If E has complex multiplication then j(E) is algebraic.

Proof. Let z ∈ ℂ. If z is algebraic (meaning algebraic over ℚ), then z has only finitely many conjugates, i.e., as σ ranges over the automorphisms of ℂ, σz ranges over a finite set. The converse of this is also true: if z is transcendental, then σz takes on uncountably many different values (if z′ is any other transcendental number, there is an isomorphism ℚ[z] → ℚ[z′] which can be extended to an automorphism of ℂ).

Now consider j(E). As σ ranges over ℂ, σE ranges over finitely many isomorphism classes, and so σj(E) ranges over a finite set. This shows that j(E) is algebraic.

Corollary 12.9. Let j be the (usual) modular function for Γ(1), and let z ∈ ℍ be such that ℚ[z] is a quadratic imaginary number field. Then j(z) is a algebraic.

Proof. The function j is defined so that j(z) = j(E(Λ)), where Λ = ℤ + ℤz. Suppose ℚ[z] is a quadratic imaginary number field. Then

{α ∈ ℂ | α(ℤ + ℤz) ⊂ ℤ + ℤz}

is an order R in ℚ[z], and E(Λ) has complex multiplication by R, from which the statement follows.

The integrality of j. Let E be an elliptic curve over a field k and let R be an order in a quadratic imaginary number field K. When we are given a isomorphism i : R → End(E), we say that E has complex multiplication by R (defined over k). Then R and ℤ_{ℓ} act on T_{ℓ}E, and therefore R ⊗_{ℤ} ℤ_{ℓ} acts on T_{ℓ}E; moreover, K ⊗_{ℚ} ℚ_{ℓ} acts on V_{ℓ}E =^{df} T_{ℓ}E ⊗ ℚ. These actions commute with the actions of Gal(k^{al}/k) on the modules.

Let α be an endomorphism of an elliptic curve E over a field k. Define,

Tr(α) = 1 + deg(α) − deg(1 − α) ∈ ℤ,

and define the characteristic polynomial of α to be

f_{α}(X) = X^{2} − Tr(α)X + deg(α) ∈ ℤ[X].

Proposition 12.10. (a) The endomorphism f_{α}(α) of E is zero.

(b) For all ℓ ≠ char(k), f_{α}(X) is the characteristic polynomial of α acting on V_{ℓ}E.

Proof. Part (b) is proved in Silverman 1986, 2.3, p134. Part (a) follows from (b), the Cayley-Hamilton theorem, and the fact that the End(E) acts faithfully on V_{ℓ}E (Silverman 1986, 7.4, p92).

Corollary 12.11. If E has complex multiplication by R ⊂ K, then V_{ℓ}E is a free K ⊗ ℚ_{ℓ}-module.

Proof. When the ground field k = ℂ, this is obvious because V_{ℓ}E = Λ ⊗_{ℤ} ℚ_{ℓ}, and Λ ⊗_{ℤ} ℚ_{ℓ} = (Λ ⊗_{ℤ} ℚ) ⊗_{ℚ} ℚ_{ℓ} = K ⊗_{ℚ} ℚ_{ℓ}. When K ⊗_{ℚ} ℚ_{ℓ} is a field, it is again obvious (every module over a field is free). Otherwise K ⊗_{ℚ} ℚ_{ℓ} = K_{v} ⊕ K_{w} where v and w are the primes of K lying over p, and we have to see that V_{ℓ}E is isomorphic to the K ⊗ ℚ_{ℓ}-module K_{v} ⊕ K_{w} (rather than K_{v} ⊕ K_{v} for example). But for α ∈ K, α ∉ ℚ, the proposition shows that characteristic polynomial of α acting on V_{ℓ}E is the minimum polynomial of α over K, and this implies what we want.

Remark 12.12. In fact T_{ℓ}E is a free R ⊗ ℤ_{ℓ}-module (see J-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. 88, 1968, pp 492-517, p502).

Proposition 12.13. The action of Gal(k^{al}/k) on V_{ℓ}E factors through K ⊗ ℚ_{ℓ}, i.e., there is a homomorphism ρ_{ℓ} : Gal(k^{al}/k) → (K ⊗ ℚ_{ℓ})^{×} such that

ρ_{ℓ}(σ) · x = σx, all σ ∈ Gal(k^{al}/k), x ∈ V_{ℓ}A.

Proof. The action of Gal(k^{al}/k) on V_{ℓ}E commutes with the action R (because we are assuming that the action of R is defined over k). Therefore the image of Gal(k^{al}/k) lies in End_{K⊗ℚ_{ℓ}}(V_{ℓ}E), which equals K ⊗ ℚ_{ℓ}, because V_{ℓ}E is free K ⊗ ℚ_{ℓ}-module of rank 1.

In particular, we see that the image of ρ_{ℓ} is abelian, and so the action of Gal(k^{al}/k) factors through Gal(k^{ab}/k)—all the ℓ^{m}-torsion points of E are rational over k^{ab} for all m. As Gal(k^{al}/k) is compact, Im(ρ_{ℓ}) ⊂ O_{ℓ}^{×}, where O_{ℓ} is the ring of integers in K ⊗_{ℚ} ℚ_{ℓ} (O_{ℓ} is either a complete discrete valuation ring or the product of two such rings).

Theorem 12.14. Let E be an elliptic curve over a number field k having complex multiplication by R over k. Then E has potential good reduction at every prime v of k.

Proof. Let ℓ be a prime number not divisible by v. According to Silverman 1986, VII.7.3, p186, we have to show that the action of the inertia group I_{v} at v on T_{ℓ}A factors through a finite quotient. But we know that it factors through the inertia subgroup J_{v} of Gal(k^{ab}/k), and class field theory tells us that there is a surjective map

O_{v}^{×} → J_{v}

where O_{v} is the ring of integers in k_{v}. Thus we obtain a homomorphism

O_{v}^{×} → J_{v} → O_{ℓ}^{×} ⊂ Aut(T_{ℓ}E),

where O_{ℓ} is the ring of integers in K⊗ℚ_{ℓ} . I claim that any homomorphism O_{v}^{×} → O_{ℓ}^{×} automatically factors through a finite quotient. In fact algebraic number theory shows that O_{v}^{×} has a subgroup U^{1} of finite index which is a pro-p-group, where p is the prime lying under v. Similary, O_{ℓ}^{×} has a subgroup of finite index V which is a pro-ℓ-group. Any map from a pro-p group to a pro-ℓ-group is zero, and so Ker(U^{1} → O^{×}) = Ker(U^{1} → O^{×}/V), which shows that the homomorphism is zero on a subgroup of finite index of U^{1}.

Corollary 12.15. If E is an elliptic curve over a number field k with complex multiplication, then j(E) ∈ O_{K}.

Proof. An elementary argument shows that, if E has good reduction at v, then j(E) ∈ O_{v} (cf. Silverman 1986, VII.5.5, p181).

Corollary 12.16. Let j be the (usual) modular function for Γ(1), and let z ∈ ℍ be such that ℚ[z] is a quadratic imaginary number field. Then j(z) is an algebraic integer.

Remark 12.17. There are analytic proofs of the integrality of j(E), but they are less illuminating.

Statement of the main theorem (first form). Let K be a quadratic imaginary number field, with ring of integers O_{K}, and let Ell(O_{K}) be the set of isomorphism classes of elliptic curves over ℂ with complex multiplication by O_{K}. For any fractional O_{K}-ideal Λ in K, we write j(Λ) for j(ℂ/Λ). (Thus if Λ = ℤω_{1} + ℤω_{2} where z = ω_{1}/ω_{2} lies in the upper half plane, then j(Λ) = j(z), where j(z) is the standard function occurring in the theory of elliptic modular functions.)

Theorem 12.18. (a) For any elliptic curve E over ℂ with complex multipli-

cation by O_{K}, K[j(E)] is the Hilbert class field K^{hcf} of K.

(b) The group Gal(K^{hcf}/K) permutes the set {j(E) | E ∈ Ell(O_{K})} transitively.

(c) For each prime ideal 𝔭 of K, Frob(𝔭)(j(Λ)) = j(Λ · 𝔭^{−1}).

The proof will occupy the next few subsections.

The theory of 𝔞-isogenies. Let R be an order in K, and let 𝔞 be a proper ideal in R. For an elliptic curve E over a field k with complex multiplication by R, we define

Ker(𝔞) = ∩_{a∈𝔞} Ker(a : E → E).

Note that if 𝔞 = (a_{1}, …, a_{n}), then Ker(𝔞) = ∩ Ker(a_{i} : E → E). Let Λ be a proper R-ideal, and consider the elliptic curve E(Λ) over ℂ. Then Λ · 𝔞^{−1} is also a proper ideal.

Lemma 12.19. There is a canonical map E(Λ) → E(Λ · 𝔞^{−1}) with kernel Ker(𝔞).

Proof. Since Λ ⊂ Λ · 𝔞^{−1}, we can take the map to be z + Λ ↦ z + Λ · 𝔞^{−1}.

Proposition 12.20. Let E be an elliptic curve over k with complex multiplication by R, and let 𝔞 be a proper ideal in R. Assume k has characteristic zero. Then there is an elliptic curve 𝔞 · E and a homomorphism map φ_{𝔞} : E → 𝔞 · E whose kernel is Ker(𝔞). The pair (𝔞 · E, φ_{𝔞}) has the following universal property: for any homomorphism φ : E → E′ with Ker(φ) ⊃ Ker(𝔞), there is a unique homomorphism ψ : 𝔞 · E → E′ such that ψ ∘ φ_{𝔞} = φ.

Proof. When k = ℂ, we write E = E(Λ) and take 𝔞 · E = E(Λ · 𝔞^{−1}). If k is a field of characteristic zero, we define 𝔞 · E = E(Λ · 𝔞^{−1}) (see Silvermann 1986, 4.12, 4.13.2, p78).

We want to extend the definition of 𝔞 · E to the case where k need not have characteristic zero. For this, we define 𝔞 · E to be the image of the map

x ↦ (a_{1}x, ..., a_{n}x) : E → E^{n}, 𝔞 = (a_{1}, …, a_{n}),

and φ_{𝔞} to be this map. We call the isogeny φ_{𝔞} : E → 𝔞 · E (or any isogeny that differs from it by an isomorphism) an 𝔞-isogeny. The degree of an 𝔞-isogeny is ℕ(𝔞) =^{df} (O_{K} : 𝔞).

We obtain an action, (𝔞, E) ↦ 𝔞 · E, of Cl(R) on Ell_{k}(R), the set of isomorphism classes of elliptic curves over k having complex multiplication by R.

Proposition 12.21. The action of Cl(R) on Ell_{k}(R) makes Ell_{k}(R) into a principal homogeneous space for Cl(R), i.e., for any x_{0} ∈ Ell(R), the map 𝔞 ↦ 𝔞 · x_{0} : Cl(R) → Ell(R) is a bijection.

Proof. When k = ℂ, this is a restatement of an earlier result (before we implicitly took x_{0} to be the isomorphism of class of ℂ/R, and considered the map Cl(R) → Ell(R), 𝔞 → 𝔞^{−1} · x_{0}). We omit the proof of the general case, although this is a key point.

Reduction of elliptic curves. Let E be an elliptic curve over a number field k with good reduction at a prime ideal v of k. For simplicity, assume that 𝔭 does not divide 2 or 3. Then E has an equation

Y^{2}Z = X^{3} + aXZ^{2} + bZ^{3}

with coefficients in O_{v} whose discriminant Δ is not divisible by 𝔭_{𝔳}.

Reduction of the tangent space. Recall that for a curve C defined by an equation F (X, Y) = 0, the tangent space at (a, b) on the curve is defined by the equation:

\frac{∂F}{∂X} |_{(a,b)} (X − a) + \frac{∂F}{∂X} |_{(a,b)} (X − b) = 0.

For example, for

Z = X^{3} + aXZ^{2} + bZ^{3}

we find that the tangent space to E at (0, 0) is given by the equation

Z = 0.

Now take a Weierstrass minimal equation for E over O_{v}—we can think of the equation as defining a curve E over O_{v}, and use the same procedure to define the tangent space T gt_{0}(E) at 0 on E—it is an O_{v}-module.

Proposition 12.22. The tangent space T gt_{0}(E) at 0 to E is a free O_{v}-module of rank one such that

T gt_{0}(E) ⊗_{O_{v}} K_{v} = T gt_{0}(E/K_{v}), T gt_{0}(E) ⊗_{O_{v}} κ(v) = T gt_{0}(E(v))

where κ(v) = O_{v}/𝔭_{𝔳} and E(v) is the reduced curve.

Proof. Obvious.

Thus we can identify T gt_{0}(E) (in a natural way) with a submodule of T gt_{0}(E), and T gt_{0}(E(v)) = T gt_{0}(E)/𝔪_{v} · T gt_{0}(E), where 𝔪_{v} is the maximal ideal of O_{v}.

Reduction of endomorphisms. Let α : E → E′ be a homomorphism of elliptic curves over k, and assume that both E and E′ have good reduction at a prime v of k. Then α defines a homomorphism α(v) : E(v) → E′(v) of the reduced curves. Moreover, α acts as expected on the tangent spaces and the points of finite order. In more detail:

(a) the map T gt_{0}(α) : T gt_{0}(E) → T gt_{0}(E′) maps T gt_{0}(E) into T gt_{0}(E′), and induces the map T gt_{0}(α(v)) on the quotient modules; (b) recall (Silverman 1986) that for ℓ ≠ char(κ(v)) the reduction map defines an

isomorphism T_{ℓ}(E) → T_{ℓ}(E_{0}); there is a commutative diagram:

T_{ℓ}E ⟶^{α} T_{ℓ}E′
↓  ↓
T_{ℓ}E(v) ⟶^{α_{0}} T_{ℓ}E′(v).

It follows from (b) and Proposition 12.10 that α and α_{0} have the same characteristic polynomial (hence the same degree).

Also, we shall need to know that the reduction of an 𝔞-isogeny is an 𝔞-isogeny (this is almost obvious from the definition of an 𝔞-isogeny).

Finally, consider an 𝔞-isogeny φ : E → E′; it gives rise to a homomorphism

T gt_{0}(E) → T gt_{0}(E′)

whose kernel is ∩T gt_{0}(a), a running through the elements of 𝔞 (this again is almost obvious from the definition of 𝔞-isogeny).

The Frobenius map. Let E be an elliptic curve over the finite field k ⊃ 𝔽_{p}. If E is defined by

Y^{2} = X^{3} + aX + b,

then write E^{(q)} for the elliptic curve

Y^{2} = X^{3} + a^{q}X + b^{q}.

Then the Frobenius map Frob_{q} is defined to be

(x, y) ↦ (x^{q}, y^{q}) : E → E^{(q)},

Proposition 12.23. The Frobenius map F rob_{p} is a purely inseparable isogeny of degree p; if φ : E → E′ is a second purely inseparable isogeny of degree p, then there is an isomorphism α : E^{(p)} → E′ such that α ∘ Frob_{p} = φ.

Proof. This is similar to Silverman 1986, 2.11, p30. We have (F rob_{p})^{∗}(k(E^{(p)})) = k(E)^{p}, which the unique subfield of k(E) such that k(E) ⊃ k(E)^{p} is a purely inseparable extension of degree p.

Remark 12.24. There is the following criterion: A homomorphism α : E → E′ is separable if and only the map it defines on the tangent spaces T gt_{0}(E) → T gt_{0}(E′) is an isomorphism.

Proof of the main theorem. The group G = Gal(ℚ^{al}/K) acts on Ell(R), and commutes with the action of Cl(R). Fix an x_{0} ∈ Ell(R), and for σ ∈ G, define φ(σ) ∈ Cl(R) by:

σx_{0} = x_{0} · φ(σ).

One checks directly that φ(σ) is independent of the choice of x_{0}, and that φ is a homomorphism. Let L be a finite extension of K such that

(a) φ factors through Gal(L/ℚ);

(b) there is an elliptic curve E defined over L with j-invariant j(𝔞), some proper R-ideal 𝔞.

Lemma 12.25. There is a set S of prime ideals of K of density one excluding those that ramify in L, such that

φ(φ_{𝔭}) = Cl(𝔭)

where φ_{𝔭} ∈ Gal(L/K) is a Frobenius element.

Proof. Let 𝔭 be a prime ideal of K such that

(i) 𝔭 is unramified in L; (ii) E has good reduction at some prime ideal 𝔓 lying over 𝔭; (iii) 𝔭 has degree 1, i.e., ℕ(𝔭) = p, a prime number.

The set of such 𝔭 has density one (conditions (i) and (ii) exclude only finitely many primes, and it is a standard result (Math 776, VI.3.2) that the primes satisfying (iii) have density one).

To prove the equation, we have to show that

φ_{𝔭}(E) ≈ 𝔭 · E.

We can verify this after reducing mod 𝔓.

We have a 𝔭-isogeny E → 𝔭 · E. When we reduce modulo 𝔭, this remains a 𝔭isogeny. It is of degree ℕ(𝔭) = 𝔭, and by looking at the tangent space, one sees that it is purely inseparable. Now φ_{𝔭}(E) reduces to E^{(p)}, and we can apply Proposition 12.23 to see that E^{(p)} is isomorphic 𝔭 · E.

We now prove the theorem. Since the Frobenius elements Frob_{𝔭} generate Gal(L/K), we see that φ is surjective; whence (a) of the theorem. Part (b) is just what we proved.

The main theorem for orders. (Outline) Let R_{f} be an order in K. Just as for the maximal order O_{K}, the ideal class group Cl(R) can be identified with a quotient of the idèle class group of K, and so class field theory shows that there is an abelian extension K_{f} of K such that the Artin reciprocity map defines an isomorphism

ϕ : Cl(R_{f}) → Gal(K_{f}/K).

Of course, when f = 1, K_{f} is the Hilbert class field. The field K_{f} is called the ring class field. Note that in general Cl(R_{f}) is much bigger than Cl(O_{K}).

The same argument as before shows that if E_{f} has complex multiplication by R_{f}, then K[j(E_{f})] is the ring class field for K. Kronecker predicted (I think)^{8} that K^{ab} should equal K^{∗} =^{df} ℚ^{cyc} · K′, where K′ = ∪K(j(E_{f})) (union over positive integers). Note that

K′ = ∪K(j(τ)) (union over τ ∈ K, τ ∈ ℍ),

and so K^{∗} is obtained from K by adjoining the special values j(z) of j and the special values e^{2πim/n} of e^{z}.

Theorem 12.26. The Galois group Gal(K^{ab}/K∗) is a product of groups of order 2.

Proof. Examine the kernel of the map 𝕀_{K} → Gal(K^{∗}/K).

8Actually, it is not too clear exactly what Kronecker predicted—see the articles of Schappacher.)

Points of order m. (Outline) We strengthen the main theorem to take account of the points of finite order. Fix an m, and let E be an elliptic curve over ℂ with complex multiplication by O_{K}. For any σ ∈ Aut(ℂ) fixing K, there is an isogeny α : E → σE, which we may suppose to be of degree prime to m. Then α maps E_{m} into σE_{m}, and we can choose α so that α(x) ≡ σx mod m) for all x ∈ T_{f}E(=^{df} ∏ T_{ℓ}E). We know that α will be an 𝔞-isogeny for some 𝔞, and under our assumptions 𝔞 is relatively prime to m.

Write Id(m) for the set of ideals in K relatively prime to m, and Cl(m) for the corresponding ideal class group. The above construction gives a homomorphism

Aut(ℂ/K) → Cl(m).

Let K_{m} be the abelian extension of K (given by class field theory) with Galois group Cl(m).

Theorem 12.27. The homomorphism factors through Gal(K_{m}/K), and is the reciprocal of the isomorphism given by the Artin reciprocity map.

Proof. For m = 1, this is the original form of the main theorem. A similar argument works in the more general case.

Adelic version of the main theorem. Omitted.

Index

affine algebra, 86 affine variety, 87 algebraic variety, 89 arithmetic subgroup, 24 automorphy factor, 53 Bernoulli numbers, 49 bounded on vertical strips, 97 bounded symmetric domain, 34 canonical model, 91 class group, 116 commensurable, 24 compatible, 10 complex multiplication, 117 complex structure, 1,10 conductor, 116 congruence subgroup, 22 continuous, 8 coordinate covering, 10 coordinate neighbourhood, 1,10 correspondence, 76,100 course moduli variety, 93 cusp form, 43 cusp, 25 cyclic subgroup, 94 degree of a point, 88 degree, 16 differential form, 12,45 dimension, 89 discontinuous, 21 divisor of a function, 15 divisors, 15 doubly periodic, 35 Eisenstein series, general, 60 Eisenstein series, normalized, 72 Eisenstein series, restricted, 60 elliptic curve, 5,40,93 elliptic modular curve, 2,34 elliptic point, 25 elliptic, 25 equivalent, 28 field of constants, 89 field of rational functions, 3,4 fine moduli variety, 93,95 first kind, 13 freely, act, 22 Frobenius map, 101 Fuchsian group, 22 function field, 89 fundamental domain, 26 fundamental parallelogram, 35 geometric conductor, 106 Hecke algebra, 76 Hecke correspondence, 76 holomorphic, 1,10,11 homogeneous coordinate ring, 4 hyperbolic, 25 integral, 28 isogeny, 120 isomorphism, 11 isotropy group, 8 Jacobian variety, 109 lattice, 5 level-N structure, 95 linear fractional transformation, 24 linearly equivalent, 16 loxodromic, 25 Mellin transform, 97 meromorphic modular form, 43 meromorphic, 1,10 model, 87 modular elliptic curve, 34,111 modular form and multiplier, 98 modular form, 3,43 modular function, 2,42 moduli problem, 92 moduli variety, 92 morphism of prevarieties, 87 nonsingular, 4 orbit, 8 order, 23,116 parabolic, 25 Petersson inner product, 58 plane affine algebraic curve, 3 plane projective curve, 4 Poincaré series, 55 point, 88 positive divisor, 15 presheaf, 86 prevariety, 87 principal congruence subgroup, 2 principal divisor, 16 principal ideal, 116 proper ideal, 116 properly discontinuous, 21 purely inseparable, 101 Ramanujan function, 51,62 ramification points, 14 reduced, 28 regular, 89 Riemann sphere, 1,10 Riemann surface, 1,10 ring class field, 123 ring of correspondences, 100 ring of regular functions, 3 ringed space, 11,86 second kind, 13 separable, 101 separated, 87 sheaf of algebras, 11 sheaf, 86 Shimura variety, 34 singular point, 4 solution to a moduli problem, 92 special orthogonal group, 19 special unitary group, 20 stabilizer, 8 standard ringed space, 11 topological group, 8 valence, 14 variety, 87 width of a cusp, 42 Z-structure, 74 zero, 45