MODULAR CURVES, HECKE CORRESPONDENCES, AND L-FUNCTIONS

David E. Rohrlich In memory of my father George F. Rohrlich January 6, 1914 – August 21, 1995

These notes on Eichler-Shimura theory are intended for a reader who is familiar with elliptic curves and perhaps slightly acquainted with modular forms. The primary sources are [8], [19], and [20]. I am deeply indebted to Jaap Top for taking my place at the conference on very short notice and to Glenn Stevens for making the necessary arrangements with tact and understanding. To both of them I express my sincerest thanks.

1. Modular curves

Throughout, the term “curve” will mean “absolutely irreducible variety of dimension one”. If 𝕜 is a field, then 𝕜(t) denotes the field of rational functions over 𝕜.

1.1. The modular curve X_{0}(N)

Let N be a positive integer. The modular curve X_{0}(N) may be defined as follows. First choose an elliptic curve E over ℚ(t) such that j(E) = t. Then choose a point of order N on E and let C be the cyclic group which it generates. The subfield of \overline{ℚ(t)} fixed by the group

\{σ ∈ Gal(\overline{ℚ(t)}/ℚ(t)) : σ(C) = C\}

is a finite extension K of ℚ(t), and it turns out that K contains no proper algebraic extension of ℚ: in other words, if we think of \overline{ℚ} as the algebraic closure of ℚ inside an algebraic closure of K, then \overline{ℚ} ∩ K = ℚ. It follows that K is the function field of a smooth projective curve over ℚ; this is X_{0}(N).

The simplest nonvacuous example is the case N = 2. Let us choose E to be the curve

y^{2} = 4x^{3} − \frac{27t}{t − 1728}x − \frac{27t}{t − 1728},

so that K is the extension of ℚ(t) generated by a root of the equation

x^{3} − \frac{27t}{4(t − 1728)}x − \frac{27t}{4(t − 1728)} = 0.

Viewed as a cubic in x, the left-hand side is an Eisenstein polynomial at the place t = 0 of ℚ(t) with discriminant 2^{2}3^{12}t^{2}/(t − 1728)^{3} ∉ ℚ(t)^{×2}. Therefore K is a nonnormal cubic extension of ℚ(t). We also see that the place t = 0 is totally ramified in K while the places t = 1728 and t = ∞ each split into two places, one ramified of degree 2 and the other unramified. A calculation

Typeset by AMS-TEX

using the Hurwitz genus formula then shows that the genus of X_{0}(2) is 0. By itself, this says little, because over ℚ there are infinitely many mutually nonisomorphic smooth projective curves of genus 0. However, it is easy to see that X_{0}(2) has a rational point: for example, observe that at either place of K above t = ∞, the residue class field is ℚ. It follows that X_{0}(2) is isomorphic to P^{1} over ℚ.

Returning to the general case, we must still verify that \overline{ℚ}∩K = ℚ and that up to isomorphism K is independent of the choice of E and C. The verification will ultimately lead us to modular functions.

We begin with some notation and conventions. Let 𝕜 be a field of characteristic not dividing N. Given a Galois extension 𝕜′ of 𝕜 containing the group μ_{N} of N-th roots of unity, we shall write κ : Gal(𝕜′/𝕜) → (ℤ/Nℤ)^{×} for the character giving the action of Gal(𝕜′/𝕜) on μ_{N}:

σ(ζ) = ζ^{κ(σ)} (σ ∈ Gal(𝕜′/𝕜), ζ ∈ μ_{N}).

Suppose now that E is an elliptic curve over 𝕜. Let E[N] ⊂ E(\overline{𝕜}) denote the subgroup of points of order dividing N, and write 𝕜(E[N]) for the finite Galois extension of 𝕜 generated by the coordinates relative to some generalized Weierstrass equation for E over 𝕜 of the affine points on E of order dividing N. After fixing an ordered basis for E[N] over ℤ/Nℤ, we may identify the natural embedding of Gal(𝕜(E[N])/𝕜) in Aut(E[N]) with a faithful representation

ρ : Gal(𝕜(E[N])/𝕜) ↪ GL(2, ℤ/Nℤ).

The formalism of the Weil pairing shows that 𝕜(E[N]) contains μ_{N} and that the determinant of ρ is κ. In particular, if 𝕜 itself contains μ_{N} then κ is trivial and ρ is a representation

Gal(𝕜(E[N])/𝕜) ↪ SL(2, ℤ/Nℤ).

Theorem 1. If E is an elliptic curve over ℂ(t) with j(E) = t, then the representation on N-division points is an isomorphism Gal(ℂ(t, E[N])/ℂ(t)) ≅ SL(2, ℤ/Nℤ).

Theorem 1 will be proved later. For now we derive consequences. The first consequence is that if E is an elliptic curve over ℚ(μ_{N}) with invariant t, then it is still true that the representation on N-division points is an isomorphism

Gal(ℚ(t, E[N])/ℚ(μ_{N})(t)) ≅ SL(2, ℤ/Nℤ).

For we know that the left-hand side is embedded in the right-hand side, but on field-theoretic grounds we also have

[ℚ(t, E[N]) : ℚ(μ_{N})(t)] ≥ [ℂ(t, E[N]) : ℂ(t)].

Hence the conclusion follows from Theorem 1. Next suppose that E is an elliptic curve over ℚ with invariant t. Then E can be viewed as an elliptic curve over ℚ(μ_{N}), and consequently the representation on N-division points

ρ : Gal(ℚ(t, E[N])/ℚ(t)) ↪ GL(2, ℤ/Nℤ)

sends Gal(ℚ(t, E[N])/ℚ(μ_{N})(t)) onto SL(2, ℤ/Nℤ). Since det ρ = κ, the image of ρ also contains a set of coset representatives for SL(2, ℤ/Nℤ) in GL(2, ℤ/Nℤ). Therefore the image of ρ is all of GL(2, ℤ/Nℤ), and we obtain the first part of the following assertion:

Corollary. If E is an elliptic curve over ℚ(t) with invariant t, then the representation on Ndivision points is an isomorphism

Gal(ℚ(t, E[N])/ℚ(t)) ≅ GL(2, ℤ/Nℤ),

and \overline{ℚ} ∩ ℚ(t, E[N]) = ℚ(μ_{N}).

The equation \overline{ℚ} ∩ ℚ(t, E[N]) = ℚ(μ_{N}) also follows from the argument just given. Indeed, put L = \overline{ℚ} ∩ ℚ(t, E[N]), and suppose that L is strictly larger than ℚ(μ_{N}). Then L(t) is strictly larger than ℚ(μ_{N})(t), and consequently

[ℚ(t, E[N]) : L(t)] < |SL(2, ℤ/Nℤ)|.

But as before, the left-hand side of this inequality is a priori ≥ [ℂ(t, E[N]) : ℂ(t)], and we have a contradiction.

We can now clarify some points in the definition of X_{0}(N). Recall that the recipe for the function field K of X_{0}(N) was as follows: Choose an elliptic curve E over ℚ(t) with j(E) = t, and then choose a cyclic subgroup C of E[N]; write G for the Galois group of ℚ(t, E[N]) over ℚ(t) and H for the subgroup of G preserving C; then K is the fixed field of H. Now any point of order N on E is the second basis vector in some ordered basis for E[N] over ℤ/Nℤ, and consequently we may identify G with GL(2, ℤ/Nℤ) in such a way that H is identified with the lower triangular group

\{( \begin{matrix}a & 0 \\ b & d \end{matrix} ) : a, d ∈ (ℤ/Nℤ)^{×}, b ∈ ℤ/Nℤ\}.

Since the determinant maps the lower triangular group onto (ℤ/Nℤ)^{×}, we have ℚ(μ_{N}) ∩ K = ℚ. Substituting ℚ(μ_{N}) = \overline{ℚ} ∩ ℚ(t, E[N]) in this equation, we obtain \overline{ℚ} ∩ K = ℚ, as claimed earlier. We also see that up to isomorphism over ℚ(t), the field K is independent of the choice of C: for if we change the basis of E[N] over ℤ/Nℤ then we conjugate H inside G, and hence we conjugate (in the field-theoretic sense) K inside ℚ(t, E[N]). It remains to examine the dependence of K on E. Quite generally, if E is an elliptic curve over a field 𝕜 of characteristic not dividing N, let 𝕜(E[N]/±) denote the extension of 𝕜 generated by the x-coordinates of the affine points on E of order dividing N. Then an elementary argument shows that 𝕜(E[N]/±) is the fixed field of

{σ ∈ Gal(𝕜(E[N])/𝕜) : σ(P) = ±P for every P ∈ E[N]}.

Returning to the case at hand, we can say that ℚ(t, E[N]/±) is the subfield of ℚ(t, E[N]) corresponding to the subgroup {±I} of GL(2, ℤ/Nℤ); thus

Gal(ℚ(t, E[N]/±)/ℚ(t)) ≅ GL(2, ℤ/Nℤ)/{±I}.

Suppose now that E′ is another elliptic curve over ℚ(t) with j(E′) = t. Then E and E′ differ by a quadratic twist, and consequently so do the associated representations

Gal(\overline{ℚ(t)}/ℚ(t)) ⟶ GL(2, ℤ/Nℤ)

provided that bases for E[N] and E′[N] are chosen compatibly. It follows that the fields ℚ(t, E[N]/±) and ℚ(t, E′[N]/±) are equal and that the associated isomorphisms

Gal(ℚ(t, E[N]/±)/ℚ(t)) = Gal(ℚ(t, E′[N]/±)/ℚ(t)) ⟶ GL(2, ℤ/Nℤ)/{±I}

are identical. Now the lower triangular subgroup of GL(2, ℤ/Nℤ) contains {±I}, so that K is a subfield of ℚ(t, E[N]/±). Hence K can be characterized as the subfield of ℚ(t, E[N]/±) fixed by the image of the lower triangular subgroup in GL(2, ℤ/Nℤ)/{±I}, and this characterization is independent of the choice of E.

1.2. Other modular curves

More generally, let H be any subgroup of GL(2, ℤ/Nℤ) satisfying two conditions:

(i) −I ∈ H.

(ii) The determinant H ⟶ (ℤ/Nℤ)^{×} is surjective.

If K is the subfield of ℚ(t, E[N]) fixed by H, then the same argument shows that K is the function field of a smooth projective curve X(H) over ℚ which up to isomorphism is independent of the choice of E. As we have just seen, X_{0}(N) corresponds to the choice

H = \{( \begin{matrix}a & 0 \\ b & d \end{matrix} ) : a, d ∈ (ℤ/Nℤ)^{×}, b ∈ ℤ/Nℤ\};

another example is the curve X_{1}(N) corresponding to

H = \{( \begin{matrix}a & 0 \\ b & ±1 \end{matrix} ) : a ∈ (ℤ/Nℤ)^{×}, b ∈ ℤ/Nℤ\}.

These two examples will be the primary focus throughout, and while we shall initially emphasize X_{0}(N) it will ultimately be X_{1}(N) which provides the broader context for the application to L-functions. We note in passing that if P is a point of order N on E and C is the subgroup which it generates then the function field of X_{1}(N) can be identified with the subfield of \overline{ℚ(t)} fixed by \{σ ∈ Gal(\overline{ℚ(t)}/ℚ(t)) : σ(P) = ±P \}, just as the function field of X_{0}(N) coincides with the subfield of \overline{ℚ(t)} fixed by \{σ ∈ Gal(\overline{ℚ(t)}/ℚ(t)) : σ(C) = C\}.

By construction, a modular curve X(H) with function field K comes equipped with a distinguished morphism to P^{1} over ℚ, namely the morphism corresponding to the inclusion of ℚ(t) in K. It is conventional to refer to the finite set of points on X(H) lying over the point t = ∞ of P^{1} as cusps. If we remove the cusps from X(H) then we obtain an affine curve Y (H), which is usually denoted Y_{0}(N) in the case of X_{0}(N) and Y_{1}(N) in the case of X_{1}(N). To illustrate the notion of a cusp, let us prove that X_{0}(N) has at least one cusp rational over ℚ. Choose an elliptic curve E over ℚ(t) with invariant t and split multiplicative reduction at the place t = ∞ of ℚ(t). For example, E could be the curve

y^{2} + xy = x^{3} − \frac{36}{t − 1728}x − \frac{1}{t − 1728},

because the covariants c_{4} and c_{6} of this equation satisfy −c_{4}/c_{6} = 1. Denote the place t = ∞ of ℚ(t) simply by ∞, identify the completion of ℚ(t) at ∞ with ℚ((1/t)), and pick an extension of ∞ to a place of \overline{ℚ(t)}, so that the corresponding decomposition subgroup Gal(\overline{ℚ(t)}/ℚ(t))_{∞} of Gal(\overline{ℚ(t)}/ℚ(t)) is identified with the Galois group of \overline{ℚ((1/t))} over ℚ((1/t)). By the theory of the Tate curves, there is an isomorphism of Gal(\overline{ℚ(t)}/ℚ(t))_{∞}-modules

E(\overline{ℚ((1/t))}) ≅ \overline{ℚ((1/t))}^{×}/q^{ℤ},

where q ∈ ℚ((1/t)) is a uniformizer and q^{ℤ} denotes the infinite cyclic group generated by q. It follows in particular that

E[N] ≅ μ_{N} ⊕ (q^{1/N})^{ℤ}/q^{ℤ}

as Gal(\overline{ℚ(t)}/ℚ(t))_{∞}-modules. Let \{P_{1}, P_{2}\} be the basis for E[N] which corresponds under the preceding isomorphism to the basis {q^{1/N} mod q^{ℤ}, ζ}, where ζ is a generator of μ_{N}. Relative to this basis, the action of Gal(\overline{ℚ(t)}/ℚ(t))_{∞} on E[N] is represented by matrices of the form

( \begin{matrix}1 & 0 \\ ∗ & κ(g) \end{matrix} ) (g ∈ Gal(\overline{ℚ(t)}/ℚ(t))_{∞}).

In particular, the cyclic group C generated by P_{2} is preserved by the decomposition group Gal(\overline{ℚ(t)}/ℚ(t))_{∞}, whence the latter is contained in the subgroup of Gal(\overline{ℚ(t)}/ℚ(t)) fixing the function field of X_{0}(N). Thus the residue class degree of the restriction of ∞ to this function field is 1, and so the field of rationality of the corresponding cusp is ℚ.

1.3. Moduli interpretation of X_{0}(N) and X_{1}(N)

Let 𝕜 be an algebraically closed field, and consider pairs (E, C) consisting of an elliptic curve E over 𝕜 and a cyclic subgroup C ⊂ E[N] of order N. An isomorphism from a pair (E_{1}, C_{1}) to a pair (E_{2}, C_{2}) is an isomorphism from E_{1} to E_{2} sending C_{1} to C_{2}. We write [E, C] for the isomorphism class containing (E, C) and Ell_{0}(N)(𝕜) for the set of all isomorphism classes. Also, if S is any subset of P^{1}(𝕜) then Ell_{0}(N)(𝕜)_{S} denotes the set of all isomorphism classes [E, C] ∈ Ell_{0}(N)(ℂ) such that j(E) ∉ S.

One can also consider pairs of the form (E, P), where P is a point of order N on E. An isomorphism from (E_{1}, P_{1}) to (E_{2}, P_{2}) is an isomorphism from E_{1} to E_{2} sending P_{1} to P_{2}. Note the equality of isomorphism classes [E, P] = [E, −P]. We write Ell_{1}(N)(𝕜) for the set of isomorphism classes and Ell_{1}(N)(𝕜)_{S} for the subset consisting of those [E, P] such that j(E) ∉ S.

Next observe that if X is a modular curve and E is an elliptic curve over ℚ(t) with invariant t then E can be viewed as an elliptic curve over the function field of X, because the latter field is naturally an extension of ℚ(t). In particular, since a point x ∈ X(ℂ) determines a discrete valuation ring O_{x} of the complex function field of X, one can ask whether E has good reduction at the maximal ideal 𝔪_{x} of O_{x}: if so, then reduction modulo 𝔪_{x} yields an elliptic curve E_{x} over ℂ. After extending O_{x} to a discrete valuation ring of ℂ(t, E[N]) in some way, we can also consider the reduction map on N-division points E[N] ⟶ E_{x}[N], which is injective. Thus if P is a point of order N on E and C is the cyclic subgroup which it generates then the reductions P_{x} and C_{x} are defined, and are still of order N.

Given a subset S of P^{1}(ℂ), let P^{1}(ℂ)_{S} denote the complement of S in P^{1}(ℂ) and X(ℂ)_{S} the inverse image of P^{1}(ℂ)_{S} in X(ℂ). For example, if S = {∞}, then X_{0}(N)(ℂ)_{S} = Y_{0}(N)(ℂ).

Proposition 1. Let E be an elliptic curve over ℚ(t) with j(E) = t, and let S be a subset of P^{1}(ℂ) containing all places where E has bad reduction. Fix an ordered basis for E[N] over ℤ/Nℤ, let P be the second element of this basis, and let C be the cyclic group of order N generated by P. Then the map x ↦ [E_{x}, C_{x}] defines a bijection of X_{0}(N)(ℂ)_{S} onto Ell_{0}(N)(ℂ)_{S}. The same is true if X_{0}(N), Ell_{0}(N), and C are replaced by X_{1}(N), Ell_{1}(N), and P respectively.

Proof. First of all, S necessarily contains 0, 1728, and ∞. Indeed E differs from the curve

y^{2} + xy = x^{3} − \frac{36}{t − 1728}x − \frac{1}{t − 1728}

by a quadratic twist over ℚ(t), and therefore the discriminant of any equation for E differs from the discriminant of the above equation by a sixth power in ℚ(t)^{×}. Since the above equation has discriminant t^{2}/(t − 1728)^{3}, it follows that E has bad reduction at 0, 1728, and ∞.

As usual, we identify Gal(ℚ(t, E[N])/ℚ(t)) with GL(2, ℤ/Nℤ) and Gal(ℂ(t, E[N])/ℂ(t)) with SL(2, ℤ/Nℤ). Let K ⊂ ℚ(t, E[N]) be the subfield corresponding to

H = \{( \begin{matrix}a & 0 \\ b & d \end{matrix} ) : a, d ∈ (ℤ/Nℤ)^{×}, b ∈ ℤ/Nℤ\},

so that ℂK is the function field of X_{0}(N) over ℂ. Also let X be a curve with complex function field ℂ(t, E[N]) and π : X ⟶ X_{0}(N) the corresponding morphism. Given a point t_{o} ∈ P^{1}(ℂ)_{S}, a point x_{o} ∈ X_{0}(N)(ℂ) lying over t_{o}, and a point x̂_{o} ∈ X(ℂ) lying over x_{o}, we have a bijection

(1)

SL(2, ℤ/Nℤ)/(H ∩ SL(2, ℤ/Nℤ)) ⟶ {fiber of X_{0}(N)(ℂ) over t_{0}}
g(H ∩ SL(2, ℤ/Nℤ)) ↦ π(gx̂_{o}),

because the extension ℂ(t, E[N])/ℂ(t) is unramified outside the places of ℂ(t) where E has bad reduction, and hence in particular outside S. On the other hand, the maps

(2)

SL(2, ℤ/Nℤ)/(H ∩ SL(2, ℤ/Nℤ)) ⟶ GL(2, ℤ/Nℤ)/H
g(H ∩ SL(2, ℤ/Nℤ)) ↦ gH

and

(3)

GL(2, ℤ/Nℤ)/H ⟶ {cyclic subgroups of E of order N}
gH ↦ gC

are also bijections, as is the map

(4)  {cyclic subgroups of E of order N} ⟶ {cyclic subgroups of E_{x_{o}} of order N}

afforded by reduction modulo 𝔪_{x_{o}}. Now if x = π(gx̂_{o}) then E_{x} = E_{x_{o}} and (gC)_{x_{o}} = C_{x} (note that these identifications are meaningful since the residue class field ℂ is a subring of O_{x_{o}} and O_{x}). Therefore composing the inverse of (1) with (2), (3), and (4), we see that the map

{fiber of X_{0}(N)(ℂ) over t_{0}} ⟶ {cyclic subgroups of E_{x_{o}} of order N}
x ↦ C_{x}

is a bijection. But j(E_{x_{o}}) = t_{o} ≠ 0, 1728, whence Aut(E_{x_{o}}) = {±1}. Thus an automorphism of E_{x_{o}} sends each cyclic subgroup of E_{x_{o}} to itself. Consequently

{cyclic subgroups of E_{x_{o}} of order N} ⟶ {[E, C] ∈ Ell_{0}(N)(ℂ)_{S} : j(E) = t_{o}}
C ↦ [E_{x_{o}}, C],

is also a bijection, and composing this map with the previous one we see that x ↦ [E_{x}, C_{x}] is a bijection from the fiber of X_{0}(N)(ℂ) over t_{o} ∈ P^{1}(ℂ)_{S} to the subset of Ell_{0}(N)(ℂ)_{S} consisting of isomorphism classes with j-invariant t_{o}.

The argument for X_{1}(N) is similar. The only additional wrinkle is that P is not actually defined over the function field of X_{1}(N)(ℂ) but only over a quadratic extension. Thus P_{x} depends on a choice of a valuation ring lying over O_{x}. However, if we make the alternate choice then we simply replace P_{x} by −P_{x}. Since [E_{x}, P_{x}] = [E_{x}, −P_{x}] the map x ↦ [E_{x}, P_{x}] is still well defined.

1.4. Proof of Theorem 1: a preliminary reduction

We claim that to prove Theorem 1 it suffices to prove the following:

Proposition 2. There exists an elliptic curve E over ℂ(t) with invariant t such that

[ℂ(t, E[N]/±) : ℂ(t)] = |SL(2, ℤ/Nℤ)/{±I}|.

Granting Proposition 2, let us see how to deduce Theorem 1. Suppose that E′ is any elliptic curve over ℚ(t) with invariant t. Then E′ differs from E by at most a quadratic twist, so that ℂ(t, E[N]/±) = ℂ(t, E′[N]/±). Consequently

[ℂ(t, E′[N]/±) : ℂ(t)] = [ℂ(t, E[N]/±) : ℂ(t)] = |SL(2, ℤ/Nℤ)/{±I}|,

and the embedding Gal(ℂ(t, E′[N]/±)/ℂ(t)) ⟶ SL(2, ℤ/Nℤ)/{±I} is an isomorphism. Hence the image of the representation Gal(ℂ(t, E′[N])/ℂ(t)) ⟶ SL(2, ℤ/Nℤ) contains a set of coset representatives for {±I} in SL(2, ℤ/Nℤ) and so in particular contains either the matrix

( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )

or its negative. Since the square of this matrix is −I we find that the image is all of SL(2, ℤ/Nℤ), and Theorem 1 follows.

1.5. Modular functions

It remains to prove Proposition 2. The argument will use the theory of modular functions. We begin by recalling the relevant definitions.

Let ℌ denote the complex upper half-plane, and let GL^{+}(2, ℝ) denote the subgroup of GL(2, ℝ) consisting of matrices with positive determinant. We consider the usual action of GL^{+}(2, ℝ) on ℌ by fractional linear transformations: for

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL^{+}(2, ℝ)

and z ∈ ℌ put

γz = \frac{az + b}{cz + d}.

If f is a function on ℌ we write f ∘ γ for the function z ↦ f(γz). Now suppose that Γ is a subgroup of finite index in SL(2, ℤ). A modular function for Γ is a meromorphic function f on ℌ satisfying two conditions:

(i) f ∘ γ = f for γ ∈ Γ.

(ii) Given δ ∈ SL(2, ℤ), let M be a positive integer such that (f ∘ δ)(z + M) = (f ∘ δ)(z). (Such an integer exists by (i), because Γ has finite index in SL(2, ℤ) and consequently some

power of the matrix δ ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) δ^{−1} belongs to Γ.) Let F be the meromorphic function on

the punctured unit disk D^{∘} = {q ∈ ℂ : 0 < |q| < 1} defined by

f(δz) = F (e^{2πiz/M}) (z ∈ ℌ).

Then F extends to a meromorphic function on the full unit disk D = {q ∈ ℂ : |q| < 1}.

Like any meromorphic function on the punctured disk, the function F in (ii) is represented by a Laurent series

F (q) = ∑_{n∈ℤ} a(n)q^{n}

for q near 0, and the content of (ii) is that F does not have an essential singularity at q = 0. Thus (ii) is equivalent to the condition that for every δ ∈ SL(2, ℤ), the function f ∘ δ has a Fourier series expansion of the form

f(δz) = ∑_{n≥n_{o}} a(n)e^{2πinz/M} (Im(z) ≫ 0)

with n_{o} ∈ ℤ. Of course if Γ = SL(2, ℤ) then f ∘ δ = f by (i), and to verify (ii) it suffices to check that f itself has such a Fourier expansion.

The usual operations of addition and multiplication of meromorphic functions make the set of modular functions for Γ into a field 𝔐(Γ). For the proof of Proposition 2 we are interested in the cases where Γ is the group SL(2, ℤ) (also denoted Γ(1)) or the group Γ(N), the kernel of the reduction-modulo-N map SL(2, ℤ) → SL(2, ℤ/Nℤ). The quotient group

Γ(1)/{±I}Γ(N) ≅ SL(2, ℤ/Nℤ)/{±I}

acts as a group of automorphisms of 𝔐(Γ(N)) with fixed field 𝔐(Γ(1)), and therefore 𝔐(Γ(N)) is a Galois extension of 𝔐(Γ(1)) with Galois group isomorphic to SL(2, ℤ/Nℤ)/{±I}. For the record, we make an explicit identification

θ : SL(2, ℤ/Nℤ)/{±I} ⟶ Gal(𝔐(Γ(N))/𝔐(Γ(1)))

by declaring that if γ ∈ SL(2, ℤ) and f ∈ 𝔐(Γ(N)) then

θ([γ])(f) = f ∘ γ^{t},

where [γ] denotes the image of γ in SL(2, ℤ/Nℤ)/{±I} and γ^{t} denotes the transpose of γ.

An essential point for the proof of Proposition 2 is that 𝔐(Γ(1)) is generated over ℂ by a single nonconstant function, the j-function. Like the functions g_{2} and g_{3} appearing in the formula

j = 1728\frac{g_{2}^{3}}{g_{2}^{3} − 27g_{3}^{2}}

which defines it, the j-function should be viewed in the first instance as a function of a lattice variable L ⊂ ℂ. We obtain functions of z ∈ ℌ by putting g_{2}(z) = g_{2}(L_{z}), g_{3}(z) = g_{3}(L_{z}), and j(z) = j(L_{z}), where

L_{z} = zℤ ⊕ ℤ.

Now as functions of lattices, g_{2} and g_{3} are defined by the formulas

g_{2}(L) = 60 ∑_{\begin{matrix}ω∈L \\ ω≠0 \end{matrix}} ω^{−4}

and

g_{3}(L) = 140 ∑_{\begin{matrix}ω∈L \\ ω≠0 \end{matrix}} ω^{−6},

which give at once the behavior of g_{2} and g_{3} under homothety:

g_{2}(λL) = λ^{−4}g_{2}(L), g_{3}(λL) = λ^{−6}g_{3}(L)

for λ ∈ ℂ^{×}. It follows that as a function of lattices, j is invariant under homothety. Furthermore,

if z ∈ ℌ and γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ) then

L_{z} = zℤ ⊕ ℤ = (az + b)ℤ ⊕ (cz + d)ℤ = λL_{γz}

with λ = cz +d, whence j(γz) = j(z). This is condition (i) in the definition of modular functions; to verify (ii) we write z = x + iy and observe that

lim_{y→∞} g_{2}(x + iy) = 120 ∑_{n≥1} n^{−4}

and

lim_{y→∞} g_{3}(x + iy) = 280 ∑_{n≥1} n^{−6}

uniformly in x. Thus the holomorphic functions G_{2} and G_{3} on D^{∘} such that g_{2}(z) = G_{2}(e^{2πiz}) and g_{3}(z) = G_{3}(e^{2πiz}) extend holomorphically to D, and consequently the function J = G_{2}^{3}/(G_{2}^{3}− 27G_{3}^{2}) extends at least meromorphically to D. Hence j is a modular function for SL(2, ℤ). Now the calculation

(120 ∑_{n≥1} n^{−4})^{3} − 27(280 ∑_{n≥1} n^{−6})^{2} = (120π^{4}/90)^{3} − 27(280π^{6}/945)^{2} = 0

shows that J(q) actually has a pole at q = 0, and a more thorough analysis reveals that the pole is simple with residue 1. Therefore j has an expansion of the form

j(z) = \frac{1}{q} + power series in q

for q = e^{2πiz} near 0. In fact the Fourier expansion of j holds for all q in the unit disk, i. e. for all z ∈ ℌ, because j is holomorphic on ℌ: indeed the properties of the Weierstrass ℘-function show that g_{2}^{3} − 27g_{3}^{2} is nowhere vanishing as a function of lattices and hence also as a function of z ∈ ℌ. From the fact that j is holomorphic on ℌ with only a simple pole as a Laurent series in q, one deduces that for any f ∈ 𝔐(Γ(1)) there exist polynomials P (t), Q(t) ∈ ℂ(t) with P (t) ≠ 0 such that P (j)f − Q(j) is holomorphic on ℌ and

lim_{y→∞} P (j(z))f(z) − Q(j(z)) = 0

uniformly in x. An application of the maximum principle on a suitably truncated fundamental domain for SL(2, ℤ)\ℌ then gives P (j)f = Q(j), whence 𝔐(Γ(1)) = ℂ(j) as claimed.

1.6. Elliptic functions

To summarize, 𝔐(Γ(1)) = ℂ(j) and 𝔐(Γ(N)) is a Galois extension of ℂ(j) with Galois group isomorphic to SL(2, ℤ/Nℤ)/{±I}. Consider the elliptic curve

E : y^{2} = 4x^{3} − \frac{27j}{j − 1728}x − \frac{27j}{j − 1728}

over ℂ(j). We will show that ℂ(j, E[N]/±) coincides with 𝔐(Γ(N)), thereby completing the proof of Proposition 2 and hence of Theorem 1.

The additional ingredient needed at this point is the Weierstrass parametrization of elliptic curves over ℂ. Let L be a lattice in ℂ and consider the elliptic curve

E^{Wst} : Y^{2} = 4X^{3} − g_{2}(L)X − g_{3}(L).

We recall that the Weierstrass ℘-function

℘(u; L) = \frac{1}{u^{2}} + ∑_{\begin{matrix}ω∈L \\ ω≠0 \end{matrix}} \frac{1}{(u + ω)^{2}} − \frac{1}{ω^{2}}

affords a complex analytic group isomorphism

\begin{matrix}ℂ/L & ⟶ E^{Wst}(ℂ) \\ u + L & ↦ (℘(u; L), ℘′(u; L)), \end{matrix}

where (℘(u; L), ℘′(u; L)) is to be interpreted as the point at infinity if u ∈ L. For present purposes we must modify the classical normalization slightly. Assume that j(L) ≠ 0, 1728 and consider the elliptic curve

E : y^{2} = 4x^{3} − \frac{27j(L)}{j(L) − 1728}x − \frac{27j(L)}{j(L) − 1728}.

Let (g_{2}(L)/g_{3}(L))^{3/2} denote a fixed square root of (g_{2}(L)/g_{3}(L))^{3}. On rewriting the relation j(L) = 1728g_{2}(L)^{3}/(g_{2}(L)^{3} − 27g_{3}(L)^{2}) in the form

\frac{g_{2}(L)^{3}}{g_{3}(L)^{2}} = \frac{27j(L)}{j(L) − 1728},

we see that the change of variables

X = (g_{3}(L)/g_{2}(L))x, Y = (g_{3}(L)/g_{2}(L))^{3/2}y

transforms the equation for E into the equation for E^{Wst}. Thus we can replace the map u + L ↦ (℘(u; L), ℘′(u; L)) by the map

u + L ↦ (\frac{g_{2}(L)}{g_{3}(L)}℘(u; L), (\frac{g_{2}(L)}{g_{3}(L)})^{3/2} ℘′(u; L))

to obtain a complex analytic group isomorphism of ℂ/L onto E(ℂ). In particular, if we fix a basis {ω_{1}, ω_{2}} for L, then the numbers

x_{r,s}(L) = \frac{g_{2}(L)}{g_{3}(L)}℘(\frac{rω_{1} + sω_{2}}{N}; L) (r, s ∈ ℤ, (r, s) ≢ (0, 0) mod N)

are the x-coordinates of the affine N-division points on E. Now as a function of u, ℘(u; L) is periodic with respect to L, even, and of degree 2 when viewed as a map ℂ/L ⟶ P^{1}(ℂ). Therefore

x_{r,s}(L) = x_{r′,s′}(L) ⟺ (r, s) ≡ ±(r′, s′) mod N.

Letting R denote the set of orbits of (ℤ/Nℤ)^{2} − {(0, 0)} under the negation map, we see that if (r, s) runs over a set of representatives in ℤ^{2} for the distinct elements of R then the numbers x_{r,s}(L) are distinct.

Now let P (w; A, B) ∈ ℤ[w, A, B] be the N-th division polynomial, a universal polynomial with the property that P (w_{0}; A, B) = 0 if and only if w_{0} is the x-coordinate of an affine N-division point on the elliptic curve y^{2} = 4x^{3} + Ax + B. Applying this property to the elliptic curve E, we find that

P (x_{r,s}(L); \frac{27j(L)}{j(L) − 1728}, \frac{27j(L)}{j(L) − 1728}) = 0

whenever j(L) ≠ 0, 1728. In particular, let us take L = L_{z}, where j(z) ≠ 0, 1728. Setting

f_{r,s}(z) = \frac{g_{2}(z)}{g_{3}(z)}℘(\frac{r + sz}{N}; L_{z}) (r, s ∈ ℤ, (r, s) ≢ (0, 0) mod N),

we have f_{r,s}(z) = x_{r,s}(L_{z}) and consequently

P (f_{r,s}(z); \frac{27j(z)}{j(z) − 1728}, \frac{27j(z)}{j(z) − 1728}) = 0.

Since this equation holds for all z such that j(z) ≠ 0, 1728, it holds identically; in other words

P (f_{r,s}; \frac{27j}{j − 1728}, \frac{27j}{j − 1728}) = 0

in the field of meromorphic functions on ℌ. Therefore the functions f_{r,s} are x-coordinates of affine N-division points on the curve E over ℂ(j) with which we started. In fact the functions f_{r,s} comprise all such x-coordinates:

Proposition 3. The set of x-coordinates of affine N-division points on the elliptic curve

E : y^{2} = 4x^{3} − \frac{27j}{j − 1728}x − \frac{27j}{j − 1728}

coincides with the set of functions f_{r,s}(z) = (g_{2}(z)/g_{3}(z))℘(\frac{r + sz}{N}; L_{z}). Therefore

ℂ(j, E[N]/±) = ℂ(j, {f_{r,s}}).

Proof. As (r, s) runs over a set of representatives for R the functions f_{r,s} are all distinct, because their values are distinct at any z such that j(z) ≠ 0, 1728. Since each function f_{r,s} is the xcoordinate of an affine N-division points on E, and since the number of such x-coordinates, like the number of functions f_{r,s}, is |R|, we conclude that the functions f_{r,s} are precisely the x-coordinates of the affine N-division points on E.

1.7. Completion of the proof

The proof of Proposition 2 and hence of Theorem 1 is completed by combining Proposition 3 with the following:

Proposition 4. 𝔐(Γ(N)) = ℂ(j, {f_{r,s}}).

Proof. Let us use the notations f_{r,s} and f_{(r,s)} interchangeably. The proof rests on two assertions:

(i) f_{(r,s)} ∘ γ = f_{(r,s)γ} for γ ∈ SL(2, ℤ).

(ii) The meromorphic function F_{r,s} on D^{∘} defined by f_{r,s}(z) = F_{r,s}(e^{2πiz/N}) extends to a meromorphic function on D.

Assertion (i) follows after a calculation from the relations g_{2}(cL) = c^{−4}g_{2}(L), g_{3}(cL) = c^{−6}g_{3}(L) already noted, together with the analogous property of ℘: ℘(cu, cL) = c^{−2}℘(u, L). For (ii) one uses the definition of ℘(u, L) as a sum over lattice points to show that lim_{y→∞} f_{r,s}(x + iy) exists uniformly in x. Now (i) implies that

f_{r,s} ∘ γ = f_{r,s}

for γ ∈ Γ(N), while (i) and (ii) together imply that if δ ∈ SL(2, ℤ) then the meromorphic function F on D^{∘} defined by

f_{r,s}(δz) = F (e^{2πiz/N})

extends to a meromorphic function on D (put (r′, s′) = (r, s)δ; then F = F_{r′,s′}). Therefore f_{r,s} belongs to 𝔐(Γ(N)). To see that the f_{r,s} actually generate 𝔐(Γ(N)), we use (i) again: if the field inclusion ℂ(j, {f_{r,s}}) ⊂ 𝔐(Γ(N)) were proper, then ℂ(j, {f_{r,s}}) would be fixed by a nontrivial subgroup of the Galois group

Γ(1)/{±I}Γ(N) ≅ SL(2, ℤ/Nℤ)/{±I}.

But a subgroup of SL(2, ℤ/Nℤ)/{±I} which acts trivially on R is trivial.

1.8. A normalized basis

The arguments just completed lead to a nearly canonical choice of basis for E[N] and hence to a nearly canonical identification of Gal(ℂ(j, E[N])/ℂ(j)) with SL(2, ℤ/Nℤ) for any elliptic curve E over ℂ(j) with invariant j. To formulate the result, let us say that E has good reduction at a point z ∈ ℌ if E has good reduction at the place j = j(z) of ℂ(j). The reduction of E, or of a point P on E, will be denoted by a subscript z: E_{z}, P_{z}. Recall that we have fixed an isomorphism

θ : SL(2, ℤ/Nℤ)/{±I} ⟶ Gal(𝔐(Γ(N))/ℂ(j))

by requiring that for γ ∈ SL(2, ℤ) and f ∈ 𝔐(Γ(N)),

θ([γ])(f) = f ∘ γ^{t},

where [γ] denotes the image of γ in SL(2, ℤ/Nℤ)/{±I}.

Proposition 5. Let E be an elliptic curve over ℂ(j) with invariant j.

(i) ℂ(j, E[N]/±) = 𝔐(Γ(N)).

(ii)There is a basis {P_{1}, P_{2}} for E[N], unique up to replacement by {−P_{1}, −P_{2}}, with the following properties:

(a) Let ρ : Gal(ℂ(j, E[N])/ℂ(j)) ⟶ SL(2, ℤ/Nℤ) be the isomorphism corresponding to {P_{1}, P_{2}} and ρ^{±} : Gal(ℂ(j, E[N]/±)/ℂ(j)) ⟶ SL(2, ℤ/Nℤ)/{±I} the induced isomorphism. Then ρ^{±} = θ^{−1}.

(b) If z ∈ ℌ is a point where E has good reduction, then there is a complex analytic group isomorphism of ℂ/L_{z} onto E_{z}(ℂ) sending 1/N + L_{z} to (P_{2})_{z}.

Proof. (i) Since ℂ(j, E[N]/±) depends on E only up to quadratic twist, this follows from Propositions 3 and 4.

(ii) Choose an equation for E over ℂ(j) of the form

c(j)y^{2} = 4x^{3} − \frac{27j}{j − 1728}x − \frac{27j}{j − 1728},

where c(t) ∈ ℂ[t] is a polynomial with simple zeros. Then E has good reduction at z ∈ ℌ if and only if j(z) ≠ 0, 1728 and c(j(z)) ≠ 0. Now we have seen (in the case c(t) = 1, and hence in general) that the x-coordinates of the affine points of order N on E are the functions f_{r,s} with (r, s) ∈ ℤ^{2} and (r, s) ≢ 0 mod N. Thus for each such pair (r, s) there is a point P_{r,s} ∈ E[N] such that x(P_{r,s}) = f_{r,s}. Of course the definition of P_{r,s} represents an arbitrary choice from among two possibilities. We also make an arbitrary choice of square roots (g_{2}(z)/g_{3}(z))^{3/2} and c(j(z))^{1/2} at each point z ∈ ℌ where E has good reduction, and we let λ_{z} : ℂ/L_{z} → E_{z}(ℂ) denote the complex analytic group isomorphism afforded by the map

u ↦ (\frac{g_{2}(z)}{g_{3}(z)}℘(u; L_{z}), (\frac{g_{2}(z)}{g_{3}(z)})^{3/2} \frac{℘′(u; L_{z})}{c(j(z))^{1/2}} ) .

Now choose any point z_{o} ∈ ℌ where E has good reduction, and let P_{1} and P_{2} be the preimages of λ_{z_{o}}(z/N + L_{z_{o}}) and λ_{z_{o}}(1/N + L_{z_{o}}) respectively under the isomorphism P ↦ P_{z_{o}} of E[N] onto E_{z_{o}}[N]. Then {(P_{1})_{z_{o}}, (P_{2})_{z_{o}}} is a basis for E_{z_{o}}[N] and a fortiori {P_{1}, P_{2}} is a basis for E[N]. We claim that

rP_{1} + sP_{2} = ±P_{r,s}.

Since the reduction map is injective on torsion, it suffices to check that

(rP_{1} + sP_{2})_{z_{o}} = ±(P_{r,s})_{z_{o}}.

The left-hand side is λ_{z_{o}}(\frac{rz_{o} + s}{N} + L_{z_{o}}) while the right-hand side has x-coordinate f_{r,s}(z_{o}). Therefore equality holds.

To verify (a), take σ ∈ Gal(ℂ(j, E[N]/±)/ℂ(j)), let σ̃ ∈ Gal(ℂ(j, E[N])/ℂ(j)) be an element which restricts to σ, and choose γ ∈ SL(2, ℤ) so that the image of γ in SL(2, ℤ/Nℤ) is ρ(σ̃). Then ρ^{±}(σ) = [γ], and the identity to be proved is θ([γ]) = σ. Since the f_{r,s} generate 𝔐(Γ(N)) over ℂ(j) (Proposition 4), it suffices to check that θ([γ])(f_{r,s}) = σ(f_{r,s}). Write

γ ( \begin{matrix}r \\ s \end{matrix} ) = ( \begin{matrix}r′ \\ s′ \end{matrix} ) .

As we have seen in the proof of Proposition 4,

θ([γ])(f_{r,s}) = f_{r,s} ∘ γ^{t} = f_{(r′,s′)}.

On the other hand, r′P_{1} + s′P_{2} = ±P_{r′,s′}, whence

f_{r′,s′} = x(r′P_{1} + s′P_{2}) = x(ρ(σ̃)(rP_{1} + sP_{2})) = σ(x(rP_{1} + sP_{2})).

Since rP_{1} + sP_{2} = ±P_{r,s}, the last term is σ(f_{r,s}), and (a) follows.

For (b), suppose that E has good reduction at z. Since the x-coordinate of (P_{2})_{z} is f_{0,1}(z), we have λ_{z}(1/N + L_{z}) = ±(P_{2})_{z}. Hence either λ_{z} or −λ_{z} sends 1/N + L_{z} to (P_{2})_{z}.

Finally, suppose that {P′_{1}, P′_{2}} is another basis for E[N] with properties (a) and (b). Choose a point z ∈ ℌ where E has good reduction, and let λ′_{z} : ℂ/L_{z} ⟶ E_{z}(ℂ) be a complex analytic group isomorphism sending 1/N + L_{z} to (P′_{2})_{z}. Then λ_{z}^{−1} ∘ λ′_{z} ∈ Aut(ℂ/L_{z}). Since E has good reduction at z we have j(z) ≠ 0, 1728 and consequently Aut(ℂ/L_{z}) = {±1}. Hence after replacing {P_{1}, P_{2}} by {−P_{1}, −P_{2}} if necessary, we may assume that P′_{2} = P_{2}. Then the change-of-basis matrix sending {P_{1}, P_{2}} to {P′_{1}, P′_{2}} is a lower triangular matrix with 1 in the lower right-hand entry. Furthermore, conjugation by this matrix induces the identity on SL(2, ℤ/Nℤ)/{±I}, because {P′_{1}, P′_{2}} also has property (a). It follows that the change-of-basis is the identity, as desired.

1.9. Quotients of the upper half-plane

We will use Proposition 5 to realize the modular curves as compactified quotients of ℌ. First we must recall how such quotients are given the structure of a Riemann surface.

Put

ℌ^{∗} = ℌ ∪ P^{1}(ℚ).

The action of SL(2, ℤ) on ℌ by fractional linear transformations extends to an action on ℌ^{∗} preserving P^{1}(ℚ), and if Γ is any subgroup of finite index in SL(2, ℤ), then we denote the respective orbit spaces of ℌ^{∗}, ℌ, and P^{1}(ℚ) under Γ by Γ\ℌ^{∗}, Γ\ℌ, and Γ\P^{1}(ℚ). Thus

Γ\ℌ^{∗} = (Γ\ℌ) ∪ (Γ\P^{1}(ℚ)) .

Since Γ has finite index in SL(2, ℤ) and SL(2, ℤ) acts transitively on P^{1}(ℚ) the set Γ\P^{1}(ℚ) is finite.

We would like to put a topology on Γ\ℌ^{∗}. First we put a topology on ℌ^{∗} itself. Given y_{0} > 0 and c ∈ P^{1}(ℚ), choose a matrix δ ∈ SL(2, ℤ) such that c = δ∞, and put

U_{y_{0}} = {x + iy : x ∈ ℝ, y > y_{0}} ⊂ ℌ,
U_{c,y_{0}}^{∘} = δ(U_{y_{0}}),

and

U_{c,y_{0}} = U_{c,y_{0}}^{∘} ∪ {c}.

The sets U_{c,y_{0}}^{∘} and U_{c,y_{0}} depend only on c and y_{0}, not on the choice of δ, because U_{y_{0}} is preserved

by the stabilizer of ∞ in SL(2, ℤ), namely \{± ( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) : n ∈ ℤ\}. We make ℌ^{∗} into a topological

space by choosing as a basis of open sets all sets of the following two types:

(a) open subsets U of H,

(b) subsets of ℌ^{∗} of the form U_{c,y_{0}}.

Then the quotient toplology on Γ\ℌ^{∗} corresponding to the natural projection

π : ℌ^{∗} ⟶ Γ\ℌ^{∗}

makes Γ\ℌ∗ into a compact Hausdorff space.

The next step is to make Γ\ℌ^{∗} into a compact Riemann surface. Let F be the sheaf of continuous complex-valued functions on Γ\ℌ^{∗}, and F_{x} the stalk at a point x. We think of F_{x} as the set of equivalence classes of germs (f, V), where V is an open neighborhood of x and f is a continuous complex-valued function on V, two germs (f, V) and (g, W) being equivalent if f and g coincide on V ∩ W. To make Γ\ℌ^{∗} into a Riemann surface, we must define a subsheaf O of F to serve as the complex structure sheaf. We define O by specifying that its stalk O_{x} at x is the subring of F_{x} consisting of those equivalence classes which contain a germ (f, V) of one of the following two types:

(a) There exists z ∈ ℌ and an open neighborhood U of z in ℌ such that x = π(z), V = π(U), and f ∘ π is holomorphic on U.

(b) There exists c ∈ P^{1}(ℚ) and y_{0} > 0 such that x = π(c), V = π(U_{c,y_{0}}), and f ∘ π satisfies the following condition. Choose δ ∈ SL(2, ℤ) such that c = δ∞, and let M be a positive integer such that (f ∘ π ∘ δ)(z + M) = (f ∘ π ∘ δ)(z) for z ∈ U_{y_{0}}. (Such an integer exists because Γ has finite index in SL(2, ℤ) and π is invariant under Γ.) Put r = e^{−2πy_{0}/M} and let F be the function on the punctured disk D^{∘}(r) = {q ∈ ℂ : 0 < |q| < r} such that

(f ∘ π ∘ δ)(z) = F (e^{2πiz/M}).

Then F is holomorphic on D^{∘}(r) and extends to a holomorphic function on the full disk D(r) = {q ∈ ℂ : |q| < r}.

One can check that with this definition of O, every point x of Γ\ℌ^{∗} has an open neighborhood V such that the ringed space (V, O|_{V}) is isomorphic to the ringed space of an open disk in ℂ. (The verification requires a little care if x is the image of an elliptic fixed point of Γ, i. e. if x = π(z) for some z ∈ ℌ which is fixed by an element of Γ different from ±I.) Granting that this is so, we conclude that O gives Γ\ℌ^{∗} the structure of a Riemann surface. Furthermore – and this is now the key point – the definitions have been constructed in such a way that the map

f ↦ (f ∘ π)|ℌ

identifies the function field of Γ\ℌ^{∗} with 𝔐(Γ). Note that both Γ\ℌ^{∗} and 𝔐(Γ) depend only on \overline{Γ}, the image of Γ in SL(2, ℤ)/\{±I\}.

1.10. Modular curves as quotients of the upper half-plane

Given a modular curve X(H), we shall now produce a subgroup Γ of SL(2, ℤ) such that the Riemann surfaces X(H)(ℂ) and Γ\ℌ^{∗} are isomorphic. By assumption, H is a subgroup of GL(2, ℤ/Nℤ) satisfying two conditons: −I ∈ H and det : H → (ℤ/Nℤ)^{×} is surjective. We let Γ ⊂ SL(2, ℤ) be the transpose of the inverse image of H ∩ SL(2, ℤ/Nℤ) under the reduction map SL(2, ℤ) → SL(2, ℤ/Nℤ).

Proposition 6. With H and Γ as above, the Riemann surfaces X(H)(ℂ) and Γ\ℌ^{∗} are isomorphic.

Proof. Let E be an elliptic curve over ℚ(j) with invariant j, and identify Gal(ℚ(j, E[N])/ℚ(j)) with GL(2, ℤ/Nℤ) using a basis for E[N] as in Proposition 5. The function field of X(H) over ℚ is the subfield K of ℚ(j, E[N]/±) fixed by H, whence the function field of the Riemann surface X(H)(ℂ) is ℂK. Now our identification of Gal(ℚ(j, E[N])/ℚ(j)) with GL(2, ℤ/Nℤ) affords an identification of Gal(ℂ(j, E[N]/±)/ℂ(j)) with SL(2, ℤ/Nℤ)/{±I}, and the hypotheses on H imply that ℂK is the subfield of ℂ(j, E[N]/±) fixed by (H ∩ SL(2, ℤ/Nℤ))/{±I}. Applying parts (i) and (ii)(a) of Proposition 5 we deduce that ℂK = 𝔐(Γ), whence the result follows from the fact that a compact Riemann surface is determined up to isomorphism by its function field.

In particular, put

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ) : c ≡ 0 mod N\}

and

Γ_{1}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ) : c ≡ 0 mod N, a, d ≡ 1 mod N\}.

Then X_{0}(N)(ℂ) ≅ Γ_{0}(N)\ℌ^{∗} and X_{1}(N)(ℂ) ≅ Γ_{1}(N)\ℌ^{∗} (in the latter case we use the fact that Γ\ℌ^{∗} depends only on \overline{Γ}). Now consider pairs (T , C) consisting of a one-dimensional complex torus T and a cyclic subgroup C of T of order N. An isomorphism from one pair (T_{1}, C_{1}) to another (T_{2}, C_{2}) is a complex analytic group isomorphism from T_{1} to T_{2} sending C_{1} onto C_{2}. We denote the isomorphism class containing (T , C) by [T , C] and the set of all isomorphism classes by Tori_{0}(N). For a point P of order N on T we make the analogous definitions of (T , P), [T , P], and Tori_{1}(N).

Proposition 7. Let E be an elliptic curve over ℚ(j) with invariant j, and let S be a subset of P^{1}(ℂ) containing all places where E has bad reduction. Fix an ordered basis for E[N] over ℤ/Nℤ, let P be the second element of this basis, and let C be the cyclic group of order N generated by P. Then there is an isomorphism of Riemann surfaces X_{0}(N)(ℂ) ≅ Γ_{0}(N)\ℌ^{∗} such that the diagram

X_{0}(N)(ℂ)_{S} ⟶ Ell_{0}(N)(ℂ)_{S}
↓  ↓
Γ_{0}(N)\ℌ ⟶ Tori_{0}(N)

commutes, where:

– The top horizontal arrow is the bijection x ↦ [E_{x}, C_{x}] of Proposition 1.

– The bottom horizontal arrow is a bijection and has the form

[z] ↦ [ℂ/L_{z}, ⟨1/N + L_{z}⟩] (z ∈ ℌ),

where [z] denotes the class of z in Γ_{0}(N)\ℌ and ⟨1/N + L_{z}⟩ denotes the cyclic subgroup of ℂ/L_{z} generated by the coset of L_{z}.

– The left vertical arrow is the restriction to X_{0}(N)(ℂ)_{S} of the isomorphism X_{0}(N)(ℂ) ≅ Γ_{0}(N)\ℌ^{∗}.

– The right vertical arrow is the restriction to Ell_{0}(N)(ℂ)_{S} of the bijection from Ell_{0}(N)(ℂ)

to Tori_{0}(N) given by [E, C] ↦ [E(ℂ), C]. The same is true if X_{0}(N), Ell_{0}(N), Γ_{0}(N), Tori_{0}(N), C, and ⟨1/N +L_{z}⟩ are replaced by X_{1}(N), Ell_{1}(N), Γ_{1}(N), Tori_{1}(N), P, and 1/N + L_{z}.

Proof. Without loss of generality we may assume that P is the second basis vector in a basis for E[N] chosen as in Proposition 5. Then the only statement requiring proof is the bijectivity of the bottom horizontal arrow. The cases Γ_{0}(N) and Γ_{1}(N) are similar; we deal with the latter. Suppose that

[ℂ/L_{z}, 1/N + L_{z}] = [ℂ/L_{z′}, 1/N + L_{z′}]

Then there exists ω ∈ ℂ^{×} such that L_{z′} = ωL_{z} and 1/N ≡ ω/N (mod ωL_{z}). The first condition implies that {ω, ωz} is a basis for L_{z′}. Hence we can write

\{ \begin{matrix}z′ = ω(az + b) \\ 1 = ω(cz + d) \end{matrix}

with integers a, b, c, d satisfying ad − bc = ±1. Since z′ = (az + b)/(cz + d) and z and z′ both have positive imaginary part, it follows that ad − bc = 1. Substituting 1 = ω(cz + d) in the congruence 1/N ≡ ω/N (mod ωL_{z}), we find that c ≡ 0 (mod N) and d ≡ 1 (mod N), whence a ≡ 1 (mod N) also since ad − bc = 1. Thus z′ = γz with

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{1}(N),

and consequently [z] = [z′]. Next suppose that [T , P] ∈ Tori_{1}(N). Write T = ℂ/L and P = ω/N + L with ω ∈ L. After replacing ω by another element of ω + NL, we may assume that ω is primitive, so that ω is part of a basis {ω′, ω} for L. Put z = ±ω′/ω, where the sign is chosen so that Im(z) > 0. Multiplication by ω^{−1} gives an isomorphism of (ℂ/L, P + L) onto (ℂ/L_{z}, 1/N + L_{z}), whence [T , P] coincides with [ℂ/L_{z}, 1/N + L_{z}].

2. Hecke correspondences

By a correspondence on a smooth projective curve X we shall mean a triple T = (Z, φ, ψ), where Z is a smooth projective curve and φ and ψ are nonconstant morphisms Z → X. We say that T is defined over a field 𝕜 if X, Z, φ, and ψ are all defined over 𝕜. We view an automorphism δ of X as a special case of a correspondence by putting Z = X, φ = id_{X}, and ψ = δ.

2.1. The Hecke correspondences on X_{0}(N)

Let N be a positive integer, p a prime number, and M the least common multiple of N and p. Choose an elliptic curve E over ℚ(t) with invariant t, and fix a basis for E[M] over ℤ/Mℤ, whence an identification of Gal(ℚ(t, E[M])/ℚ(t)) with GL(2, ℤ/Mℤ). We consider the subgroup

H_{p} = \{( \begin{matrix}a & c \\ b & d \end{matrix} ) ∈ GL(2, ℤ/Mℤ) : c ≡ 0 mod N, b ≡ 0 mod p\}.

Since −I ∈ H_{p} and det(H_{p}) = (ℤ/Mℤ)^{×}, the fixed field of H_{p} is the function field of a smooth projective curve over ℚ, which we shall denote X_{0}(N, p). The Hecke correspondence T_{p} on X_{0}(N) is a correspondence over ℚ of the form

T_{p} = (X_{0}(N, p), φ_{p}, ψ_{p}),

where the morphisms φ_{p}, ψ_{p} : X_{0}(N, p) → X_{0}(N) must now be defined.

The definition of φ_{p} is straightforward. Let K_{p} and K denote the fixed fields of H_{p} and

H = \{( \begin{matrix}a & c \\ b & d \end{matrix} ) ∈ GL(2, ℤ/Mℤ) : c ≡ 0 mod N\}

respectively. Then H_{p} ⊂ H, whence K ⊂ K_{p}. The latter inclusion is an inclusion of function fields and so corresponds to a morphism of curves

φ_{p} : X_{0}(N, p) = X(H_{p}) ⟶ X(H).

But X(H) is X_{0}(N), because the kernel of the reduction map GL(2, ℤ/Mℤ) → GL(2, ℤ/Nℤ) is a subgroup of H and the image of H in GL(2, ℤ/Nℤ) is the lower triangular group. Therefore φ_{p} is a morphism from X_{0}(N, p) to X_{0}(N).

The definition of ψ_{p} is more subtle. It corresponds to an inclusion of function fields K′ ↪ K_{p}, where K′ is a subfield of K_{p} which is isomorphic to but distinct from K. To define K′, let us recall once again that our identification of Gal(ℚ(t, E[M])/ℚ(t)) with GL(2, ℤ/Mℤ) rests on a choice of basis for E[M] over ℤ/Mℤ and hence in particular on a decomposition of E[M] as a direct sum of cyclic subgroups of order M:

E[M] = C_{1} ⊕ C_{2}.

Let C denote the cyclic subgroup of C_{2} of order N, and let Π denote the cyclic subgroup of C_{1} of order p. Then C and Π are stable under H_{p}, hence defined over K_{p}. In particular, since Π is defined over K_{p} there is an elliptic curve E/Π defined over K_{p} together with an isogeny

λ : E ⟶ E/Π

over K_{p} with kernel Π. Furthermore, E/Π has a cyclic subgroup of order N defined over K_{p}, namely the subgroup λ(C). Now the property that E/Π has a cyclic subgroup of order N defined over K_{p} is inherited by every quadratic twist of E/Π over K_{p}, because it is equivalent to the property that the Galois representation on N-division points is lower triangular (relative to an appropriate basis). Consequently these equivalent properties hold for any elliptic curve over K_{p} with invariant

t′ = j(E/Π) ∈ K_{p},

because t′ is transcendental and hence in particular ≠ 0, 1728. It follows that if E′ is an elliptic curve over ℚ(t′) with invariant t′ then E′ has a cyclic subgroup C′ of order N which is defined over K_{p}. Then K_{p} contains the field K′ fixed by

\{σ ∈ Gal(\overline{ℚ(t′)}/ℚ(t′)) : σ(C′) = C′\}.

Since K′ is isomorphic to the function field of X_{0}(N) we obtain the desired morphism ψ_{p} from X_{0}(N, p) to X_{0}(N).

2.2. The Hecke correspondences on X_{1}(N)

Mutatis mutandis, the same construction yields a correspondence

T_{p} = (X_{1}(N, p), φ_{p}, ψ_{p})

on X_{1}(N), where X_{1}(N, p) is the modular curve determined by the subgroup

H_{p} = \{( \begin{matrix}a & c \\ b & d \end{matrix} ) ∈ GL(2, ℤ/Mℤ) : c ≡ 0 mod N, b ≡ 0 mod p, d ≡ ±1 mod N\}

of GL(2, ℤ/Mℤ). Put

H = \{( \begin{matrix}a & c \\ b & d \end{matrix} ) ∈ GL(2, ℤ/Mℤ) : c ≡ 0 mod N, d ≡ ±1 mod N\}

and write K_{p} and K for the subfields of ℚ(t, E[M]) fixed by H_{p} and H. Then φ_{p} is the morphism X_{1}(N, p) → X_{1}(N) corresponding to the inclusion of K in K_{p}. To define ψ_{p}, let {P_{1}, P_{2}} be our chosen basis for E[M] and put P = (M/N)P_{2}. Also let Π be the group of order p generated by (M/p)P_{1}. As before, there is an elliptic curve E/Π over K_{p} and an isogeny λ : E → E/Π over K_{p} with kernel Π. Since K is contained in K_{p} and the set {±P } is stable under Gal(ℚ(t, E[M])/K) it follows that {±λ(P)} is stable under Gal(ℚ(t, E[M])/K_{p}). Arguing as before, we conclude that any elliptic curve over K_{p} with invariant t′ = j(E/Π) has a point P′ of order N such that {±P′} is defined over K_{p}. Thus if E′ is any elliptic curve over ℚ(t′) with invariant t′ then there is a point P′ of order N on E′ such that {±P′} is defined over K_{p}. Hence K_{p} contains K′, the field fixed by

\{σ ∈ Gal(\overline{ℚ(t′)}/ℚ(t′)) : σ(P′) = ±P′\}.

Since K′ is isomorphic to the function field of X_{1}(N) we obtain a morphism ψ_{p} from X_{1}(N, p) to X_{1}(N).

2.3. Moduli interpretation of the Hecke correspondences

We denote the free abelian group on a set W by Div(W). In particular, if X is a smooth projective curve over an algebraically closed field 𝕜 then Div(X(𝕜)) is the usual group of divisors on X(𝕜). Given a correspondence T = (Z, φ, ψ) on X, we use the same letter T to denote the map

\begin{matrix}X(𝕜) & ⟶ Div(X(𝕜)) \\ x & ↦ ∑_{\begin{matrix}z∈Z \\ φ(z)=x \end{matrix}}(mult_{z}φ)ψ(z), \end{matrix}

where mult_{z}φ is the ramification index of φ at z. In the case of the Hecke correspondence T_{p} we shall give a formula for this map which displays its effect on isomorphism classes [E, C] and [E, P]. First a point of notation.

Suppose that E is an elliptic curve over an algebraically closed field 𝕜 and Λ is a subgroup of E of order p. In keeping with our usage thus far, we shall write E/Λ for an elliptic curve which is the image of a separable isogeny with domain E and kernel Λ. Note that E/Λ is unique up to isomorphism. Now if λ : E → E/Λ is a separable isogeny with kernel Λ and C is a cyclic subgroup of E of order N which intersects Λ trivially (a vacuous condition if N is prime to p), then we obtain a well-defined isomorphism class

[E/Λ, (C + Λ)/Λ] ∈ Ell_{0}(N)(𝕜)

by putting [E/Λ, (C + Λ)/Λ] = [λ(E), λ(C)]. To see that [E/Λ, (C + Λ)/Λ] is independent of the choice of λ, suppose that λ′ : E → E/Λ is another such isogeny. Then there is an automorphism θ of E/Λ such that λ′ = θ ∘ λ, whence [λ′(E), λ′(C)] = [λ(E), λ(C)]. Similarly, if P is a point of order N on E such that the cyclic subgroup ⟨P⟩ generated by P intersects Λ trivially, then we define

[E/Λ, P + Λ] ∈ Ell_{1}(N)(𝕜)

by putting [E/Λ, P + Λ] = [λ(E), λ(P)].

Proposition 8. Let E be an elliptic curve over ℚ(t) with invariant t. Let S, S′, and S′′ be subsets of P^{1}(ℂ) containing all places where E has bad reduction and such that

φ_{p}^{−1}(X_{0}(N)(ℂ)_{S}) ⊂ X_{0}(N, p)(ℂ)_{S′}

and

ψ_{p}(X_{0}(N, p)(ℂ)_{S′}) ⊂ X_{0}(N)(ℂ)_{S′′}.

Fix an ordered basis for E[N] over ℤ/Nℤ, let P be the second element of this basis, and let C be the cyclic group generated by P. Then the diagram

X_{0}(N)(ℂ)_{S} ⟶^{T_{p}} Div(X_{0}(N)(ℂ)_{S′′})
↓  ↓
Ell_{0}(N)(ℂ) ⟶ Div(Ell_{0}(N)(ℂ))

commutes, where the left vertical arrow is the map x ↦ [E_{x}, C_{x}] of Proposition 1, the right vertical arrow is the corresponding homomorphism between free abelian groups, and the bottom horizontal arrow is the map

[E, C] ↦ ∑_{\begin{matrix}[E[p]:Λ]=p \\ C∩Λ=\{0\} \end{matrix}}[E/Λ, (C + Λ)/Λ],

the sum being taken over subgroups Λ of index p in E[p] which intersect C trivially. The same is true if X_{0}(N) is replaced by X_{1}(N), Ell_{0}(N) by Ell_{1}(N), the left vertical arrow by x ↦ [E_{x}, P_{x}], and the bottom horizontal arrow by

[E, P] ↦ ∑_{\begin{matrix}[E[p]:Λ]=p \\ ⟨P⟩∩Λ=\{0\} \end{matrix}}[E/Λ, P + Λ],

where ⟨P⟩ denotes the cyclic subgroup generated by P.

Proof. For x ∈ X_{0}(N)(ℂ)_{S} the formula

T_{p}(x) = ∑_{\begin{matrix}z∈Z \\ φ_{p}(z)=x \end{matrix}}(mult_{z}φ_{p})ψ_{p}(z)

can be written simply as

T_{p}(x) = ∑_{z∈φ_{p}^{−1}(x)} ψ_{p}(z),

because the morphism φ_{p} : X_{0}(N, p) → X_{0}(N) is unramified outside S: indeed the corresponding extension of function fields K_{p}/K is contained inside the extension ℚ(t, E[M])/ℚ(t) and is therefore unramified outside the places where E has bad reduction. Here M denotes the least common multiple of N and p, as before.

Consider triples (E, C, Λ), where E is an elliptic curve over ℂ, C is a cyclic subgroup of E of order N, and Λ is a cyclic subgroup of E of order p which intersects C trivially. We write [E, C, Λ] for the isomorphism class of (E, C, Λ) and Ell_{0}(N, p)(ℂ) for the set of isomorphism classes. If we define maps φ and ψ from Ell_{0}(N, p)(ℂ) to Ell_{0}(N)(ℂ) by

φ([E, C, Λ]) = [E, C]

and

ψ([E, C, Λ]) = [E/Λ, (C + Λ)/Λ],

then the map

[E, C] ↦ ∑_{\begin{matrix}[E[p]:Λ]=p \\ C∩Λ=\{0\} \end{matrix}}[E/Λ, (C + Λ)/Λ],

in the statement of the proposition has the form

x ↦ ∑_{z∈φ^{−1}(x)} ψ(z).

Therefore it suffices to check that the diagrams

X_{0}(N, p)(ℂ)_{S′} ⟶^{φ_{p}} X_{0}(N)(ℂ)_{S}
↓  ↓
Ell_{0}(N, p)(ℂ) ⟶_{φ} Ell_{0}(N)(ℂ)

and

X_{0}(N, p)(ℂ)_{S′} ⟶^{ψ_{p}} X_{0}(N)(ℂ)_{S′′}
↓  ↓
Ell_{0}(N, p)(ℂ) ⟶_{ψ} Ell_{0}(N)(ℂ)

commute, where the right vertical arrows are the maps x ↦ [E_{x}, C_{x}] of Proposition 1 and the left vertical arrows are given by

\begin{matrix}X_{0}(N, p)(ℂ)_{S′} & ⟶ Ell_{0}(N, p)(ℂ) \\ z & ↦ [E_{z}, C_{z}, Π_{z}]. \end{matrix}

As before, Π denotes a subgroup of E of order p which intersects C trivially, and the subscript z indicates reduction modulo the maximal ideal of the discrete valuation ring corresponding to z in the function field of X_{0}(N, p). Now the commutativity of the first diagram amounts to the equation

[E_{z}, C_{z}] = [E_{φ_{p}(z)}, C_{φ_{p}(z)}],

which follows from the compatibility of reduction at a good place with base extension. To verify that the second diagram commutes put t′ = j(E/Π), choose an elliptic curve E′ over ℚ(t′) with invariant t′, and let C′ be a subgroup of E′ of order N which is defined over the function field K_{p} of X_{0}(N, p). We must check that

(1)  [E_{z}/Π_{z}, (C_{z} + Π_{z})/Π_{z}] = [E′_{ψ_{p}(z)}, C′_{ψ_{p}(z)}].

We verify this equation in two steps. First,

(2)  [E_{z}/Π_{z}, (C_{z} + Π_{z})/Π_{z}] = [(E/Π)_{z}, ((C + Π)/Π)_{z}]

by the compatibility of reduction with isogenies. Next let L be a quadratic extension of K_{p} such that E/Π and E′ are isomorphic over L, and let θ be an isomorphism from E/Π to E′ over L. If σ is the nontrivial element of Gal(L/K_{p}) then σ ∘ θ ∘ σ^{−1} = −θ, whence the image under θ of a subgroup of E/Π which is defined over K_{p} is again defined over K_{p}. Thus without loss of generality we may assume that C′ = θ((C + Π)/Π). Let θ_{z} denote the reduction of θ modulo a place of L above z. Then θ_{z} gives an isomorphism from (E/Π)_{z} to (E′)_{ψ_{p}(z)} sending ((C +Π)/Π)_{z} to (C′)_{ψ_{p}(z)}, whence

(3)  [(E/Π)_{z}, ((C + Π)/Π)_{z}] = [(E′)_{ψ_{p}(z)}, (C′)_{ψ_{p}(z)}].

Together, (2) and (3) give (1). The argument for X_{1}(N) is similar.

2.4. The Hecke correspondences on the upper half-plane

Given d ∈ (ℤ/Nℤ)^{×} we write ⟨d⟩ to denote any element of Γ_{0}(N) with lower right-hand entry congruent to d modulo N. Also, if d ∈ ℤ is an integer prime to N then we write ⟨d⟩ for ⟨d mod N⟩.

Proposition 9. There is a commutative diagram

X_{0}(N)(ℂ) ⟶^{T_{p}} Div(X_{0}(N)(ℂ))
↓  ↓
Γ_{0}(N)\ℌ^{∗} ⟶ Div(Γ_{0}(N)\ℌ^{∗}),

where the left vertical arrow is the isomorphism of Proposition 7, the right vertical arrow is the corresponding homomorphism of free abelian groups, and the bottom horizontal arrow is the map

[z] ↦ \{ \begin{matrix}∑_{ν=0}^{p−1}[(z + ν)/p] + [pz] & if p ∤ N \\ ∑_{ν=0}^{p−1}[(z + ν)/p] & if p|N. \end{matrix}

The same is true if X_{0}(N) is replaced by X_{1}(N) and Γ_{0}(N) by Γ_{1}(N) provided that the bottom horizontal arrow is modified as follows: in the case p ∤ N, the term [pz] is replaced by [⟨p⟩pz].

Proof. By a continuity argument it suffices to check that the diagram commutes when X_{0}(N)(ℂ) is replaced by X_{0}(N)(ℂ)_{S} for some finite set S. Propositions 7 and 8 then reduce the problem to the following: Given [T , C] ∈ Tori_{0}(N) with T = ℂ/L_{z} and C = ⟨1/N + L_{z}⟩, show that

∑_{\begin{matrix}[T [p]:Λ]=p \\ C∩Λ=\{0\} \end{matrix}}[T /Λ, (C + Λ)/Λ]

coincides with

∑_{ν=0}^{p−1}[ℂ/L_{(z+ν)/p}, ⟨1/N + L_{(z+ν)/p}⟩] + [ℂ/L_{pz}, ⟨1/N + L_{pz}⟩],

the last term being omitted if p divides N. Now the subgroups of order p in ℂ/L_{z} which intersect ⟨1/N + L_{z}⟩ trivially are

⟨(z + ν)/p + L_{z}⟩ (0 ≤ ν ≤ p − 1)

and also

⟨1/p + L_{z}⟩

if p ∤ N. Furthermore, for T = ℂ/L_{z} and Λ = ⟨(z + ν)/p + L_{z}⟩ we have T /Λ ≅ ℂ/L_{(z+ν)/p}. Hence the only point to check is that if p ∤ N then

[ℂ/(zℤ ⊕ p^{−1}ℤ), ⟨1/N + (zℤ ⊕ p^{−1}ℤ)⟩] = [ℂ/L_{pz}, ⟨1/N + L_{pz})⟩].

This holds because multiplication by p maps ℂ/(zℤ ⊕ p^{−1}ℤ) isomorphically onto ℂ/L_{pz} and because the cosets of 1/N and p/N generate the same subgroup of ℂ/L_{pz} for p ∤ N.

The argument for Γ_{1}(N) is much the same, except that in the case p ∤ N one must check that

[ℂ/L_{pz}, p/N + L_{pz}] = [ℂ/L_{⟨p⟩pz}, 1/N + L_{⟨p⟩pz}].

Put w = pz and γ = ⟨p⟩, and write γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) . Since L_{w} = wℤ ⊕ ℤ and L_{γw} = \frac{aw+b}{cw+d}ℤ ⊕ ℤ,

multiplication by cw + d defines an isomorphism from ℂ/L_{γw} to ℂ/L_{w} sending 1/N + L_{γw} to (cw + d)/N + L_{w}. The latter coset coincides with p/N + L_{w}, because

(cw + d)/N − p/N = (c/N)w + (d − p)/N ∈ L_{w}.

2.5. The diamond automorphisms

The next construction pertains only to X_{1}(N). Choose an elliptic curve E over ℚ(t) with invariant t, and make the usual identification of Gal(ℚ(t, E[N])/ℚ(t)) with GL(2, ℤ/Nℤ), and of the function field K of X_{1}(N) with the fixed field of

H = \{( \begin{matrix}a & 0 \\ b & ±1 \end{matrix} ) ∈ GL(2, ℤ/Mℤ) : a ∈ (ℤ/Nℤ)^{×}, b ∈ (ℤ/Nℤ)^{×}\}.

Since H is normal in the lower triangular subgroup B, the quotient group B/H acts as a group of automorphisms of K and hence of X_{1}(N). We shall identify B/H with (ℤ/Nℤ)^{×}/{±1} via the

map sending the coset of ( \begin{matrix}a & 0 \\ b & d \end{matrix} ) modulo H to the coset of d modulo \{±1\}. The automorphism

of X_{1}(N) corresponding to the coset of d ∈ (ℤ/Nℤ)^{×} modulo {±1} will be denoted ⟨d⟩. Of course we have already used the symbol ⟨d⟩ to denote an element of Γ_{0}(N). The second of the following two propositions shows that the two usages are consistent. The proofs are left to the reader.

Proposition 10. Let E be an elliptic curve over ℚ(t) with invariant t, and let S be a subset of P^{1}(ℂ) containing all places where E has bad reduction. Fix an ordered basis for E[N] over ℤ/Nℤ and let P be the second element of this basis. For d ∈ (ℤ/Nℤ)^{×} the diagram

X_{1}(N)(ℂ)_{S} ⟶^{⟨d⟩} X_{1}(N)(ℂ)_{S}
↓  ↓
Ell_{1}(N)(ℂ) ⟶ Ell_{1}(N)(ℂ)

commutes, where the bottom horizontal arrow is the map [E, P] ↦ [E, dP] and the vertical arrows are the map x ↦ [E_{x}, C_{x}] of Proposition 1.

Proposition 11. There is a commutative diagram

X_{1}(N)(ℂ) ⟶^{⟨d⟩} X_{1}(N)(ℂ)
↓  ↓
Γ_{1}(N)\ℌ^{∗} ⟶ Γ_{1}(N)\ℌ^{∗},

where the bottom horizontal arrow is the map [z] ↦ [⟨d⟩z] and the vertical arrows are the isomorphism of Proposition 7.

For the application to L-functions toward which we are now heading it is enough to consider the curve X_{1}(N), and henceforth the curve X_{0}(N) will drop out of sight.

2.6. Hecke correspondences and the Frobenius automorphism

Let p be a prime not dividing N, and fix a prime ideal 𝔭 of \overline{ℚ} lying over p. Write \overline{𝔽}_{p} for the residue class field of 𝔭. Reduction modulo 𝔭 will be denoted by a tilde: for example, if E is an elliptic curve over \overline{ℚ} with good reduction at 𝔭 then Ẽ denotes the elliptic curve over \overline{𝔽}_{p} obtained from E by reduction modulo 𝔭. We define sets Ell_{1}(N)(\overline{ℚ}) and Ell_{1}(N)(\overline{𝔽}_{p}) by replacing ℂ by \overline{ℚ} or \overline{𝔽}_{p} respectively in the definition of Ell_{1}(N)(ℂ). Thus Ell_{1}(N)(\overline{ℚ}) can be identified with the subset of Ell_{1}(N)(ℂ) consisting of classes [E, ∗] such that j(E) ∈ \overline{ℚ}. Also, we let Ell_{1}(N)(\overline{ℚ})_{gd} denote the subset of Ell_{1}(N)(\overline{ℚ}) consisting of classes [E, ∗] such that E has good reduction at 𝔭. Under our assumption that p does not divide N we have a well-defined map

Ell_{1}(N)(\overline{ℚ})_{gd} ⟶ Ell_{1}(N)(\overline{𝔽}_{p})
[E, P] ↦ [Ẽ, P̃],

because reduction modulo 𝔭 is injective on N-torsion.

We recall that an elliptic curve E over \overline{ℚ} is said to have ordinary good reduction at 𝔭 if Ẽ[p] has order p. Since the reduction map

E[p] ⟶ Ẽ[p]

is surjective, its kernel is a subgroup of E[p] of order (or index) p. In addition E[p] has exactly p other subgroups of index p.

We use a superscript p to indicate the image of an object under the Frobenius automorphism of \overline{𝔽}_{p} and a superscript p^{−1} to indicate the image under the inverse of the Frobenius automorphism.

Proposition 12. Let E be an elliptic curve over \overline{ℚ} with ordinary reduction at 𝔭, and let Λ_{o} be the kernel of the reduction map

E[p] ⟶ Ẽ[p].

Let P be a point of order N on E. If Λ is a subgroup of E[p] of index p, then

[Ẽ/Λ, { \atop P + Λ}] = \{ \begin{matrix}[Ẽ^{p}, P̃^{p}] & if Λ = Λ_{o} \\ [Ẽ^{p^{−1}}, pP̃^{p^{−1}}] & if Λ ≠ Λ_{o}. \end{matrix}

Proof. Let λ_{o} : E → E/Λ_{o} be an isogeny with kernel Λ_{o} and μ_{o} : E/Λ_{o} → E the dual isogeny. The image of (E/Λ_{o})[p] under μ_{o} is Λ_{o}. Indeed, since μ_{o} is a p-isogeny, the image of (E/Λ_{o})[p] under μ_{o} is a group of order p; but if this group were not contained in Λ_{o} then the image of (E/Λ_{o})[p] under λ_{o} ∘ μ_{o} would not be zero, contradicting the fact that λ_{o} ∘ μ_{o} is multiplication by p. Now consider the commutative diagram

E[p] ⟶^{λ_{o}} (E/Λ_{o})[p] ⟶^{μ_{o}} E[p]
↓  ↓  ↓
Ẽ[p] \underrightarrow{λ̃_{o}} { \atop E/Λ_{o}}[p] \underrightarrow{μ̃_{o}} Ẽ[p],

where the vertical arrows are reduction modulo 𝔭 and hence surjective. Since the image of (E/Λ_{o})[p] under μ_{o} is Λ_{o}, the commutativity of the right-hand square shows that μ̃_{o} is zero on { \atop E/Λ_{o}}[p]. Now Ẽ is ordinary by assumption, and since { \atop E/Λ_{o}} is isogenous to Ẽ, it too is ordinary. Hence the fact that μ̃_{o} is zero on { \atop E/Λ_{o}}[p] means that μ̃_{o} is a separable p-isogeny. But multiplication by p is inseparable. Consequently λ̃_{o} is an inseparable (and hence purely inseparable) isogeny of degree p, and we can write λ̃_{o} = θ ∘ β, where β is the Frobenius endomorphism of degree p and θ : E^{p} → { \atop E/Λ_{o}} is an isomorphism. Therefore

[λ̃_{o}(Ẽ), λ̃_{o}(P̃)] = [β(Ẽ), β(P̃)],

But the left-hand side is [{ \atop E/Λ_{o}}, { \atop P + Λ_{o}}], while the right-hand side is [Ẽ^{p}, P̃^{p}]. Hence we get the stated equality.

Now suppose that Λ ≠ Λ_{o}. Choose an isogeny λ : E → E/Λ with kernel Λ, and consider the curve λ(E) (= E/Λ) together with its subgroup λ(Λ_{o}) of order p. Since λ(E) is isogenous to E, it has ordinary reduction at 𝔭, and consequently the kernel of reduction mod 𝔭 on λ(E)[p] is a subgroup of order p. The calculation

{ \atop λ(Λ_{o})} = λ̃(Λ̃_{o}) = λ̃(\{0\}) = \{0\}

shows that λ(Λ_{o}) is contained in this subgroup and hence coincides with it, since both have order p. Therefore we can apply to λ(E) and λ(Λ_{o}) what we have already proved for E and Λ_{o}:

(1)  [{ \atop λ(E)/λ(Λ_{o})}, { \atop λ(P) + λ(Λ_{o})}] = [λ̃(E)^{p}, { \atop λ(P)}^{p}].

But if μ : λ(E) → λ(E)/λ(Λ_{o}) is any isogeny with kernel λ(Λ_{o}) then

[λ(E)/λ(Λ_{o}), P + Λ_{o}] = [(μ ∘ λ)(E), (μ ∘ λ)(P)]

by definition. We choose μ to be the isogeny dual to λ, and then [(μ ∘ λ)(E), (μ ∘ λ)(P)] becomes [E, pP]. Thus the left-hand side of (1) is [Ẽ, pP̃], and (1) can be rewritten

(2)  [Ẽ, pP̃] = [Ẽ/Λ^{p}, { \atop P + Λ}^{p}].

Applying the inverse Frobenius automorphism to (2) we obtain the stated formula.

Let Ell_{1}(N)(\overline{ℚ})_{ord} be the subset of Ell_{1}(N)(\overline{ℚ})_{gd} consisting of classes [E, ∗] such that E has ordinary reduction at 𝔭. Reduction of isomorphism classes

Ell_{1}(N)(\overline{ℚ})_{ord} ⟶ Ell_{1}(N)(\overline{𝔽}_{p})
[E, P] ↦ [Ẽ, P̃],

extends uniquely to a homomorphism

red_{𝔭} : Div(Ell_{1}(N)(\overline{ℚ})_{ord}) ⟶ Div(Ell_{1}(N)(\overline{𝔽}_{p})).

Proposition 13. Let σ_{𝔭} ∈ Gal(\overline{ℚ}/ℚ) be a Frobenius element at 𝔭. Then

T_{p} = σ_{𝔭} + p⟨p⟩σ_{𝔭}^{−1}

when both sides are regarded as maps

Ell_{1}(N)(\overline{ℚ})_{ord} ⟶ Div(Ell_{1}(N)(\overline{ℚ})_{ord})/Ker(red_{𝔭}).

Proof. Let E be an elliptic curve over \overline{ℚ} with ordinary reduction at 𝔭, and let P a point of order N on E. We must show that T_{p}([E, P]) and (σ_{𝔭} + p⟨p⟩σ_{𝔭}^{−1})([E, P]) have the same image under red_{𝔭}. Now Proposition 8 gives

T_{p}([E, P]) = ∑_{Λ}[E/Λ, P + Λ],

where the sum runs over subgroups Λ ⊂ E[p] of index p which have trivial intersection with the cyclic group generated by P. On the other hand,

(σ_{𝔭} + p⟨p⟩σ_{𝔭}^{−1})([E, P]) = [E^{σ_{𝔭}}, P^{σ_{𝔭}}] + p[E^{σ_{𝔭}^{−1}}, pP^{σ_{𝔭}^{−1}}].

Thus we must check that

∑_{Λ}[Ẽ/Λ, { \atop P + Λ}] = [Ẽ^{p}, P̃^{p}] + p[Ẽ^{p^{−1}}, pP̃^{p^{−1}}].

This follows from Proposition 12, because the sum on the left-hand side has p + 1 terms, exactly one of which coincides with the kernel of E[p] → Ẽ[p].

Given a smooth projective curve X and a correspondence T = (Z, φ, ψ) on X, we use the same letter T to denote the endomorphism ψ_{∗} ∘ φ^{∗} of its Jacobian variety Jac(X). If 𝕜 is an algebraically closed field then the map on points Jac(X)(𝕜) → Jac(X)(𝕜) can be obtained from T : X(𝕜) → Div(X(𝕜)) by extending the latter map to Div(X(𝕜)), restricting to the subgroup Div^{0}(X(𝕜)) of divisors of degree 0, and then passing to divisor classes.

We are concerned with the case X = X_{1}(N), T = T_{p}. We denote the Jacobian variety Jac(X_{1}(N)) simply by J_{1}(N). Also, the automorphism of J_{1}(N) induced by the diamond automorphism ⟨d⟩ of X_{1}(N) will be denoted by the same symbol ⟨d⟩. If ℓ is a prime and n a positive integer then the ring of endomorphisms of J_{1}(N) acts on the abelian group J_{1}(N)[ℓ^{n}]. So does Gal(\overline{ℚ}/ℚ), because J_{1}(N) is defined over ℚ. In sketching a proof of the next statement we shall simply quote what we need from the work of Igusa [10], in particular the fact that X_{1}(N) has good reduction at primes not dividing N.

Theorem 2. Let σ_{𝔭} ∈ Gal(\overline{ℚ}/ℚ) be a Frobenius element at 𝔭. Then for ℓ ≠ p and n ≥ 1,

T_{p} = σ_{𝔭} + p⟨p⟩σ_{𝔭}^{−1}

as endomorphisms of J_{1}(N)[ℓ^{n}].

Proof. Let { \atop X_{1}(N)} denote the reduction of X_{1}(N) modulo p. As a map on points, the reduction map X_{1}(N)(\overline{ℚ}) → { \atop X_{1}(N)}(\overline{𝔽}_{p}) is compatible with the map [E, P] ↦ [Ẽ, P̃] in a sense which we shall now describe.

Let ℤ[t]_{(p)} denote the localization of ℤ[t] at the prime ideal generated by p. We say that an elliptic curve E over ℚ(t) has good reduction at p if there is a generalized Weierstrass equation for E over ℤ[t]_{(p)} with discriminant a unit of ℤ[t]_{(p)}. The reduction of this equation modulo pℤ[t]_{(p)} then defines an elliptic curve Ẽ over 𝔽_{p}(t). Now let E be an elliptic curve over ℚ(t) with invariant t and good reduction at p. For example we can take E to be the curve defined by the equation

y^{2} + xy = x^{3} − \frac{36}{t − 1728}x − \frac{1}{t − 1728}

of discriminant t^{2}/(t − 1728)^{3}. Let P be a point of order N on E, let P̃ ∈ Ẽ[N] be its reduction modulo 𝔭, and let K̃ be the fixed field of \{σ ∈ Gal(\overline{𝔽_{p}(t)}/𝔽_{p}(t)) : σ(P) = ±P \}, where \overline{𝔽_{p}(t)}

denotes a separable algebraic closure of 𝔽_{p}(t). Then { \atop X_{1}(N)} is characterized up to isomorphism as the smooth projective curve over 𝔽_{p} with function field K̃. Furthermore, by viewing Ẽ as an elliptic curve over K̃ one obtains a reduction map

\begin{matrix}Ẽ(\overline{𝔽}_{p})_{S′} & ⟶ Ell_{1}(N)(\overline{𝔽}_{p})_{S′} \\ x & ↦ [(Ẽ)_{x}, (P̃)_{x}] \end{matrix}

for any subset S′ of P^{1}(\overline{𝔽}) containing the places where Ẽ has bad reduction. Let S be the inverse image of S′ under the reduction map P^{1}(\overline{ℚ}) ⟶ P^{1}(\overline{𝔽}). Then the diagram of reduction maps

X_{1}(N)(\overline{ℚ})_{S} ⟶ Ell_{1}(N)(\overline{ℚ})
↓  ↓
{ \atop X_{1}(N)}(\overline{𝔽}_{p})_{S′} ⟶ Ell_{1}(N)(\overline{𝔽}_{p})

commutes.

Henceforth we take S′ to be the set of places where Ẽ has bad or supersingular reduction. Note that S′ is a finite set. The commutativity of the above diagram allows us to replace Ell_{1}(N)(\overline{ℚ})_{ord}

and Ell_{1}(N)(\overline{𝔽}) in the statement of Proposition 13 by X_{1}(N)(\overline{ℚ})_{S} and { \atop X_{1}(N)(\overline{𝔽}_{p})_{S′}} respectively.

Now let { \atop J_{1}(N)} denote the reduction of J_{1}(N) modulo p, identifiable with the Jacobian of { \atop X_{1}(N)}. There is a commutative diagram

Div^{0}(X_{1}(N)(\overline{ℚ})_{S}) ⟶ J_{1}(N)(\overline{ℚ})
↓  ↓
Div^{0}({ \atop X_{1}(N)}(\overline{𝔽}_{p})_{S′}) ⟶^{α} { \atop J_{1}(N)}(\overline{𝔽}_{p})

in which the vertical arrows are reduction modulo 𝔭 and the horizontal arrows send a divisor to the point on the Jacobian representing its divisor class. Since S′ is finite α is surjective. Let L ∈ J_{1}(N)(\overline{ℚ}) be a torsion point of ℓ-power order; we must show that

(T_{p} − σ_{𝔭} − p⟨p⟩σ_{𝔭}^{−1})(L) = 0.

In fact it is enough to show that this equation holds after reduction modulo 𝔭, because reduction mod 𝔭 is injective on ℓ-torsion. Write L̃ = α(D̃) with D ∈ Div^{0}(X_{1}(N)(\overline{ℚ})_{S′}). According to Proposition 13, the divisor (T_{p} − σ_{𝔭} − p⟨p⟩σ_{𝔭}^{−1})(D) reduces to 0 modulo 𝔭, and consequently so does the divisor (T_{p} − σ_{𝔭} − p⟨p⟩σ_{𝔭}^{−1})(L).

3. L-functions

The significance of Theorem 2 is that it leads to an expression for certain Hasse-Weil Lfunctions in terms of L-functions of modular forms. We begin by putting this application in a larger context.

3.1. The Hasse-Weil conjecture

Originally conceived of as an assertion about the zeta function of a smooth projective variety over a number field, the conjecture has since evolved into a more general statement about Lfunctions of motives. Here we shall restrict our attention to motives of a very special kind, namely motives afforded by H^{1} of an abelian variety over ℚ and more generally products of such motives with Artin motives. To begin with we take the Artin motive to be trivial. Let A be an abelian variety of dimension g defined over ℚ, and recall that for every prime number ℓ one has an ℓ-adic representation

ρ_{ℓ} : Gal(\overline{ℚ}/ℚ) → Aut(V_{ℓ}(A)) ≅ GL(2g, ℚ_{ℓ}),

where V_{ℓ}(A) = ℚ_{ℓ} ⊗_{ℤ_{ℓ}} T_{ℓ}(A) and T_{ℓ}(A) is the Tate module of A:

T_{ℓ}(A) = \underleftarrow{lim}_{n} A[ℓ^{n}].

We let ρ_{ℓ}^{∗} denote the contragredient representation on the dual space V_{ℓ}^{∗}(A) of V_{ℓ}(A). Given a prime number p, one defines a polynomial P_{p}(A, t) ∈ ℤ[t] by the formula

P_{p}(A, t) = det (1 − tρ_{ℓ}^{∗}(σ_{𝔭}^{−1})|V_{ℓ}^{∗}(A)^{I(𝔭)}) ,

where ℓ is any prime number different from p, I(𝔭) and σ_{𝔭} denote respectively the inertia group and a Frobenius element of some prime ideal 𝔭 of \overline{ℚ} lying over p, and

V_{ℓ}^{∗}(A)^{I(𝔭)} = {v ∈ V_{ℓ}^{∗}(A) : ρ_{ℓ}^{∗}(g)v = v for all g ∈ I(𝔭)}.

That P_{p}(A, t) is independent of the choice of 𝔭 and σ_{𝔭} follows by a straightforward verification from the conjugacy under Gal(\overline{ℚ}/ℚ) of the prime ideals lying over a given rational prime. Far deeper is the fact that P_{p}(A, t) belongs to ℤ[t] and is independent of the choice of ℓ ≠ p. Indeed we are able to make this assertion for all p, and not just for the p where A has good reduction, precisely because we have confined ourselves to the case of abelian varieties, for which Grothendieck’s semistable reduction theorem [9] is available: in the case of an arbitrary smooth projective variety, the analogues of P_{p}(A, t) – defined using ℓ-adic cohomology groups H_{ℓ}^{i}(∗) rather than the Tate module – are not yet known to be independent of ℓ when p is a prime of bad reduction and i > 1 (for i = 1 the ℓ-adic cohomology group is dual to the Tate module of the Albanese, so we are back to the case of abelian varieties). Now write

P_{p}(A, t) = ∏_{i=1}^{2g}(1 − α_{i,p}t)

with complex numbers α_{i,p}. One has

|α_{i,p}| ≤ \sqrt{p} (1 ≤ i ≤ 2g)

with equality if p is a prime of good reduction, whence the Euler product

L(A, s) = ∏_{p} P_{p}(A, p^{−s})^{−1}

converges in the region Re(s) > 3/2. Another consequence of the semistable reduction theorem is that one can associate to A a well-defined conductor N(A) and sign W (A) = ±1 (cf. [17]). The definition of the “root number” W (A) requires the theory of local epsilon factors [6].

Conjecture 1. Put

Λ(A, s) = N(A)^{s/2}((2π)^{−s}Γ(s))^{g}L(A, s).

Then Λ(A, s) has an analytic continuation to an entire function of order one satisfying the functional equation

Λ(A, s) = W (A)Λ(A, 2 − s).

It is also useful to have at hand a slightly less precise formulation of the conjecture, evocative of the state of affairs which prevails when H_{ℓ}^{1}(A) is replaced by the cohomology of an arbitrary smooth projective variety:

Conjecture 1^{∗}. There exist:

– a finite set S of prime numbers containing all primes where A has bad reduction,

– for each p ∈ S, a polynomial

P_{p}^{∗}(A, t) = ∏_{i=1}^{2g}(1 − α_{i,p}^{∗}t) ∈ ℤ[t]

with |α_{i,p}^{∗}| < p for all i,

– a positive integer N^{∗}(A), and

– a sign W^{∗}(A) ∈ {±1}, such that if

L^{∗}(A, s) = ∏_{p∉S} P_{p}^{∗}(A, p^{−s})^{−1} · ∏_{p∈S} P_{p}^{∗}(A, p^{−s})^{−1}

and

Λ^{∗}(A, s) = (N^{∗})^{s/2}((2π)^{−s}Γ(s))^{g}L^{∗}(s, A)

then Λ^{∗}(A, s) has an analytic continuation to an entire function of order one satisfying the functional equation

Λ^{∗}(A, s) = W^{∗}(A)Λ^{∗}(A, 2 − s).

We have included a bound on α_{i,p}^{∗} in the statement of Conjecture 1^{∗} to ensure that if Conjecture 1 is true then N^{∗}(A), W^{∗}(A), and P_{p}^{∗}(A, t) coincide respectively with N(A), W (A), and P_{p}(t, A). Indeed for all good p (and hence in particular for all p ∉ S) we already have the stronger information that |α_{i,p}| = \sqrt{p}, so that the stated bound on α_{i,p}^{∗} affords a uniform estimate

p > \{ \begin{matrix}|α_{i,p}| & (p ∉ S) \\ |α_{i,p}^{∗}| & (p ∈ S); \end{matrix}

but a remark of Deligne-Serre ([7], p. 515, Lemme 4.9) then shows that N^{∗}(A), W^{∗}(A), and the P_{p}^{∗}(A, t) are uniquely determined by the functional equation, whence these quantities coincide with the corresponding quantities in Conjecture 1 whenever the latter conjecture is satisfied.

3.2. Modular forms

Quite apart from its significance for the arithmetic of abelian varieties, Conjecture 1 asserts the existence of a class of Dirichlet series with Euler products and functional equations. Such Dirichlet series arise naturally in the theory of modular forms.

Let k be a positive integer. Given a holomorphic function f on ℌ and a matrix

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL^{+}(2, ℝ),

we put

(f|_{k}γ)(z) = \frac{det(γ)}{(cz + d)^{k}} f(γz).

This formula defines a right action of GL^{+}(2, ℝ) on the space of holomorphic functions on ℌ. Now let Γ be a subgroup of finite index in SL(2, ℤ). A modular form of weight k for Γ is a holomorphic function f on ℌ satisfying two conditions:

(i) f|_{k}γ = f for γ ∈ Γ.

(ii) For every δ ∈ SL(2, ℤ) the function f|_{k}δ has a Fourier expansion of the form

(f|_{k}δ)(z) = ∑_{n≥0} a(n)e^{2πinz/M}.

If for every δ ∈ SL(2, ℤ) the coefficient a(0) in (ii) is 0 then f called a cusp form. The vector space of modular forms of weight k for Γ will be denoted M_{k}(Γ) and the subspace of cusp forms S_{k}(Γ). These spaces are finite-dimensional. We remark in passing that in condition (ii) the phrase “δ ∈ SL(2, ℤ)” can be replaced by “δ ∈ GL^{+}(2, ℚ)”, where GL^{+}(2, ℚ) = GL(2, ℚ) ∩ GL^{+}(2, ℝ). This is simply a matter of writing an element of GL^{+}(2, ℚ) as the product of an element of SL(2, ℤ) and an upper triangular matrix. It follows in particular that if Γ is normalized by a matrix δ ∈ GL^{+}(2, ℚ) then the spaces M_{k}(Γ) and S_{k}(Γ) are stable under the map f ↦ f|_{k}δ.

Let us now specialize to the case Γ = Γ_{1}(N). In this case we denote the spaces M_{k}(Γ) and S_{k}(Γ) simply by M_{k}(N) and S_{k}(N). Furthermore, given a character χ of (ℤ/Nℤ)^{×} we let M_{k}(N, χ) and S_{k}(N, χ) be the subspaces of M_{k}(N) and S_{k}(N) consisting of f such that

f|_{k}γ = χ(d)f

for

γ = ( \begin{matrix}a & b \\ cN & d \end{matrix} ) ∈ Γ_{0}(N).

(Implicit in the notation χ(d) is the usual identification of characters of (ℤ/Nℤ)^{×} with Dirichlet characters modulo N.) Another way to describe the subspaces M_{k}(N, χ) and S_{k}(N, χ) is to say that they are the χ-eigenspaces for the “diamond operators” f ↦ f|_{k}⟨d⟩. In this approach d denotes an element of (ℤ/Nℤ)^{×}, and the operator ⟨d⟩ is defined by setting

f|_{k}⟨d⟩ = f|_{k}γ

for any γ ∈ Γ_{0}(N) which reduces modulo N to a matrix with d as lower right-hand entry. In view of the isomorphism

Γ_{0}(N)/Γ_{1}(N) ⟶ (ℤ/Nℤ)^{×}
coset of ( \begin{matrix}a & b \\ cN & d \end{matrix} ) ↦ d (mod N)

the diamond operators give a well-defined action of (ℤ/Nℤ)^{×} on M_{k}(N) and S_{k}(N), and consequently we have eigenspace decompositions

M_{k}(N) = ⊕_{χ}M_{k}(N, χ)

and

S_{k}(N) = ⊕_{χ}S_{k}(N, χ)

where χ runs over Dirichlet characters modulo N. Note that if χ is the trivial character then M_{k}(N, χ) and S_{k}(N, χ) coincide with M_{k}(Γ_{0}(N)) and S_{k}(Γ_{0}(N)) respectively.

Henceforth we restrict our attention to cusp forms. To see why cusp forms give rise to Dirichlet series with functional equations, observe that the matrix

W_{N} = ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} )

normalizes Γ_{1}(N), whence f|_{k}W_{N} ∈ S_{k}(N) if f ∈ S_{k}(N). In fact

( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ) ( \begin{matrix}a & b \\ cN & d \end{matrix} ) ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} )^{−1} = ( \begin{matrix}d & −c \\ −bN & a \end{matrix} ) ,

so that f|_{k}W_{N} ∈ S_{k}(N, \overline{χ}) if f ∈ S_{k}(N, χ). Now write

f(z) = ∑_{n≥1} a(n)e^{2πinz}

and

(f|_{k}W_{N})(z) = ∑_{n≥1} b(n)e^{2πinz},

and put

A(s) = N^{s/2}(2π)^{−s}Γ(s) ∑_{n≥1} a(n)n^{−s}

and

B(s) = N^{s/2}(2π)^{−s}Γ(s) ∑_{n≥1} b(n)n^{−s}.

As Hecke observed, these Dirichlet series converge absolutely in some right half-plane and can be analytically continued by a method which goes back to Riemann’s paper on the the Riemann zeta function: The usual interchange of summation and integration shows that

A(s) = ∫_{0}^{∞} f(it/\sqrt{N})t^{s} \frac{dt}{t},

whence

\begin{aligned}A(s) & = ∫_{0}^{1} f(it/\sqrt{N})t^{s} \frac{dt}{t} + ∫_{1}^{∞} f(it/\sqrt{N})t^{s} \frac{dt}{t} \\ & = ∫_{1}^{∞}(f(i/(t\sqrt{N}))t^{−s} + f(it/\sqrt{N})t^{s})\frac{dt}{t} \end{aligned}

on making the change of variables t ↦ 1/t in the integral from 0 to 1. Since

f (\frac{i}{t\sqrt{N}} ) = (it)^{k}(f|_{k}W_{N})(it/\sqrt{N})

we obtain

A(s) = ∫_{1}^{∞}(i^{k}(f|_{k}W_{N})(it/\sqrt{N})t^{k−s} + f(it/\sqrt{N})t^{s})\frac{dt}{t}.

But (W_{N})^{2} = −NI, and consequently f|_{k}(W_{N})^{2} = (−1)^{k}f. Hence one can repeat the preceding calculation with A(s) replaced by B(s), f by f|_{k}W_{N}, and f|_{k}W_{N} by (−1)^{k}f, and a comparison of the resulting expressions for A(s) and B(s) yields:

Proposition 14. The functions A(s) and B(s) have analytic continuations to entire functions of order one satisfying the functional equation A(s) = i^{k}B(k − s).

We have avoided referring to the Dirichlet series ∑ a(n)n^{−s} and ∑ b(n)n^{−s} as L-functions, because as yet we have imposed no condition to guarantee the existence of an Euler product. For this we need the Hecke operators.

3.3. Hecke operators

Given a prime number p, let Δ_{p} denote the set of 2 × 2 matrices with integer coefficients and determinant p which are congruent modulo N to a matrix of the form

( \begin{matrix}1 & ∗ \\ 0 & p \end{matrix} ) .

It is immediate from the definition that Δ_{p} is stable under left and right multiplication by Γ_{1}(N), and elementary calculations show that if Γ = Γ_{1}(N) and

δ_{p} = ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )

then Δ_{p} has the one-term double-coset decomposition

Δ_{p} = Γδ_{p}Γ

and the following decomposition as a disjoint union of right cosets:

Δ_{p} = \{ \begin{matrix}⋃_{ν=0}^{p−1} Γ ( \begin{matrix}1 & ν \\ 0 & p \end{matrix} ) ∪ Γ⟨p⟩ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , & if p ∤ N \\ ⋃_{ν=0}^{p−1} Γ ( \begin{matrix}1 & ν \\ 0 & p \end{matrix} ) , & if p|N \end{matrix}

(recall that if p does not divide N then ⟨p⟩ denotes an arbitrary element of Γ_{0}(N) with lower right-hand entry congruent to p modulo N). Of course if γ ∈ Γ_{1}(N) and {δ} is any set of representatives for the distinct right cosets of Γ_{1}(N) in Δ_{p} then {δγ} is another such set, because Δ_{p} is stable under right multiplication by Γ_{1}(N).

The p-th Hecke operator

T_{p} : S_{k}(N) ⟶ S_{k}(N)

is defined by the formula

f|_{k}T_{p} = p^{k/2−1} ∑_{δ} f|_{k}δ,

where δ runs over a set of representatives for the distinct right cosets of Γ_{1}(N) in Δ_{p}. The definition is independent of the choice of coset representatives because f ∈ S_{k}(N). Furthermore, f|_{k}T_{p} does belong to S_{k}(N), because right multiplication by any γ ∈ Γ_{1}(N) sends one set of right coset representatives to another. For much the same reason, T_{p} commutes with the diamond operators ⟨d⟩, whence each subspace S_{k}(N, χ) is stable under T_{p}: since Γ_{0}(N) normalizes both Δ_{p} and Γ_{1}(N), conjugation by an element of Γ_{0}(N) sends one set of right coset representatives for Γ_{1}(N) in Δ_{p} to another. To exhibit the effect of T_{p} on Fourier expansions, suppose that f ∈ S_{k}(N, χ) and write

f(z) = ∑_{n≥1} a(n)q^{n}

with q = e^{2πiz}. A straightforward calculation using the right coset representatives listed above gives

(f|_{k}T_{p})(z) = ∑_{n≥1} a(pn)q^{n} + χ(p)p^{k−1} ∑_{n≥1} a(n)q^{pn}.

Note that if p divides N then χ(p) is to be interpreted as 0 in keeping with the usual conventions for Dirichlet characters modulo N. In the literature T_{p} is often denoted U_{p} in this case and the preceding formula is written

(f|_{k}U_{p})(z) = ∑_{n≥1} a(pn)q^{n} (p|N).

The notation U_{p} has the advantage of forestalling an ambiguity which in principle could arise when N = pM, p ∤ M, and f ∈ S_{k}(M): in this situation the expression f|_{k}T_{p} can have two possible meanings depending on whether we regard f as belonging to S_{k}(M) or to S_{k}(N). Nevertheless, we shall continue to use the notation T_{p} for all primes p, leaving the appropriate interpretation to context.

By a Hecke eigenform we shall mean a nonzero element of S_{k}(N, χ) which is an eigenvector of the operators T_{p} for all primes p. If f = ∑ a(n)q^{n} is a Hecke eigenform and λ_{p} is the eigenvalue of T_{p} on f, then the above formula for f|_{k}T (p) gives

a(pn) − λ_{p}a(n) + χ(p)p^{k−1}a(n/p) = 0 (n ≥ 1),

where a(n/p) is understood to be 0 if n is not divisible by p. Taking n = 1 we see that a(p) = λ_{p}a(1), so that a(1) = 0 implies a(p) = 0. More generally, using induction on the total number of prime factors of n one finds that if a(1) = 0 then a(n) = 0 for all n ≥ 1, whence f = 0. Therefore:

Proposition 15. If f = ∑_{n≥1} a(n)q^{n} is a Hecke eigenform then a(1) ≠ 0.

A Hecke eigenform f = ∑ a(n)q^{n} is said to be normalized if a(1) = 1. The proposition implies that if f is any Hecke eigenform then some scalar multiple of f is normalized. For a normalized eigenform the relation a(p) = λ_{p}a(1) becomes λ_{p} = a(p), whence the recursion formula for a(n) becomes

a(pn) − a(p)a(n) + χ(p)p^{k−1}a(n/p) = 0.

Taking n = p^{ν−1} with ν ≥ 1 one sees that

a(p^{ν}) − a(p)a(p^{ν−1}) + χ(p)p^{k−1}a(p^{ν−2}) = 0,

and then taking n = p^{ν−1}m with m relatively prime to p one deduces by induction on ν that a(p^{ν}m) = a(p^{ν})a(m). A further induction on the number of distinct prime factors of some l relatively prime to m shows that a(lm) = a(l)a(m). In other words, the function n ↦ a(n) is multiplicative; the associated formal Dirichlet series has an Euler product:

∑_{n≥1} a(n)n^{−s} = ∏_{p}(∑_{ν≥0} a(p^{ν})p^{−νs}).

On the other hand, the recursion relation for a(p^{ν}) amounts to the formal identity

∑_{ν≥0} a(p^{ν})p^{−νs} = (1 − a(p)p^{−s} + χ(p)p^{k−1}p^{−2s})^{−1},

and substitution in the preceding equation gives one direction of the following equivalence (the other is obtained by reversing the argument):

Proposition 16. For an element f(z) = ∑_{n≥1} a(n)e^{2πinz} of S_{k}(N), the following are equivalent:

(i) f is a normalized Hecke eigenform.

(ii) ∑_{n≥1} a(n)n^{−s} = ∏_{p}(1 − a(p)p^{−s} + χ(p)p^{k−1−2s})^{−1}.

If f is a normalized Hecke eigenform then the Dirichlet series in (ii) is called the L-function of f and denoted L(f, s). ¿From Proposition 14 we know that there is a functional equation relating the L-function of f to a Dirichlet series associated to f|_{k}W_{N}, but we do not know that the latter Dirichlet series has an Euler product. Thus it remains to find conditions under which both f and f|_{k}W_{N} are Hecke eigenforms. Such conditions are provided by the theory of new forms. The starting point is to define a suitable inner product on S_{k}(N).

3.4. The Petersson inner product

Put

S_{k} = ⋃_{Γ} S_{k}(Γ),

where the union is taken over all subgroups of finite index in SL(2, ℤ). We define an inner product (∗, ∗) on S_{k} as follows. Given f, g ∈ S_{k}, choose a subgroup Γ of finite index in SL(2, ℤ) such that f and g both belong to S_{k}(Γ), and put

(f, g) = [SL(2, ℤ) : Γ]^{−1} ∫_{Γ\GL^{+}(2,ℝ)}(f|_{k}r)(i)\overline{(g|_{k}r)(i)}dr,

where dr denotes the measure on Γ\GL^{+}(2, ℝ) afforded by a Haar measure on GL^{+}(2, ℝ) (recall that GL^{+}(2, ℝ) is unimodular – a left Haar measure is a right Haar measure). Using the fact that f and g are cusp forms, one can check that the integral is absolutely convergent. Furthermore, the factor [SL(2, ℤ) : Γ]^{−1} in front of the integral guarantees that the value of (f, g) is independent of the choice of Γ. Now if

r = ( \begin{matrix}y & x \\ 0 & 1 \end{matrix} )

with y > 0 then

(f|_{k}r)(i)\overline{(f|_{k}r)(i)} = |f(x + iy)|^{2}y^{k},

and consequently (f, f) > 0 if f ≠ 0. Thus (∗, ∗) is in fact an inner product. Since we have not specified a choice of Haar measure on GL^{+}(2, ℝ), we have defined (∗, ∗) only up to a scalar multiple; this suffices for our purposes.

Next we observe that if f, g ∈ S_{k} and δ ∈ GL^{+}(2, ℚ) then f|_{k}δ and g|_{k}δ^{−1} both belong to S_{k} and

(f|_{k}δ, g) = (f, g|_{k}δ^{−1}).

Indeed choose Γ of finite index in SL(2, ℤ) so that f, g ∈ S_{k}(Γ). Then the groups Γ′ = Γ ∩ δ^{−1}Γδ and Γ′′ = Γ ∩ δΓδ^{−1} also have finite index in SL(2, ℤ) and satisfy δΓ′δ^{−1} = Γ′′. Hence we can express (f|_{k}δ, g) and (f, g|_{k}δ^{−1}) as integrals over Γ′\GL^{+}(2, ℝ) and Γ′′\GL^{+}(2, ℝ) respectively, and the stated formula follows from the left-invariance of Haar measure on GL^{+}(2, ℝ). More generally, taking γ, γ′ ∈ Γ and replacing δ by δγ′, we find that

(f|_{k}γδγ′, g) = (f, g|_{k}δ^{−1}),

because f|_{k}γ = f and g|_{k}(γ′)^{−1} = g.

Let us apply the preceding formula with Γ = Γ_{1}(N) and δ ∈ Δ_{p}, where p is a prime not dividing N. Since Δ_{p} is equal to a single double coset of Γ, we have Δ_{p} = ΓδΓ and consequently

(f|_{k}δ′, g) = (f, g|_{k}δ^{−1})

for any δ′ ∈ Δ_{p}. It follows that

(1)  (f|_{k}T_{p}, g) = (p + 1)p^{k/2−1}(f|_{k}, g|_{k}δ^{−1}),

because f|_{k}T_{p} is the sum of p+1 terms of the form p^{k/2−1}f|_{k}δ′. Take δ = δ_{p} in (1), and as usual, let ⟨p⟩ denote any element of Γ_{0}(N) with lower right-hand entry congruent to p modulo N (and hence with upper left-hand entry congruent to p^{−1} modulo N). Since

(pI)⟨p⟩(δ_{p})^{−1} ∈ Δ_{p}

formula (1) becomes

(2)  (f|_{k}T_{p}, g) = (p + 1)p^{k/2−1}(f|_{k}, (g|_{k}⟨p⟩^{−1})|_{k}δ)

with some new element δ of Δ_{p}. Now repeat the previous argument with g|_{k}⟨p⟩^{−1} in place of f. We see that (f|_{k}, (g|_{k}⟨p⟩^{−1})|_{k}γδγ′) is independent of γ, γ′ ∈ Γ, and consequently that

(3)  (p + 1)p^{k/2−1}(f|_{k}, (g|_{k}⟨p⟩^{−1})|_{k}δ) = (f|_{k}, (g|_{k}⟨p⟩^{−1})|_{k}T_{p}).

Together, (2) and (3) give

Ť_{p} = ⟨p⟩^{−1}T_{p},

where Ť_{p} denotes the adjoint of T_{p} on S_{k}(N) with respect to (∗, ∗). Since the diamond operators commute with the Hecke operators, we conclude that for p not dividing N the operators T_{p} are normal. We also obtain:

Proposition 17. Let f ∈ S_{k}(N, χ) be a Hecke eigenform and p a prime not dividing N. If λ_{p} is the eigenvalue of T_{p} on f then \overline{λ_{p}} = \overline{χ}(p)λ_{p}.

As a commuting family of normal operators, the operators T_{p} (p ∤ N) are simultaneously diagonalizable on S_{k}(N). However, simultaneous diagonalization of the T_{p} for all primes p, including those dividing N, is a more delicate matter and is possible in general only on a subspace of S_{k}(N), the subspace of new forms.

3.5. New forms

Consider positive integers M and r such that M divides N properly and r divides N/M, and put

V_{r} = ( \begin{matrix}r & 0 \\ 0 & 1 \end{matrix} ) .

The calculation

( \begin{matrix}r & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & b \\ cN & d \end{matrix} ) ( \begin{matrix}r & 0 \\ 0 & 1 \end{matrix} )^{−1} = ( \begin{matrix}a & br \\ cN/r & d \end{matrix} )

shows that the map f ↦ f|_{k}V_{r} sends S_{k}(M) to S_{k}(N), indeed each subspace S_{k}(M, χ) to the corresponding subspace of S_{k}(N). In fact a glance at Fourier expansions shows that if p does not divide N then (f|_{k}T_{p})|_{k}V_{r} = (f|_{k}V_{r})|_{k}T_{p}, so that f ↦ f|_{k}V_{r} sends eigenvectors of T_{p} to eigenvectors of T_{p}. The need for a distinction between “old forms” and “new forms” arises because this last assertion fails for p dividing N.

The space of old forms of level N is by definition the subspace S_{k}(N)^{old} of S_{k}(N) spanned by the images of the maps f ↦ f|_{k}V_{r} as M and r vary over all integers satisfying the divisibility conditions stated above. In other words,

S_{k}(N)^{old} = span ( ⋃_{\begin{matrix}M|N \\ M<N \end{matrix}} ⋃_{r|N/M}\{f|_{k}V_{r} : f ∈ S_{k}(M)\}) .

A Hecke eigenform belonging to S_{k}(N)^{old} is called an old form of level N. The space of new forms of level N, denoted S_{k}(N)^{new}, is the orthogonal complement of S_{k}(N)^{old} in S_{k}(N) relative to the Petersson inner product. A Hecke eigenform belonging to S_{k}(N)^{new} is called a new form of level N, and a normalized new form of level N is called a primitive form of level N. Let Prim_{k}(N) denote the set of primitive forms of weight k and level N. One of the main theorems of the theory of new forms is that Prim_{k}(N) is a basis for S_{k}(N)^{new}; as a corollary one deduces that the set

⋃_{M|N} ⋃_{r|N/M}{f|_{k}V_{r} : f ∈ Prim_{k}(M)}

is a basis for all of S_{k}(N). Results such as these are important to mention here because they show that the theory of new forms is nonvacuous, but for present purposes the result of primary interest is the following theorem, which will lead us to a functional equation for the L-function of a primitive form:

Theorem 3. Given f ∈ Prim_{k}(N), g ∈ S_{k}(N), and a finite set S of prime numbers such that g is an eigenvector of T_{p} for p ∉ S, suppose that the eigenvalues of T_{p} on f and g coincide for p ∉ S. Then g is a scalar multiple of f.

For the proof, the reader is referred to the literature on new forms: Atkin-Lehner [2], Casselman [4], Li [14], and Miyake [15]. The application to L-functions starts from the observation that if f ∈ S_{k}(N, χ) and

f̌(z) = \overline{f(−\overline{z})}

then f̌ ∈ S_{k}(N, \overline{χ}). This follows from the identity −\overline{γz} = γ′(−\overline{z}), where γ ∈ GL^{+}(2, ℝ) and γ′ is obtained from γ by negating the diagonal entries. One also verifies that the map f ↦ f̌ is unitary with respect to (∗, ∗) and preserves S_{k}(N)^{old}, whence it preserves S_{k}(N)^{new} as well. Now at the level of Fourier expansions the map f ↦ f̌ has the form

∑_{n≥1} a(n)e^{2πinz} ↦ ∑_{n≥1} \overline{a(n)}e^{2πinz}.

Hence on applying complex conjugation to the formal identity in part (ii) of Proposition 17, we see that if f is a normalized Hecke eigenform, then so is f̌. Since S_{k}(N)^{new} is stable under f ↦ f̌ we conclude that Prim_{k}(N) is stable under this map also.

Suppose now that f ∈ S_{k}(N, χ). We shall compare f̌ and f|_{k}W_{N}. For a prime p not dividing N, let Δ′_{p} denote the set consisting of 2 × 2 matrices with integer coefficients and determinant p which are congruent modulo N to a matrix of the form

( \begin{matrix}p & ∗ \\ 0 & 1 \end{matrix} ) .

A calculation shows that

W_{N}Δ_{p}W_{N}^{−1} = Δ′_{p} = Δ_{p}⟨p⟩^{−1}.

Since W_{N} normalizes Γ_{1}(N) we deduce that if {δ} is a set of representatives for the distinct right cosets of Γ_{1}(N) in Δ_{p} then both {W_{N}δW_{N}^{−1}} and {δ⟨p⟩^{−1}} are sets of representatives for the distinct right cosets of Γ_{1}(N) in Δ′_{p}. It follows that

f|_{k}W_{N}|_{k}T_{p} = \overline{χ(p)}f|_{k}T_{p}|_{k}W_{N}.

Thus f is an eigenvector of T_{p} for all primes p not dividing N then so is f|_{k}W_{N}, and if λ_{p} is the eigenvalue of T_{p} on f then λ′_{p} = \overline{χ(p)}λ_{p} is the eigenvalue of T_{p} on f|_{k}W_{N}. Referring to Proposition 18, we see that λ′_{p} = \overline{λ_{p}}, and then Theorem 3 implies that f|_{k}W_{N} is a scalar multiple of f̌. We shall write the scalar in question as i^{−k}W (f), so that

f|_{k}W_{N} = i^{−k}W (f)f̌.

Then Proposition 14 gives:

Proposition 18. Given f ∈ Prim_{k}(N), put Λ(f, s) = N^{s/2}(2π)^{−s}Γ(s)L(f, s). Then Λ(f, s) has an analytic continuation to an entire function of order one satisfying the functional equation Λ(f, s) = W (f)Λ(f̌, k − s).

We have reached the limits of what can be done to suggest a possible connection between modular forms and Conjecture 1 on the basis of formal analytic properties alone. The next step is to make a connection between modular forms and modular curves, or at least between cusp forms of weight 2 and regular differentials on modular curves.

3.6. Differentials and cusp forms of weight 2

To begin with let Γ be any subgroup of finite index in SL(2, ℤ) and let π denote the restriction to ℌ of the natural map ℌ^{∗} → Γ\ℌ^{∗}. If ω is a regular differential on Γ\ℌ^{∗} then π^{∗}ω = f(z)dz for some function f on ℌ. We claim that the functions f which arise in this way are characterized by the following conditions:

(o) f is holomorphic.

(i) f(γz)d(γz) = f(z)dz for γ ∈ Γ.

(ii) Suppose that δ ∈ SL(2, ℤ), and let M be a positive integer such that

(f ∘ δ)(z + M)d(δ(z + M)) = (f ∘ δ)(z)d(δz)

(such an integer exists by (i)). Let F be the holomorphic function on the punctured unit disk D^{∘} = {q ∈ ℂ : 0 < |q| < 1} defined by

f(δz)\frac{d}{dz} δz = F (e^{2πiz/M}) (z ∈ ℌ).

Then F extends to a holomorphic function on the full unit disk D = {q ∈ ℂ : 0 < |q| < 1} vanishing at 0.

Indeed (i) says that the differential f(z)dz on ℌ descends to a differential on Γ\ℌ, while (o) is the condition for the descended differential to be holomorphic (at an elliptic fixed point of Γ the equivalence between the holomorphy of ω and the holomorphy of f requires a small verification). As for (ii), its content is that the descended differential extends holomorphically from Γ\ℌ to Γ\ℌ^{∗}. Again there is a small verification: according to the definition of the complex structure on Γ\ℌ^{∗}, the change of variable q = e^{2πiδ^{−1}z/M} defines a local parameter at δ∞, and condition

(ii) is a consequence of the fact that dz = \frac{dq}{q} · \frac{d}{dz} δz · \frac{M}{2πi}. Thus (o), (i), and (ii) do characterize

the functions on ℌ obtained by pulling back regular differentials from Γ\ℌ. Now if γ = ( \begin{matrix}a & b \\ c & d \end{matrix} )

is an element of GL^{+}(2, ℝ) then

d(γz) = \frac{det(γ)}{(cz + d)^{2}} dz.

Therefore condition (i) can be rewritten

f|_{2}γ = f (γ ∈ Γ),

while in (ii) the requirement is the existence, for any δ ∈ SL(2, ℤ), of a Fourier series expansion of the form

(f|_{2}δ)(z) = ∑_{n≥1} a(n)e^{2πinz/M}.

Returning to the equation π^{∗}ω = f(z)dz, we conclude that as ω runs over the space of regular differentials on Γ\ℌ^{∗} the function f runs over the space of cusp forms of weight 2 for Γ.

Let us now specialize to the case Γ = Γ_{1}(N). We shall write H^{0}(Ω_{X_{1}(N)}^{1}) for the space of regular differentials on X_{1}(N) defined over ℚ and H^{0}(Ω_{X_{1}(N)/ℂ}^{1}) for the corresponding space over ℂ, so that

H^{0}(Ω_{X_{1}(N)/ℂ}^{1}) = ℂ ⊗_{ℚ} H^{0}(Ω_{X_{1}(N)}^{1}).

The isomorphism just described gives an identification

H^{0}(Ω_{X_{1}(N)/ℂ}^{1}) ≅ S_{2}(Γ_{1}(N)),

and on the right-hand side we have an action of the Hecke operators T_{p}. As we shall now explain, the Hecke correspondences determine operators on the left-hand side (to be denoted T_{p} also) such that the above isomorphism respects the action of T_{p}. Quite generally, if T = (Z, φ, ψ) is a correspondence on a smooth projective curve X, then T gives rise to the operator

\begin{matrix}H^{0}(Ω_{X}^{1}) & ⟶ H^{0}(Ω_{X}^{1}) \\ ω & ↦ tr_{φ}(ψ^{∗}ω), \end{matrix}

where tr_{φ} is the trace on differentials associated to the morphism φ: if K ⊂ L is the inclusion of function fields afforded by φ then any α ∈ H^{0}(Ω_{Z}^{1}) has the form α = udv with u ∈ L and v ∈ K, and by definition, tr_{φ}(α) = tr_{L/K}(u)dv. Returning to the case at hand, we see that the Hecke correspondence T_{p} = (X_{1}(N, p), φ_{p}, ψ_{p}) on X_{1}(N) determines an operator T_{p} on H^{0}(Ω_{X_{1}(N)}^{1}) and hence by extension of scalars an operator on H^{0}(Ω_{X_{1}(N)/ℂ}^{1}).

Proposition 19. The canonical isomorphism H^{0}(Ω_{X_{1}(N)/ℂ}^{1}) ≅ S_{2}(N) commutes with the action of T_{p}.

Proof. We begin with a general remark. Suppose that X is a smooth projective curve over a subfield of ℂ and T = (Z, φ, ψ) is a correspondence on X. Let d be the degree of φ. Since the base field is a subfield of ℂ, we can discuss the correspondence T in the language of Riemann surfaces, and in particular we can speak of the local analytic sections (φ^{−1})_{x_{o},i} (1 ≤ i ≤ d) of φ in a neighborhood of some unramified point x_{o} ∈ X(ℂ). Then

tr_{φ}(α) = ∑_{1≤i≤d}((φ^{−1})_{x_{o},i})^{∗}α

locally at x_{o}. Thus for ω ∈ H^{0}(Ω_{X/ℂ}^{1}) we have

T (ω) = ∑_{1≤i≤d}(φ_{i}^{−1})_{x_{o},i}^{∗}ψ^{∗}ω.

This formula makes it possible to compute the action of T on differentials directly from a knowledge of the map T : X(ℂ) ⟶ Div(X(ℂ)). Indeed if the latter map is given locally by

x ↦ ∑_{1≤i≤d}(ρ_{x_{o},i}(x))

with analytic functions ρ_{x_{o},i}, then after a permutation of indices we have ρ_{x_{o},i} = ψ ∘ (φ_{i}^{−1})_{x_{o},i} and consequently

T (ω) = ∑_{1≤i≤d} ρ_{x_{o},i}^{∗}ω

locally at x_{o}.

In the case at hand we can identify X_{1}(N)(ℂ) with Γ_{1}(N)\ℌ^{∗}, and in the coordinate z of ℌ the map T_{p} : X_{1}(N)(ℂ) ⟶ Div(X_{1}(N)(ℂ)) is given by

[z] ↦ \{ \begin{matrix}∑_{ν=0}^{p}[(z + ν)/p] + [⟨p⟩pz] & if p ∤ N \\ ∑_{ν=0}^{p−1}[(z + ν)/p] & if p|N \end{matrix}

(Proposition 9). This map can be written simply as

[z] ↦ ∑_{δ}[δz],

where δ runs over a set of representatives for the right cosets of Γ_{1}(N) in Δ_{p}. Now we have already observed that if f is a function on ℌ and γ ∈ GL^{+}(2, ℝ) then f(γz)d(γz) = (f|_{2}γ)(z)dz. Thus for f ∈ S_{2}(N) we can write

∑_{δ} f(δz)d(δz) = ∑_{δ}(f|_{2}δ)(z)dz = (f|_{2}T_{p})(z)dz,

the factor p^{k/2−1} in the definition of T_{p} being 1 for k = 2.

Using Proposition 11 one proves the analogous statement for the diamond operators:

Proposition 20. The canonical isomorphism H^{0}(Ω_{X_{1}(N)/ℂ}^{1}) ≅ S_{2}(N) commutes with the action of ⟨d⟩ for d ∈ (ℤ/Nℤ)^{×}.

One consequence of Propositions 19 and 20 is that S_{2}(N) has a ℚ-form stable under the operators T_{p} and ⟨d⟩, because H^{0}(Ω_{X_{1}(N)/ℂ}^{1}) has such a ℚ-form, namely H^{0}(Ω_{X_{1}(N)}^{1}). It follows that if S_{2}(N, χ) contains a Hecke eigenform on which the operator T_{p} has eigenvalue λ_{p} then for any σ ∈ Aut(ℂ) the space S_{2}(N, χ^{σ}) contains a Hecke eigenform on which the operator T_{p} has eigenvalue λ_{p}^{σ}. Now the Fourier coefficients of a normalized Hecke eigenform are polynomials with integer coefficients in the Hecke eigenvalues and the character values. Hence we can define an action of Aut(ℂ) on the set of normalized Hecke eigenforms in S_{2}(N) by the rule

f = ∑_{n≥1} a(n)q^{n} ↦ f^{σ} = ∑_{n≥1} a(n)^{σ}q^{n}.

We claim that if f is a new form of level N then so is f^{σ}. Suppose on the contrary that f^{σ} is an old form. Then there is a proper divisor M of N and an element g = ∑_{n≥1} b(n)q^{n} of Prim_{2}(M) such that a(p)^{σ} = b(p) for p ∤ N. Then a(p) = b(p)^{σ^{−1}} for p ∤ N, whence g^{σ^{−1}} is a normalized Hecke eigenform in S_{2}(M) which has the same Hecke eigenvalues as f for p ∤ N. This contradicts Theorem 3, proving the claim. We conclude that the map f ↦ f^{σ} defines an action of Aut(ℂ) on Prim_{2}(N). Since Prim_{2}(N) is finite, it follows that if f ∈ Prim_{2}(N) then the field generated by the Fourier coefficients of f has finite degree over ℚ. We denote this field 𝔼_{f}.

3.7. The Hecke algebra

Given a smooth projective curve X let Corr(X) denote the free abelian group on the set of isomorphism classes of correspondences on X. If T = (Z, φ, ψ) and T′ = (Z′, φ′, ψ′) are correspondences on X we define the product of their isomorphism classes [T′] · [T] by the formula

[T′] · [T] = ∑_{W}[W̃, φ ∘ pr_{Z} ∘ ν_{W} , ψ′ ∘ pr_{Z′} ∘ ν_{W}],

where W runs over the irreducible components of

Z′′ = {(z, z′) ∈ Z × Z′ : ψ(z) = φ′(z′)},

ν_{W} : W̃ → W is the normalization map, and pr_{Z} : Z′′ → Z, pr_{Z′} : Z′′ → Z′ are the projections. Extending this product to Corr(X) by ℤ-linearity, we make Corr(X) into a ℤ-algebra. We shall view Aut(X) as a subgroup of the multiplicative group of Corr(X) by identifying ψ ∈ Aut(X) with the isomorphism class of the correspondence (X, id_{X}, ψ) on X.

In the case of X_{1}(N) we are interested in the subalgebra 𝕋 of Corr(X_{1}(N)) generated over ℤ by the isomorphism classes of all Hecke correspondences T_{p} and all diamond automorphisms ⟨d⟩. We refer to 𝕋 as the Hecke algebra (of level N). Furthermore, we use the same symbol 𝕋 and the same term “Hecke algebra” for the image of 𝕋 under the canonical embedding of Corr(X_{1}(N)) in End(J_{1}(N)), and we likewise identify the opposite algebra 𝕋^{opp} with its image in End(H^{0}(Ω_{X_{1}(N)}^{1})). Alternatively, we can view 𝕋 itself as acting on the dual space of H^{0}(Ω_{X_{1}(N)}^{1}), or we can consider 𝕋 to be acting on H^{0}(Ω_{X_{1}(N)}^{1}) on the right. This last point of view is consistent with our identification of H^{0}(Ω_{X_{1}(N)/ℂ}^{1}) with End(S_{2}(N)) (Propositions 19 and 20), and we may therefore think of 𝕋 as the subring of End(S_{2}(N)) generated over ℤ by the Hecke operators and diamond operators on S_{2}(N). It follows in particular that 𝕋 is commutative, so that 𝕋 and 𝕋^{opp} are canonically isomorphic and every left 𝕋-module is a right 𝕋-module.

The next step is to associate a quotient ring 𝕋_{f} of 𝕋 to each f ∈ Prim_{2}(N). Consider the ring homomorphism λ_{f} : 𝕋 → ℂ such that f|_{2}T = λ_{f}(T)f for T ∈ 𝕋, and let 𝕀_{f} be the kernel of λ_{f}. Then

𝕋_{f} = 𝕋/𝕀_{f}.

Thus 𝕋_{f} is the quotient of 𝕋 by the annihilator ideal of f. If f(z) = ∑_{n≥1} a(n)q^{n} is the Fourier expansion of f and S_{2}(N, χ) is the character space to which f belongs then λ_{f} induces an isomorphism

ℚ ⊗_{ℤ} 𝕋_{f} ⟶ 𝔼_{f}

sending T_{p} + 𝕀_{f} to a(p) and ⟨d⟩ + 𝕀_{f} to χ(d). Let A_{f} be the abelian variety over ℚ defined by the formula

A_{f} = J_{1}(N)/𝕀_{f}J_{1}(N).

The action of 𝕋 on J_{1}(N) induces an action of 𝕋_{f} on A_{f} and hence on each V_{ℓ}(A_{f}).

Proposition 21. The image of 𝕋_{f} in 𝔼_{f} is an order of 𝔼_{f}, and V_{ℓ}(A_{f}) is a free module of rank two over ℚ_{ℓ} ⊗_{ℤ} 𝕋_{f}. In particular, A_{f} is an abelian variety of dimension [𝔼_{f} : ℚ], and A_{f} is an elliptic curve if and only if the Fourier coefficients of f are rational.

Proof. The second statement is contained in the first because V_{ℓ}(A_{f}) is a vector space of dimension 2 dim(A_{f}) over ℚ_{ℓ}, while

dim_{ℚ_{ℓ}} ℚ_{ℓ} ⊗_{ℤ} 𝕋_{f} = rank_{ℤ}𝕋_{f} = [𝔼_{f} : ℚ].

To prove the first statement we start with the observation that as a subring of End(J_{1}(N)), the Hecke algebra 𝕋 acts on H_{1}(J_{1}(N)(ℂ), ℤ) and consequently also on H_{1}(X_{1}(N)(ℂ), ℤ), the two homology groups being isomorphic via the map on homology induced by the embedding of X_{1}(N)(ℂ) in J_{1}(N)(ℂ). Denoting the complex dual of H^{0}(Ω_{X_{1}(N)/ℂ}^{1}) by H^{0}(Ω_{X_{1}(N)/ℂ}^{1})^{∗}, we see that the standard isomorphism of complex tori

J_{1}(N)(ℂ) ≅ \frac{H^{0}(Ω_{X_{1}(N)/ℂ}^{1})^{∗}}{H_{1}(X_{1}(N)(ℂ), ℤ)}

is actually an isomorphism of 𝕋-modules. Hence so is the isomorphism

(1)  J_{1}(N)(ℂ) ≅ S_{2}(N)^{∗}/Λ,

where Λ is the image of H_{1}(X_{1}(N)(ℂ), ℤ) when we identify H^{0}(Ω_{X_{1}(N)/ℂ}^{1})^{∗} with S(2, N)^{∗}. The fact that the lattice Λ in S_{2}(N)^{∗} is stable under 𝕋 already shows that the eigenvalues of 𝕋 on S_{2}(N) are algebraic integers, because eigenvalues are preserved under transpose. It follows that the image of 𝕋_{f} in 𝔼_{f} is an order.

Next put

V_{f} = S_{2}(N)/(S_{2}(N)|_{2}𝕀_{f}),

where S_{2}(N)|_{2}𝕀_{f} denotes the space of all g|_{2}T with g ∈ S_{2}(N) and T ∈ 𝕋. We identify V_{f}^{∗} with the quotient of S_{2}(N)^{∗} by 𝕀_{f}S_{2}(N)^{∗}, and we let Λ_{f} be the lattice in V_{f}^{∗} corresponding to Λ/𝕀_{f}Λ under this identification. Then (1) induces an isomorphism of 𝕋_{f}-modules

(2)  A_{f}(ℂ) ≅ V_{f}^{∗}/Λ_{f}.

We claim that V_{f} (hence also V_{f}^{∗}) is a free module of rank one over ℂ ⊗ 𝕋_{f}. Granting the claim, we deduce that V_{f}^{∗} is free of rank two over ℝ ⊗ 𝕋_{f}. Since V_{f}^{∗} = ℝ ⊗ Λ_{f} it follows that there is a sublattice Λ′_{f} ⊂ Λ_{f} which is free of rank two over 𝕋_{f}. But (2) gives

T_{ℓ}(A_{f}) ≅ \underleftarrow{lim}_{n}(ℓ^{−n}Λ_{f})/Λ_{f} ≅ ℤ_{ℓ} ⊗ Λ_{f}.

Therefore

V_{ℓ}(A_{f}) ≅ ℚ_{ℓ} ⊗ Λ_{f} ≅ ℚ_{ℓ} ⊗ Λ′_{f},

and the proposition follows.

It remains to prove the claim. The semisimple ring ℂ ⊗ 𝔼_{f} is canonically a product

ℂ ⊗ 𝔼_{f} = ∏_{σ} ℂ,

where the factors are indexed by the distinct embeddings of 𝔼_{f} in ℂ. Projection onto the factor corresponding to σ gives a character pr_{σ} : ℂ ⊗ 𝔼_{f} → ℂ sending T_{p} + 𝕀_{f} to a(p)^{σ} and ⟨d⟩ + 𝕀_{f} to χ(d)^{σ}, and a simple ℂ ⊗ 𝔼_{f}-module is a one-dimensional complex vector space on which ℂ ⊗ 𝔼_{f} acts through one of the characters pr_{σ}. Now as a finitely generated ℂ ⊗ 𝔼_{f}-module V_{f} is a direct sum of simple modules and is therefore spanned over ℂ by eigenvectors with eigencharacters of the form pr_{σ}. Suppose that v ∈ V_{f} is such an eigenvector. Then v is in particular an eigenvector for the family of operators T_{f} = {T_{p} + 𝕀_{f} : p ∤ N}. But the action of T_{f} on V_{f} = S_{2}(N)/(S_{2}(N)|_{2}𝕀_{f}) is induced by the action of T = {T_{p} : p ∤ N} on S_{2}(N), and as a commuting family of normal operators T acts semisimply on S_{2}(N). It follows that v is the image in V_{f} of some T -eigenvector g ∈ S_{2}(N). Then Theorem 3 implies that g is a scalar multiple of one of the cusp forms f^{σ}. It follows that the restriction to ⊕_{σ}ℂf^{σ} of the natural map of S_{2}(N) onto V_{f} is surjective. But the restriction is also injective, because

(⊕_{σ}ℂf^{σ}) ∩ (S_{2}(N)|_{2}𝕀_{f}) ⊂ (⊕_{σ}ℂf^{σ}) ∩ (S_{2}(N)^{new}|_{2}𝕀_{f}) = {0}

by the theory of new forms. Therefore V_{f} is isomorphic to ⊕_{σ}ℂf^{σ} as a ℂ ⊗ 𝔼_{f}-module and is consequently free of rank one.

We are now ready to compute the Euler factor of A_{f} at a prime of good reduction:

Theorem 4. Let f ∈ S_{2}(N, χ) be a primitive cusp form of level N, with Fourier expansion

f(z) = ∑_{n≥1} a(n)e^{2πinz}.

If p is a prime not dividing N then

P_{p}(A_{f}, t) = ∏_{σ}(1 − a(p)^{σ}t + χ(p)^{σ}pt^{2}),

where σ runs over the distinct embeddings of 𝔼_{f} in ℂ.

Proof. Fix a prime ℓ and let ρ_{ℓ} denote the natural representation of Gal(\overline{ℚ}/ℚ) on V_{ℓ}(A_{f}). It will suffice to prove that for a prime p not dividing ℓN we have

(1)  det(xI − ρ_{ℓ}(σ_{𝔭})) = ∏_{σ}(x^{2} − a(p)^{σ}x + χ(p)^{σ}p),

where x is an indeterminate, 𝔭 is a prime ideal of \overline{ℚ} lying over p, and σ_{𝔭} ∈ Gal(\overline{ℚ}/ℚ) is the Frobenius automorphism at 𝔭. Indeed the left-hand side of (1) coincides with the characteristic polynomial of ρ_{ℓ}^{∗}(σ_{𝔭}^{−1}) on V_{ℓ}(A_{f}), because a matrix and its transpose have the same characteristic polynomial. Also V_{ℓ}(A_{f}) = V_{ℓ}(A_{f})^{I(𝔭)} by the criterion of Néron-Ogg-Shafarevich [18]. Hence if x is replaced by 1/t and the equation multiplied by t^{2[𝔼_{f}:ℚ]} then (1) becomes the stated formula for P_{p}(A_{f}, t), valid for any prime p not dividing Nℓ. Since ℓ was arbitrary the stated formula follows for any prime p not dividing N.

To prove (1) we recall a fact from linear algebra. Suppose that B is an (mn) × (mn) matrix which can be written as an m × m block matrix B = (B^{ij}) with n × n blocks B^{ij}. Suppose further that the ring generated over ℤ by the matrices B^{ij} is commutative. Then

det B = det(det_{m×m}B),

where det_{m×m}(B) denotes the determinant of the m × m matrix over ℤ[B^{ij}] with ij-entry equal to B^{ij}. On replacing B by xI − B we obtain the formula

det(xI − B) = det(det_{m×m}(xI − B))

for the characteristic polynomial of B.

To apply this formula, recall that V_{ℓ}(A_{f}) is a free module of rank two over ℚ_{ℓ} ⊗𝕋_{f} and observe that ρ_{ℓ}(σ_{𝔭}) is a ℚ_{ℓ} ⊗ 𝕋_{f}-linear transformation of V_{ℓ}(A_{f}). Let det_{ℚ_{ℓ}⊗𝕋_{f}} (xI − ρ_{ℓ}(σ_{𝔭})) denote the characteristic polynomial of ρ_{ℓ}(σ_{𝔭}) as a ℚ_{ℓ} ⊗ 𝕋_{f}-linear map. Then

det(xI − ρ_{ℓ}(σ_{𝔭})) = N_{ℚ_{ℓ}⊗𝕋_{f} /ℚ_{ℓ}}(det_{ℚ_{ℓ}⊗𝕋_{f}} (xI − ρ_{ℓ}(σ_{𝔭})),

where N_{ℚ_{ℓ}⊗𝕋_{f} /ℚ_{ℓ}} is the norm from ℚ_{ℓ} ⊗ 𝕋_{f}[x] to ℚ_{ℓ}[x] (which coincides on 𝕋_{f}[x] with the norm from 𝕋_{f}[x] to ℚ[x]). To prove (1) it suffices to show that

(2)  det_{ℚ_{ℓ}⊗𝕋_{f}} (xI − ρ_{ℓ}(σ_{𝔭})) = x^{2} − T_{p}x + ⟨p⟩p,

because N_{𝕋_{f} /ℚ}(x^{2} − T_{p}x + ⟨p⟩p) is the right-hand side of (1).

Write

ℚ_{ℓ} ⊗ 𝕋_{f} = ∏_{λ|ℓ} 𝔼_{f,λ},

where λ runs over the places of 𝔼_{f} dividing ℓ and 𝔼_{f,λ} idenotes the completion of 𝔼_{f} at λ. Also put ρ_{λ} = pr_{λ} ∘ ρ_{ℓ}, where pr_{λ} is the projection map from ℚ_{ℓ} ⊗ 𝕋_{f} to the factor 𝔼_{f,λ} on the right-hand side. Then equation (2) is equivalent to a system of equations indexed by the places λ, namely the equations

(3)  det(xI − ρ_{λ}(σ_{𝔭})) = x^{2} − T_{p,λ}x + ⟨p⟩_{λ}p

with T_{p,λ} = pr_{λ}(T_{p}) and ⟨p⟩_{λ} = pr_{λ}(⟨p⟩). It follows from Theorem 2 that the right-hand side of (3) annihilates ρ_{λ}(σ_{𝔭}). Furthermore, if a nonscalar 2 × 2 matrix over a field is annihilated by a monic polynomial of degree 2 then that polynomial is its characteristic poynomial. Therefore (3) holds whenever ρ_{λ}(σ_{𝔭}) is nonscalar. Now fix a place λ_{o} dividing ℓ and let P_{o} be the set of primes p not dividing Nℓ such that ρ_{λ_{o}}(σ_{𝔭}) is scalar (note that this condition is independent of the choice of 𝔭). It remains to show that (3) holds for λ = λ_{o} and all p ∈ P_{o}.

Let e_{o} ∈ ℚ_{ℓ} ⊗ 𝕋_{f} be the idempotent which generates the kernel of the map

∏_{λ≠λ_{o}} pr_{λ} : ℚ_{ℓ} ⊗ 𝕋_{f} ⟶ ∏_{λ≠λ_{o}} 𝔼_{f,λ},

and choose an integer ν ≥ 0 such that the element d_{o} = ℓ^{ν}e_{o} belongs to ℤ_{ℓ} ⊗ 𝕋_{f}. Since V_{ℓ}(A_{f}) is free of rank two over ℚ_{ℓ} ⊗ 𝕋_{f} it follows that the ℤ_{ℓ}-module

d_{o}T_{ℓ}(A_{f}) ≅ \underleftarrow{lim}_{n} d_{o}A_{f}[ℓ^{n}]

has a ℤ_{ℓ}-submodule of finite index which is free of rank 2[𝔼_{f,λ_{o}} : ℚ_{ℓ}]. In particular, putting

A_{f}[ℓ^{∞}] = ⋃_{n≥1} A_{f}[ℓ^{n}],

we see that d_{o}A_{f}[ℓ^{∞}] is infinite.

Put L = ℚ(d_{o}A_{f}[ℓ^{∞}]). Then the torsion subgroup of A_{f}(L) contains d_{o}A_{f}[ℓ^{∞}] and is consequently infinite. Hence a theorem of Ribet [16] implies that L is not contained in the maximal cyclotomic extension of ℚ. Therefore the group G = Gal(L/ℚ) is nonabelian. Let Frob_{L}(P_{o}) be the set of Frobenius elements of prime ideals of L lying over primes in P_{o}, and let H be the closure of the subgroup of G generated by Frob_{L}(P_{o}). We claim that H is abelian, whence H is a proper subgroup of G. Indeed ρ_{λ_{o}} can be viewed as a faithful representation of G on d_{o}V_{ℓ}(A_{f}), and since the restriction of ρ_{λ_{o}} to H is scalar the claim follows. Now let \overline{P_{o}} be the complement of P_{o} in the set of prime numbers not dividing Nℓ. Also let Frob_{L}(\overline{P_{o}}) ⊂ G be the set of Frobenius elements of prime ideals of L lying over primes in \overline{P_{o}}. Then the Chebotarev density theorem implies that the set G − H is contained in the closure of Frob_{L}(\overline{P_{o}}). Since any group is generated by the complement of a proper subgroup, it follows that the subgroup generated by Frob_{L}(\overline{P_{o}}) is dense in G.

Next we consider two continuous homomorphisms Gal(\overline{ℚ}/ℚ) ⟶ 𝔼_{f,λ_{o}}^{×}. The first, to be denoted κ_{λ_{o}}, is obtained by composing the ℓ-adic cyclotomic character Gal(\overline{ℚ}/ℚ) ⟶ ℤ_{ℓ}^{×} with the inclusion of ℤ_{ℓ}^{×} in 𝔼_{f,λ_{o}}^{×}. For the second character, we compose the canonical surjection

Gal(\overline{ℚ}/ℚ) ⟶ Gal(ℚ(μ_{N})/ℚ) ≅ (ℤ/Nℤ)^{×}

with the map

\begin{matrix}(ℤ/Nℤ)^{×} & ⟶ 𝕋_{f} \\ d & ↦ ⟨d⟩ \end{matrix}

followed by pr_{λ_{o}}. This second character will be written σ ↦ ⟨σ⟩_{λ_{o}}. Note that if p is a prime not dividing Nℓ then κ_{λ_{o}}(σ_{𝔭}) = p and ⟨σ_{𝔭}⟩_{λ_{o}} = ⟨p⟩_{λ_{o}}. On the other hand, if p happens to belong to \overline{P_{o}}, then det ρ_{λ}(σ_{𝔭}) = ⟨p⟩_{λ_{o}}p, because equation (3) holds for λ = λ_{o} and p ∈ \overline{P_{o}}. Therefore, writing Frob\overline{ℚ}(\overline{P_{o}}) for the set of Frobenius elements of prime ideals of \overline{ℚ} lying over primes in \overline{P_{o}}, we have

det ρ_{λ_{o}}(σ_{𝔭}) = ⟨σ_{𝔭}⟩_{λ_{o}}κ_{λ_{o}}(σ_{𝔭})

for σ_{𝔭} ∈ Frob\overline{ℚ}(\overline{P_{o}}). Since both sides of this equation are continuous, equality holds on the closure of the subgroup of Gal(\overline{ℚ}/ℚ) generated by Frob\overline{ℚ}(\overline{P_{o}}). Let us consider the image of this subgroup under the natural map Gal(\overline{ℚ}/ℚ) → Gal(L/ℚ). The image of Frob\overline{ℚ}(\overline{P_{o}}) is Frob_{L}(\overline{P_{o}}), and we saw above that the subgroup of Gal(L/ℚ) generated by Frob_{L}(\overline{P_{o}}) is dense in Gal(L/ℚ). Thus the closure of the subgroup of Gal(\overline{ℚ}/ℚ) generated by Frob\overline{ℚ}(\overline{P_{o}}) maps onto Gal(L/ℚ). We conclude that if p is any prime not dividing Nℓ then det ρ_{λ_{o}}(σ_{𝔭}) = ⟨p⟩_{λ_{o}}p.

We can now prove that equation (3) holds for λ = λ_{o} and all p ∈ P_{o}. Indeed if B is any nonsingular 2 × 2 matrix over a field which is annihilated by a monic polynomial of degree 2, and if the constant term of that polynomial is the determinant of B, then the polynomial is the characteristic polynomial of B. This completes the proof.

3.8. Modular abelian varieties

Let us now complete the train of thought initiated in Theorem 4. Let f ∈ S_{2}(N, χ) be a primitive cusp form of level N with Fourier expansion

f(z) = ∑_{n≥1} a(n)e^{2πinz}.

For a prime p dividing N we define

P_{p}^{∗}(A_{f}, t) = ∏_{σ}(1 − a(p)^{σ}t).

We also put g = [𝔼_{f} : ℚ], N^{∗}(A_{f}) = N^{g}, and W^{∗}(A_{f}) = ∏_{σ} W (f^{σ}), so that

L^{∗}(A_{f}, s) = ∏_{σ} L(f^{σ}, s)

and

Λ^{∗}(A_{f}, s) = (N^{∗}(A_{f}))^{s/2}((2π)^{−s}Γ(s))^{g}L^{∗}(A_{f}, s).

Then

Λ^{∗}(A_{f}, s) = ∏_{σ} Λ(f^{σ}, s).

Now according to Proposition 18, each Λ(f^{σ}, s) has an analytic continuation to an entire function of order one satisfying the functional equation Λ(f^{σ}, s) = W (f^{σ})Λ(f^{σρ}, 2 − s), where ρ ∈ Aut(ℂ) denotes complex conjugation. Since composition with ρ merely permutes the distinct embeddings of 𝔼_{f} in ℂ, we deduce that Λ^{∗}(A_{f}, s) has an analytic continuation to an entire function of order one satisfying the functional equation

Λ^{∗}(A_{f}, s) = W^{∗}(A_{f})Λ^{∗}(A_{f}, 2 − s).

Consequently A_{f} satisfies Conjecture 1^{∗}. However, it follows from a theorem of Carayol [3] (completing work of Deligne [5], Ihara [11], and Langlands [12]) that P_{p}(A_{f}, t) = ∏_{σ}(1 − a(p)^{σ}t) for p dividing N, and furthermore that N(A_{f}) = N^{g} and W (A_{f}) = ∏_{σ} W (f^{σ}). Thus a stronger assertion holds:

Theorem 5. For f ∈ Prim_{2}(N) and g = [𝔼_{f} : ℚ] the invariants L(A_{f}, s), N(A_{f}), and W (A_{f}) coincide with ∏_{σ} L(f^{σ}, s), N^{g}, and ∏_{σ} W (f^{σ}) respectively, where σ runs over the distinct embeddings of 𝔼_{f} in ℂ. Consequently A_{f} satisfies Conjecture 1.

Let Prim_{2} denote the union of the set Prim_{2}(N) over all positive integers N, and let A be an abelian variety over ℚ. If A is isogenous over ℚ to a product of abelian varieties of the form A_{f} with f ∈ Prim_{2}, then we call A a modular abelian variety, or in the case of dimension one, a modular elliptic curve. Since the L-function, conductor, and root number of A depend on A only up to isogeny over ℚ, and since all three of these invariants respect products, we deduce:

Corollary. If A is a modular abelian variety then A satisfies Conjecture 1.

In the remaining paragraphs we discuss a partial converse to the corollary in the case of dimension one, the converse being contingent on a suitable strengthening of Conjecture 1.

3.9. Conjecture 1 with twists

As we have already mentioned, Conjecture 1 is a special case of a more general hypothesis about L-functions of motives. We shall now state a slight extension of Conjecture 1 (still far from the general case) in which we allow twists of the motives in Conjecture 1 by Artin motives. For the application we have in mind it would suffice to consider Artin motives corresponding to Dirichlet characters, but specializing the context in this way does not seem to simplify the formulation.

Consider as before an abelian variety A over ℚ together with its associated family of ℓ-adic representations {ρ_{ℓ}}. In addition, let τ be a continuous finite-dimensional complex representation of Gal(\overline{ℚ}/ℚ), and let 𝔼_{τ} ⊂ ℂ be a finite extension of ℚ such that τ is realizable on an 𝔼_{τ} -vector space W. If λ is a place of 𝔼_{τ} lying over some ℓ and 𝔼_{τ,λ} is the completion of 𝔼_{τ} at λ then we obtain a representation ρ_{ℓ} ⊗ τ of Gal(\overline{ℚ}/ℚ) on the 𝔼_{τ,λ}-vector space

U_{λ} = (𝔼_{τ,λ} ⊗_{ℚ_{ℓ}} V_{ℓ}(A)) ⊗ (𝔼_{τ,λ} ⊗_{𝔼_{τ}} W).

Given a prime p, we choose ℓ ≠ p and put

P_{p}(A, t) = det (1 − t(ρ_{ℓ} ⊗ τ)(σ_{𝔭})|U_{λ}^{I(𝔭)}) .

As before, the semistable reduction theorem implies that the coefficients of P_{p}(A, t) lie in 𝔼 and are independent of ℓ and λ. Furthermore, the complex numbers α_{i,p} in the factorization

P_{p}(A, τ, t) = ∏_{i=1}^{2g dim τ}(1 − α_{i,p}t).

still satisfy

|α_{i,p}| ≤ \sqrt{p} (1 ≤ i ≤ 2g dim τ),

so that the Euler product

L(A, τ, s) = ∏_{p} P_{p}(A, τ, p^{−s})^{−1}

converges for Re(s) > 3/2. Also, the conductor N(A, τ) of the compatible family {ρ_{ℓ} ⊗ τ}_{ℓ} is defined, as is the root number W (A, τ), which is a complex number of absolute value 1 (no longer necessarily equal to ±1 unless τ is equivalent to its contragredient τ^{∗}). If the conductors N(A) and N(τ) of A and τ are relatively prime, then

N(A, τ) = N(A)^{dim τ} N(τ)^{2g}

and

W (A, τ) = det τ((−1)^{g}N(A))W (A)^{dim τ} W (τ)^{2g}(−1)^{dim τ(g−1)},

where in the second equation W (τ) is the root number of τ and det τ is thought of as a Dirichlet character.

Conjecture 2. Put Λ(A, τ, s) = N(A, τ)^{s/2}((2π)^{−s}Γ(s))^{g dim τ} L(s, A, τ). Then Λ(A, τ, s) has an analytic continuation to an entire function of order one satisfying the functional equation

Λ(A, τ, s) = W (A, τ)Λ(A, τ^{∗}, 2 − s).

For A = A_{f} and certain τ with solvable image a statement along these lines follows from the Rankin-Selberg method and the theory of base change (cf. [1], [13], [21]). If A = A_{f} and τ is one-dimensional then Conjecture 2 is subsumed in the results of Carayol [3].

3.10. Epilogue: the Shimura-Taniyama conjecture

Let us now consider Conjecture 2 in the special case where dim A and dim τ are both one. Thus A is an elliptic curve and τ can be identified with a primitive Dirichlet character χ. We shall further assume that the integers N = N(A) and r = N(χ) are relatively prime, whence

N(A, χ) = Nr^{2}

and

W (A, χ) = χ(−N)W (A)W (χ)^{2}.

In this setting the assertion of Conjecture 2 has a particularly elementary formulation. To begin with, let us put

a(p) = \{ \begin{matrix}1 − |Ã(𝔽_{p})| + p & if A has good reduction at p \\ 1 & if A has split multiplicative reduction at p \\ −1 & if A has nonsplit multiplicative reduction at p \\ 0 & if A has additive reduction at p. \end{matrix}

Then the Euler factors of A are determined by the elementary rule

P_{p}(A, t) = \{ \begin{matrix}1 − a(p)t + pt^{2} & if A has good reduction at p \\ 1 − a(p)t & if A has bad reduction at p. \end{matrix}

Therefore

L(A, s) = ∏_{p∤N(A)}(1 − a(p)p^{−s} + p^{1−2s})^{−1} · ∏_{p|N(A)}(1 − a(p)p^{−s})^{−1}.

Furthermore, since we are assuming that r is relatively prime to N, the L-function L(A, χ, s) coincides with the naive twist of L(A, s) by χ: if we write L(A, s) as a Dirichlet series

L(A, s) = ∑_{n≥1} a(n)n^{−s},

then

L(A, χ, s) = ∑_{n≥1} χ(n)a(n)n^{−s}.

Thus in the case at hand Conjecture 2 asserts that the function

Λ(A, χ, s) = (Nr^{2})^{s/2}(2π)^{−s}Γ(s) ∑_{n≥1} χ(n)a(n)n^{−s}

is entire of order one and satisfies the functional equation

Λ(A, χ, s) = χ(−N)W (A)W (χ)^{2}Λ(A, \overline{χ}, 2 − s).

Now compare this assertion to condition (i) of the following result, which is a version of Weil’s converse to Hecke theory specialized to the case of weight 2 and trivial character:

Theorem 6. Let N be a positive integer and a(1), a(2), a(3), … a sequence of complex numbers satisfying the formal identity

∑_{n≥1} a(n)n^{−s} = ∏_{p∤N}(1 − a(p)p^{−s} + p^{1−2s})^{−1} · ∏_{p|N}(1 − a(p)p^{−s})^{−1}.

Suppose furthermore that

|a(p)| < \{ \begin{matrix}2p & if p ∤ N \\ p & if p|N, \end{matrix}

so that the Dirichlet series and Euler product actually converge for Re(s) > 2. Put

f(z) = ∑_{n≥1} a(n)e^{2πinz}.

Then the following are equivalent:

(i) There exists a complex number W (f) of absolute value 1 such that for every positive integer r prime to N and every primitive Dirichlet character χ modulo r, the function

Λ(f, χ, s) = (Nr^{2})^{s/2}(2π)^{−s}Γ(s) ∑_{n≥1} χ(n)a(n)n^{−s}

has an analytic continuation to an entire function of order one satisfying the functional equation

Λ(f, χ, s) = χ(−N)W (f)W (χ)^{2}Λ(f, \overline{χ}, 2 − s).

(ii) f is a primitive cusp form of weight 2 for Γ_{0}(N).

Theorem 6 can be pieced together from Weil [22], Deligne-Serre ([7], p. 515, Lemme 4.9), and the theory of new forms ([2].[4],[14],[15]). It applies in particular to the situation at hand, because if a(p) is the coefficient of p^{−s} in the L-series L(A, s) of an elliptic curve A over ℚ, then

|a(p)| ≤ \{ \begin{matrix}2\sqrt{p} & if p ∤ N \\ 1 & if p|N, \end{matrix}

which is a stronger estimate than that required by the hypothesis of the theorem. Thus conditions (i) and (ii) are equivalent for L(A, s), and if we grant Conjecture 2 then it follows that there is a primitive cusp form f for Γ_{0}(N) of weight 2 such that L(f, s) = L(A, s). Now this equation implies in particular that the Fourier coefficients of f are rational, whence 𝔼_{f} = ℚ and A_{f} is an elliptic curve. Furthermore, Theorem 3 gives L(A_{f}, s) = L(A, s), and then the isogeny theorem of Faltings implies that A is isogenous over ℚ to A_{f}. Thus A is a modular elliptic curve. To summarize, if we grant Conjecture 2, then we are forced to believe:

Conjecture 3. Every elliptic curve over ℚ is modular.

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