Math 726: L-functions and modular forms Fall 2011

Lecture 15 : Algebraic Modular Forms

Instructor: Henri Darmon Notes written by: Maxime Turgeon

Recall the following theorem:

Theorem 1. For all normalised Hecke newforms f ∈ S_{k}(Γ_{0}(N), χ), with Fourier coefficients in a number field K, there exists a compatible system {V_{f,λ}}_{λ∈Spec(O_{K})} of λ-adic K-rational representations of G_{ℚ} such that

L({V_{f,λ}}_{λ∈Spec(O_{K})}, s) = L(f, s).

More precisely,

• For all primes p ∤ Nℓ (where ℓ = Norm_{ℚ}^{K}(λ)), V_{f,λ} is unramified at p, and

det((1 − xF rob_{p})|_{V_{f,λ}}) = 1 − a_{p}(f)x + χ(p)p^{k−1}x^{2};

• For all primes p | N, V_{f,λ} is ramified at p (even when λ ∤ p), and

det((1 − xF rob_{p})|_{V_{f,λ}^{Ip}}) = 1 − a_{p}(f)x.

The main goal of the next few lectures is now to give some idea of how {V_{f,λ}} is constructed from f.

1. (Weight of f is 2 ) The representations {V_{f,λ}} are obtained from étale cohomology. Consider the congruence subgroup Γ_{1}(N). If we take the quotient Y_{1}(N) := Γ_{1}(N)\H of the upper half-plane by the action of this group, we get a Riemann surface, whose compactification is usually denoted X_{1}(N) (geometrically, X_{1}(N) is obtained from Y_{1}(N) by adding the cusps). A priori, X_{1}(N) is a curve defined over ℂ, but one can show that it actually admits a natural model over the field ℚ of rational numbers, that is, X_{1}(N) can actually be defined over ℚ. The representations we are looking for are then realised as the action of G_{ℚ} on the cohomology groups H_{ét}^{1}(X_{1}(N)\overline{ℚ}, ℚ_{ℓ}). Luckily, in this case, these groups have a nice interpretation, namely H_{ét}^{1}(X_{1}(N)\overline{ℚ}, ℚ_{ℓ}) is the dual of V_{ℓ}(J_{1}(N)\overline{ℚ}) := ℚ_{ℓ} ⊗_{ℤ_{ℓ}} \underleftarrow{lim} J_{1}(N)[ℓ^{n}](\overline{ℚ}), where J_{1}(N) is the Jacobian of the modular curve X_{1}(N), J_{1}(N)[ℓ^{n}] is the group of ℓ^{n}-torsion points of J_{1}(N), and where the inverse limit is taken with respect to the usual ℓ-power maps. This construction was first done by Eichler and Shimura (in the case of primes with good reduction), and then Igusa completed it by identifying and analysing the behaviour at the primes of bad reduction.

2. (Weight of f is strictly greater than 2 ) Recall that Y_{1}(N) is a moduli space of isomorphism classes of complex elliptic curves and N-torsion data. In this case, we have a universal elliptic curve E over Y_{1}(N)

\xymatrix{E \ar[d] \\ Y_{1}(N)}

which is also called a Kuga-Sato surface. Generalizing this situation, Deligne considered the fibered product (over Y_{1}(N)) of k − 2 copies of E:

\xymatrix{E^{k−2} \ar[d] \\ Y_{1}(N)}

which is called the open Kuga-Sato variety of dimension k − 1. Let W_{k−2}(N) denote the compactification of E^{k−2}. Then, the representations{V_{f,λ}} occur in the cohomology groups H_{ét}^{k−1}(W_{k−2}(N)\overline{ℚ}, ℚ_{ℓ}).

Note that from the Weil conjectures (more specifically, the Riemann hypothesis for varieties over finite fields), we can deduce that {V_{f,λ}} is of weight k − 1, that is, the eigenvalues of F rob_{p} are of complex absolute value p^{\frac{k−1}{2}} .

Example 2. Consider the cusp form Δ = q ∏_{n=1}^{∞}(1 − q^{n})^{24} ∈ S_{12}(SL_{2}(ℤ)), whose q-expansion is given by Δ = ∑_{n=1}^{∞} τ(n)q^{n}, where τ(n) is the Ramanujan τ-function. In this case, τ(p) is the trace of F rob_{p} acting on V_{Δ,λ}, which gives the estimate |τ(p)| < 2p^{\frac{11}{2}} , the so-called Ramanujan conjecture (compare this estimate with the “elementary” estimate |τ(p)| < Cp^{6}).

3. (Weight of f is 1 ) For modular forms of weight one, there is no direct cohomological construction of {V_{f,λ}}. In this case, Serre and Deligne related Hecke eigenforms f ∈ S_{1}(Γ_{0}(N), χ) to (complex) Artin representations, reducing this case to the result for higher weight. This reduction, which does not require any algebraic geometry but rather some useful analytic estimates on Fourier coefficients of cusp forms, will be explained in detail in this class.

We would now like to give an algebro-geometric interpretation of modular forms and modular curves; the starting point will be the lattices in ℂ. The main tool we will use is Weierstrass theory of complex elliptic curves, which gives a bijection

\{ \begin{matrix}Lattices \\ Λ in ℂ \end{matrix} \} ⟶ \{ \begin{matrix}Pairs (E, ω) \\ E is a complex elliptic curve \\ ω is a generator of Ω^{1}(E/ℂ) \end{matrix} \} ,

where, to a lattice Λ ⊂ ℂ, we attach the pair (ℂ/Λ, dz), and to a pair (E, ω), we attach the lattice

Λ_{E,ω} := {∫_{γ} ω | γ ∈ H_{1}(E(ℂ), ℤ)} .

Using this bijection, we can now view classical modular form as functions on the set of pairs (E, ω) defined over ℂ. We are thus led to the following definition.

Definition 3. A marked elliptic curve over a field k is a pair (E, ω) consisting of an elliptic curve E over k, together with a k-vector space generator ω of Ω^{1}(E/k).

Actually, one can even define marked elliptic curves over any ring R. For our purposes, this will simply be a Weierstrass equation with coefficients in R, like

E : y^{2} = x^{3} + ax + b, a, b ∈ R,

where we require that the discriminant Δ(E) be a unit in R.

Definition 4. A weakly homolomorphic algebraic modular form f of weight k and level 1 over a ring S is a rule, which to every pair (E, ω)_{R} consisting of a marked elliptic curve over an S-algebra R assigns an element f(E, ω)_{R} ∈ R, subject to the following conditions:

• (Compatibility with base change) For all homomorphisms ϕ : R → R′ of S-algebras

ϕ(f(E, ω)_{R}) = f((E, ω) ⊗_{ϕ} R′);

• For all λ ∈ R^{×}, f(E, λω)_{R} = λ^{−k}f(E, ω)_{R}.

We now give some important examples of elliptic curves over rings.

Example 5.

1. The “universal” elliptic curve over ℂ

For this example, we let

• R= ring of holomorphic functions on H = {τ ∈ ℂ | Im(τ) > 0};

• E = ℂ/⟨1, τ⟩;

• ω = dz.

Weierstrass theory constructs from this data two doubly periodic functions with poles at z = 0, namely

x = ℘_{⟨1,τ⟩}(z) = \frac{1}{z^{2}} + \begin{matrix}∑ & ′ \\ λ∈Λ \end{matrix} (\frac{1}{(z − λ)^{2}} − \frac{1}{λ^{2}} );
y = ℘′_{⟨1,τ⟩}(z) = \frac{−2}{z^{3}} + \begin{matrix}∑ & ′ \\ λ∈Λ \end{matrix} \frac{−2}{(z − λ)^{3}} .

Moreover, these two functions satisfy the following equation

y^{2} = x^{3} − g_{4}(τ) − g_{6}(τ),

where

g_{4}(τ) = 60\begin{matrix}∑ & ′ \\ (m,n)∈ℤ^{2} \end{matrix}(mτ + n)^{−4} = \frac{4π^{3}}{3} [1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n}]

g_{6}(τ) = 140\begin{matrix}∑ & ′ \\ (m,n)∈ℤ^{2} \end{matrix}(mτ + n)^{−6} = \frac{8π^{6}}{27} [1 − 504 ∑_{n=1}^{∞} σ_{5}(n)q^{n}] .

Furthermore, we have ω = \frac{dx}{y}. The data (ℂ/⟨1, τ⟩, dz) is thus a marked elliptic curve over the ring R.

2. The Tate curve over ℂ

In the above construction, we got a torus by identifying opposite sides of the (possibly tilted) rectangle given by the basis {1, τ}. By making some change of variables, namely t = e^{2πiz}, q = e^{2πiτ}, we can give a similar construction by identifying the inner ring and the outer ring of an annulus. More concretely, we define:

• R= ring of holomorphic functions on the punctured open disk D^{×} = {q ∈ ℂ^{×} | |q| < 1};

• E = ℂ^{×}/⟨q⟩;

• ω = \frac{dt}{t} = 2πidz.

Now, after choosing our new coordinate functions X, Y to be

y = 2π^{3}iY, x = π^{2}X,

we get the following equation for the Tate curve:

Y^{2} = X^{3} − \frac{1}{3} (1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n}) X + \frac{3}{27} (1 − 504 ∑_{n=1}^{∞} σ_{5}(n)q^{n}) .

Also, we get ω = \frac{dx}{y}. In this case, the discriminant is equal to

Δ(E) = q ∏_{n=1}^{∞}(1 − q^{n})^{24} ∈ R^{×}.

Remark 1. These computations reveal that the Tate curve, which was a priori defined over the ring R, can also be viewed as an elliptic curve over the ring ℤ[\frac{1}{6}]((q)).