CHAPTER 17 Lecture 17

1. Modular Curves

In this section we summarize the main properties of modular curves which will be of interest to us. We will refer primarily to [DI], which provides more extensive references and some proofs. Other standard references are (in rough order of difficulty) [Se-CA, Chapter 7], [Si-2, Chapter 1], [La-MF] and [Shi].

1.1. Definitions. Let ℌ be the upper half plane in ℂ; that is,

ℌ = {z ∈ ℂ | Im z > 0}.

Recall that there is a natural left action of SL_{2}(ℤ) on ℌ, given by

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

(In fact, −1 ∈ SL_{2}(ℤ) acts trivially, so that this action can be viewed as an action of PSL_{2}(ℤ).)

Fix an integer N and let Γ_{0}(N) be the subgroup of SL_{2}(ℤ)

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) | c ≡ 0 (mod N)\} .

Then one can show that the quotient

Γ_{0}(N)\ℌ

has the structure of a Riemann surface. (See [DI, Section 7].) In fact, this Riemann surface can be given the structure of a smooth, affine algebraic curve defined over ℚ, which we denote by Y_{0}(N). (See [DI, Section 8].) Thus

Y_{0}(N)(ℂ) ≅ Γ_{0}(N)\ℌ.

The projective closure of Y_{0}(N) is denoted by X_{0}(N); the complement of Y_{0}(N) in X_{0}(N) are called the cusps. X_{0}(N) is a smooth, projective curve defined over ℚ.

The importance of X_{0}(N) and Y_{0}(N) to us comes from the fact that they are a (partial) solution to the problem of classifying modular pairs (E/S, C_{N}), where S is some ℚ-scheme, E is an elliptic curve over S (that is, E is an abelian scheme over S of dimension 1) and C_{N} is a finite, flat subgroup scheme of E, all of whose geometric fibers are cyclic groups of order N. (If S is the spectrum of a field K, then this is just a pair of an elliptic curve defined over K and a cyclic subgroup of order N of E(K̄). For simplicity we shall always refer to C_{N} as above as “cyclic subgroups of order N”.) More precisely, we define a functor F_{N} from ℚ-schemes to sets by sending a ℚ-scheme S to the set of isomorphism classes of modular pairs (E/S, C_{N}). (Two modular pairs (E/S, C_{N}) and (E′/S, C′_{N}) are isomorphic if there is an isomorphism (over S) of E and E′ taking C_{N} to C′_{N}.) This functor is not actually representable, but Y_{0}(N) is what is called the representable hull. This means that if F_{Y_{0}(N)} is the functor from ℚ-schemes to sets

0Last modified December 17, 1999

S ↦ Hom_{Spec ℚ}(S, Y_{0}(N)) = Y_{0}(N)(S),

then there is a natural transformation of functors

j : F_{N} → F_{Y_{0}(N)}

with the following universal property: If Z is any other ℚ-scheme such that there is a natural transformation

i : F_{N} → F_{Z},

where F_{Z} is the functor sending ℚ-schemes to Hom_{Spec ℚ}(S, Z), then i factors uniquely through j:

\xymatrix{F_{N} \ar[rr]^{i} \ar[dr]_{j} &  & F_{Z} \\  & F_{Y_{0}(N)} \ar@{-->}[ur]}

Note that any natural transformation F_{Y_{0}(N)} → F_{Z} must come from some map of schemes

Y_{0}(N) → Z;

specifically, the image of the identity map in F_{Y_{0}(N)}(Y_{0}(N)) yields an element of F_{Z}(Y_{0}(N)) = Hom_{Spec ℚ}(Y_{0}(N), Z), and one easily checks that this map Y_{0}(N) → Z induces the given natural transformation F_{Y_{0}(N)} → F_{Z}.

Now, j yields a map j_{S} from pairs (E/S, C_{N}) to Y_{0}(N)(S). In general this map need be neither injective nor surjective. However, if K is an extension of ℚ (not necessarily finite), then the map j_{K} from (E/K, C_{N}) to Y_{0}(N)(K) is surjective. If K is algebraically closed, then it is a bijection.

1.2. Modular Curves over ℂ. In order to better illustrate the modular interpretation of Y_{0}(N), let us return to the case of modular curves over the complex numbers. We assume in this section that the reader is familiar with the theory of elliptic curves over ℂ as in [Si-AEC, Chapter 6].

Recall that an elliptic curve over ℂ is isomorphic to the quotient of ℂ by a lattice Λ ⊆ ℂ. From this point of view, Y_{0}(N)(ℂ) classifies isomorphism classes of pairs of lattices (Λ ⊆ Λ′) where Λ′/Λ is cyclic of order N. Specifically, to such a pair we associate the pair

(ℂ/Λ, Λ′/Λ)

of an elliptic curve and a subgroup of order N, and two pairs of lattices are isomorphic if the give rise to isomorphic pairs. By [Si-AEC, Chapter 6, Corollary 5.1.1] this occurs if and only if the pairs are homothetic.

Let us now consider pairs (Λ ⊆ Λ′) more explicitly. It is not difficult to see that one can choose a ℤ-basis ω_{1}, ω_{2} ∈ ℂ of Λ such that

Im(ω_{1}/ω_{2}) > 0

and ω_{1}, \frac{ω_{2}}{N} is a ℤ-basis of Λ′. Since lattice pairs are defined only up to homothety, we obtain a map τ from pairs (ω_{1}, ω_{2}) ∈ ℂ × ℂ with Im(ω_{1}/ω_{2}) > 0 to the upper half plane, given by

(ω_{1}, ω_{2}) ↦ \frac{ω_{1}}{ω_{2}} .

To complete our analysis, we must determine what other pairs (ω_{1}, ω_{2}) give rise to the same lattice pair

(ℤω_{1} ⊕ ℤω_{2}, ℤω_{1} ⊕ ℤ\frac{ω_{2}}{N}).

(The map τ already takes care of homothety.) If (ω′_{1}, ω′_{2}) is some other pair (with Im(ω′_{1}/ω′_{2}) > 0 as always), it is a basis for Λ if and only if there is some matrix

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ)

such that

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

(The matrix must be in SL_{2}(ℤ) rather than GL_{2}(ℤ) in order to preserve the upper half-plane condition.) Furthermore, it is easy to see that it gives rise to the same Λ′ if and only if

c ≡ 0 (mod N);

that is, if

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

Thus the set of isomorphism classes of lattice pairs is in bijection with the quotient of the upper half-plane by the action of Γ_{0}(N); that is, they are in bijection with

Γ_{0}(N)\ℌ ≅ Y_{0}(N)(ℂ).

Since isomorphism classes of lattice pairs correspond to isomorphism classes of modular pairs (E/ℂ, C_{N}), this shows that the points of Y_{0}(N)(ℂ) really do correspond to such pairs, in a canonical way.

One can also give an explicit description of the cusps

X_{0}(N)(ℂ) − Y_{0}(N)(ℂ).

To do this, one considers the extended upper half-plane

ℌ^{∗} = ℌ ∐ ℙ^{1}(ℚ).

One defines an action of Γ_{0}(N) on ℙ^{1}(ℚ) in the natural way: thinking of elements

of ℙ^{1}(ℚ) as homogeneous pairs ( \begin{matrix}a \\ b \end{matrix} ), the action is just by multiplication on the

left; thinking of ℙ^{1}(ℚ) as ℚ ∐{∞} (as we usually will), one has

( \begin{matrix}a & b \\ c & d \end{matrix} ) τ = \frac{aτ + b}{cτ + d},

with the convention that

\frac{a∞ + b}{c∞ + d} = \frac{a}{c}.

With this action, one has

X_{0}(N)(ℂ) = Γ_{0}(N)\ℌ^{∗},

so that the cusps correspond to the Γ_{0}(N)-orbits on ℙ^{1}(ℚ).

In the case that N is prime, one easily checks that there are only two cusps:

0 = \{\frac{a}{b} ∈ ℚ | b ≢ 0 (mod N)\}

∞ = \{\frac{a}{b} ∈ ℚ | b ≡ 0 (mod N)\} ∐\{∞\}.

Any time that we need a base point for X_{0}(N) (whether N is prime or not) we will use the cusp containing the class of ∞.

1.3. Functions on Modular Curves. We consider briefly here the theory of functions and differential forms on modular forms; this a moderately important area in modern number theory^{1}, and we give here a complete treatment^{2}.

Note that for any integer N, the matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) lies in Γ_{0}(N). This matrix

acts on ℌ by

τ ↦ τ + 1.

Thus any function or differential form on X_{0}(N), when considered as a Γ_{0}(N)- invariant function or differential form on ℌ^{∗}, can be expanded in terms of the parameter

q = e^{2πiτ} .

We can therefore expand functions f : X_{0}(N)(ℂ) → ℂ as Laurent series

f(τ) = ∑_{n} a_{n}q^{n}

(these are modular forms of weight 0 and level N) and differential forms fdτ as

f(τ)dτ = ∑_{n} a_{n}q^{n} \frac{dq}{q}

(these are modular forms of weight 2 and level N).

2. Operators on Modular Curves

2.1. Degeneracy operators. We are going to construct operators on modular curves by defining the operators on the underlying problems which the modular curves “classify”. It will be useful to refer to the following formal argument. Let F_{1} and F_{2} be two functors from ℚ-schemes to sets which have ℚ-schemes Y_{1} and Y_{2} as representable hulls; thus, if F_{Y_{1}} and F_{Y_{2}} are the functors represented by Y_{1} and Y_{2} respectively, we have natural transformations

j_{1} : F_{1} → F_{Y_{1}}

and

j_{2} : F_{2} → F_{Y_{2}}.

Suppose that we define a natural transformation

i : F_{1} → F_{2}.

Composition with j_{2} yields a natural transformation

j_{2} ∘ i : F_{1} → F_{Y_{2}},

1Sarcasm.

2Extreme sarcasm.

so since Y_{1} is a representable hull for F_{1} we obtain a commutative diagram of natural transformations

\xymatrix{F_{1} \ar[rr]^{j_{2}∘i} \ar[dr]^{j_{1}} &  & F_{Y_{2}} \\  & F_{Y_{1}} \ar[ur]^{i′}}

i′ in turn comes from a map

Y_{1} → Y_{2}.

Thus any natural transformation of functors yields a canonical map of their representable hulls. If in addition the Y_{i} are smooth with projective closures X_{i}, one obtains a map

X_{1} → X_{2}.

For any integer N, we let F_{N} be the functor from ℚ-schemes to sets associating to a ℚ-scheme S isomorphism classes of pairs (E, C_{N}) as in Section 1.1. Thus F_{N} has Y_{0}(N) as a representable hull. Also, for simplicity we will only give constructions over fields, or sometimes even only over algebraically closed fields, with references to [DI] for the general construction.

Now, let N be an integer dividing another integer M. Let d be an integer dividing the quotient M/N; d = 1 is permitted. We will define a map

B_{d} : X_{0}(M) → X_{0}(N)

called a degeneracy operator. We will do this by defining a natural transformation of the functors F_{M} and F_{N}. So let K be a field containing ℚ and let (E/K, C_{M}) be a pair of an elliptic curve defined over K and a cyclic subgroup of order M of E(K̄). C_{M} has a unique subgroup C_{d} of order d. Furthermore, the cyclic subgroup C_{M}/C_{d} of order M/d on the elliptic curve E/C_{d} has a unique subgroup C_{N} of order N, since N divides M/d. (See [Si-AEC, Chapter 3, Section 4] for basics on quotients of elliptic curves.) We let B_{d} (as a natural transformation of functors) be the correspondence

(E, C_{M}) ↦ (E/C_{d}, C_{N}).

As above, B_{d} induces a map, which we shall also call B_{d}, from

X_{0}(M) → X_{0}(N).

(See [DI, Sections 6.3 and 7.3] for the general case.)

In the case K = ℂ, one can ask what effect B_{d} has on q-expansions. If q_{M} is the parameter on X_{0}(M) and q_{N} is the parameter on X_{0}(N), it is not difficult to show (from the analysis in Section 1.2) that under B_{d}, q_{N} pulls back to q_{M}^{d}.

2.2. Atkin-Lehner Involutions. We now define maps from modular curves to themselves. So fix an integer N, and let N = ab be a factorization of N with a and b relatively prime. We will define a map

w_{a} : X_{0}(N) → X_{0}(N)

called an Atkin-Lehner operator, again by defining a natural transformation from F_{N} to F_{N}. (See [DI, Section 4 and Remark 8.4.1] for the general case.)

So let K be a field containing ℚ and let (E/K, C_{N}) be a modular pair. The group C_{N} has a unique decomposition (since a and b are relatively prime)

C_{N} = C_{a} + C_{b}

as a product of a cyclic group of order a and a cyclic group of order b. Also recall that the full a-torsion E[a] of E is free of rank 2 over ℤ/aℤ. We define w_{a} to be the natural transformation

(E, C_{N}) ↦ (E/C_{a}, E[a] + C_{b}/C_{a});

since E[a]/C_{a} is cyclic of order N and C_{a} ∩ C_{b} = {1} this makes sense.

We further claim that

w_{a}^{2} = 1.

To see this, note that

w_{a}^{2}(E, C_{N}) = w_{a}(E/C_{a}, E[a] + C_{b}/C_{a}) = (E/E[a], (E/C_{a})[a] + C_{b}/E[a]).

One checks easily that multiplication by a

a : E/E[a] → E

yields the desired isomorphism with (E, C_{N}).

In the case K = ℂ, we can also interpret w_{a} in terms of an action on the upper half-plane. Going through the analysis in Section 1.2, one finds that there is a commutative diagram

\xymatrix{ℌ \ar[r] \ar[d] & ℌ \ar[d] \\ Y_{0}(N)(ℂ) \ar[r]^{w_{n}} & Y_{0}(N)(ℂ)}

where the vertical maps are the quotient maps and the top horizontal map is

τ ↦ −\frac{1}{aτ} .

Note that it is completely unclear what the operation on q-expansions is; to determine it, one must find the q-expansion

e^{−2πi/aτ} ,

which is rather mysterious.

2.3. Jacobians of Curves. We briefly recall some mapping properties of jacobians of curves. (See Lecture 13, Section 1.1 for the basics on jacobians.) So let X and Y be smooth, proper, irreducible curves over some field K with jacobians J_{X} and J_{Y} respectively. Suppose that we have a non-constant map

φ : X → Y;

thus φ has finite degree. It induces both a covariant map

φ_{∗} : J_{X} → J_{Y}

and a contravariant map

φ^{∗} : J_{Y} → J_{X}.

See [Si-AEC, Chapter 2, Section 3] for the corresponding maps on divisors. The composition

φ_{∗}φ^{∗} : J_{Y} → J_{Y}

is simply multiplication by the degree of φ.

We will write J_{0}(N) for the jacobian of the modular curve X_{0}(N). We fix the embedding

X_{0}(N) ↪ J_{0}(N)

so that the cusp ∞ of X_{0}(N) maps to the origin of J_{0}(N). Thus an arbitrary point x ∈ X_{0}(N) maps to the divisor class of

(x) − (∞).

2.4. Hecke Correspondences. Fix an integer N and a prime number l not dividing N. We will define a correspondence T_{l} on X_{0}(N) called the l^{th} Hecke correspondence or the l^{th} Hecke operator. For our purposes, a correpondence will simply be a many-valued

\xymatrix{X_{0}(N) \ar@{-->}[r] & X_{0}(N).}

Alternately, and somewhat more accurately, a self-correspondence on X_{0}(N) is a subscheme T_{l} of the product X_{0}(N) ×_{Spec ℚ} X_{0}(N). See [DI, Section 3.2] for more details on correspondences, and [DI, Section 3 and Section 8.3] for details on Hecke operators.

For our first definition, we work on the level of the functor F_{N} over an algebraically closed field K. Let (E, C_{N}) be a modular pair over K. Recall that E[l] ≅ ℤ/lℤ × ℤ/lℤ. We consider the subgroups of E[l] of order l + 1; there are l + 1 of them and they are naturally classified by ℙ^{1}(𝔽_{l}). (See [Se-CA, Chapter 7, Section 5] for all combinatorial analysis in this section.) We define T_{l} by sending (E, C_{N}) to the l + 1 values

(E/C_{l}, C_{l} + C_{N}/C_{l}),

where C_{l} runs over the subgroups of E[l] of order l.

In the case K = ℂ we have X_{0}(N)(ℂ) = Γ_{0}(N)\ℌ and we can be a bit more explicit. If τ ∈ Γ_{0}(N)\ℌ, T_{l}(τ) is the set of values

\frac{τ}{l} , \frac{τ + 1}{l}, ⋯, \frac{τ + l − 1}{l}, lτ.

The advantage of this expression is that the number N does not appear; the Hecke correspondence T_{l} is thus in some sense the same on all modular curves.

We can also use the above analysis to determine the effect of T_{l} on q-expansions. We consider the case of a differential form

f \frac{dq}{q} = ∑ a_{n}q^{n} \frac{dq}{q} .

The Hecke correspondence is clearly linear, so it will suffice to consider the case of

q^{n} \frac{dq}{q} = e^{2πinτ} \frac{dq}{q} .

Under T_{l}, this goes to

\frac{1}{l} ∑_{j=0}^{l−1} e^{2πin(τ+j)/l} \frac{dq}{q} + le^{2πinlτ} \frac{dq}{q} = \frac{1}{l} ∑_{j=0}^{l−1} e^{2πinj/l} · e^{2πinτ/l} \frac{dq}{q} + le^{2πinlτ} \frac{dq}{q} .

But

∑_{j=0}^{l−1} e^{2πinj/l} = \begin{cases}0 & l ∤ n \\ l & l | n, \end{cases}

so the above expression reduces to

q^{n/l} \frac{dq}{q} + lq^{nl} \frac{dq}{q} ,

where we interpret q^{n/l} as 0 if l does not divide n. In general, then,

T_{l} (∑ a_{n}q^{n} \frac{dq}{q} ) = ∑(a_{nl} + la_{n/l})q^{n} \frac{dq}{q} ,

where we set a_{n/l} = 0 if l does not divide n.

One can also define T_{l} in terms of the operator

B_{1} : X_{0}(Nl) → X_{0}(N)

and the Atkin-Lehner involution w_{l} on X_{0}(Nl). We write π for B_{1} for this section. Consider the (many-valued) composition

X_{0}(N) ⟶^{π^{−1}} X_{0}(Nl) ⟶^{w_{l}} X_{0}(Nl) ⟶^{π} X_{0}(N).

The first map sends a modular pair (E, C_{N}) to the set of all pairs (E, C_{N} + C_{l}), where C_{l} is a cyclic subgroup of E of order l. w_{l} then sends (E, C_{N} + C_{l}) to (E/C_{l}, E[l] + C_{N}/C_{l}), which is mapped by π to (E/C_{l}, C_{l} + C_{N}/C_{l}). Thus this composition is precisely the Hecke correspondence T_{l}.

This interpretation allows us to interpret T_{l} as an endomorphism of the Jacobian J_{0}(N). Specifically, the commutative diagram

\xymatrix{X_{0}(Nl) \ar[r]^{w_{l}} \ar[d]^{π} & X_{0}(Nl) \ar[d]^{π} \\ X_{0}(N) \ar@{-->}[r]^{T_{l}} & X_{0}(N)}

is reinterpreted as a commutative diagram

\xymatrix{J_{0}(Nl) \ar[r]^{w_{l}^{∗}} & J_{0}(Nl) \ar[d]^{π_{∗}} \\ J_{0}(N) \ar[r]^{T_{l∗}} \ar[u]^{π^{∗}} & J_{0}(N)}

(That is, we take π_{∗}w_{l∗}π^{∗} as the definition of T_{l∗}.) We could also define T_{l}^{∗} by this method, but note that since w_{l} is an involution we have w_{l∗} = w_{l}^{∗}, so

T_{l}^{∗} = π_{∗}w_{l}^{∗}π^{∗} = T_{l∗}.

We will often just write T_{l} for the map on J_{0}(N) induced by the Hecke correspondence.

There is also a double coset decomposition definition of Hecke operators, which is quite useful for generalizations. Ask the local Gross student or see [DI, Section 3.1].