P-ADIC MODULAR FORMS BY KEVIN BUZZARD - LECTURE 1

1. History

Serre and Katz (1972): Understand congruences mod p and mod p^{n} between modular forms (see Antwerp). Serre (1972): Application to p-adic L-functions. Katz (1972): Wanted conceptual explanation of Atkin’s work and Serre’s paper. Hida (early 80’s): “Ordinary” subspace of space of p-adic modular forms, and constructed families. Gouvea (1987): Makes conjectures that Mazur’s deformation rings correspond to Hida’s ring of Hecke operators. Wiles and Taylor-Wiles (1994): Deformation rings = Hecke rings Taylor et al: Applications to conjecture of Artin. Gouvea-Mazur: Conjecture “forms come in big families” Coleman: Uses rigid analytic approach, proved that forms come in small families. Wan, Smithline, Emerton (more recently) Stein (more recently): Enables us to do calculations.

2. Gouvea-Mazur Conjecture

Let p be prime, N ≥ 1 an integer prime to p and S_{k}(Γ_{0}(Np)) the space of cusp forms of weight k and level Np (think of N, p fixed with k varying). Consider U_{p}, the Hecke operator, acting on S_{k}(Γ_{0}(Np)), the characteristic polynomial of U_{p} lies in ℤ[X], think of the roots, λ_{1}, ..., λ_{n} of this polynomial as elements of \overline{ℚ}_{p} so that they have valuations. For α ∈ ℚ define

d(k, α) = #{λ_{i} : v(λ_{i}) = α}

Conjecture 1. Let k_{1}, k_{2} ∈ ℤ, α ∈ ℚ, α ≥ 0 and assume that

(i) k_{1}, k_{2} > α + 1 (ii) k_{1} ≡ k_{2} mod (p − 1)p^{r} for some r ∈ ℤ, r ≥ α then d(k_{1}, α) = d(k_{2}, α).

One approach: Remove hypothesis (i) but instead let d(k, α) be the number of eigenvalues, with valuation α, of U_{p} acting on huge space of “overconvergent p-adic modular forms of weight k”. Classical mod forms of wt k ⊂ Overconvergent p-adic mod forms of wt k

3. Modular Forms

Definition 1. A modular form of level 1 and weight k ∈ ℤ is an analytic function f : H → ℂ such that

f(\frac{aτ + b}{cτ + d}) = (cτ + d)^{k}f(τ) for all (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ)

and f satisfies boundedness conditions.

If k = 0 the only modular forms are constants. One could weaken the boundedness conditions by, e.g., allowing f to have some poles, and then one gets a whole lot of interesting functions (e.g. j, which has a ”pole at infinity”). For k = 0 a modular form is a function on SL_{2}(ℤ) \ H, however if k ≠ 0 then one can’t think of a modular form as a function on SL_{2}(ℤ) \ H because of the (cτ + d)^{k} factor.

SL_{2}(ℤ) \ H is a parameter space for elliptic curves: τ ∈ H : ℂ/⟨1, τ⟩ is a 1-dimensional complex torus.

ℂ/⟨1, τ⟩ ≅ ℂ/⟨1, σ⟩ ⟺ λ⟨1, σ⟩ = ⟨1, τ⟩ for some λ ∈ ℂ

⟺ λ = cτ + d, σλ = aτ + b

⟺ σ = \frac{aτ + b}{cτ + d} for some (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ)

So let’s remember λ by considering functions f sending a pair (E, ω) to a complex number f(E, ω) where E is a complex 1-dimensional torus and ω is a non-vanishing global differential such that f(E, λω) = λ^{−k}f(E, ω) A modular form of weight k is an analytic rule sending (E, τ) to f(E, τ) such that f(E, λω) = λ^{−k}f(E, ω) and f satisfies boundedness conditions.

That’s “encoded” the functional equation, now we wish to encode the word “analytic” by allowing families of tori.

If S is a base complex manifold and π : T → S is a family of tori then we should try and make sense of f(T/S, ω) where ω is now a family of nonvanishing differentials: f(T/S, ω) should be an analytic function S → ℂ and if s ∈ ℂ then f(T/S, ω)(s) = f(T_{s}, ω_{s}).

This is now getting very non-computable but also much more algebraic.

Definition 2. Let R_{0} be a ring (commutative with identity). A modular form of level 1 and weight k defined over R_{0} is a rule, which to every pair (E/R, ω) where,

(1) R is an R_{0}-algebra

(2) E/R is an elliptic curve, π : E → Spec(R)

(3) ω ∈ H^{0}(E, Ω^{1}) is a nowhere vanishing differential an element f(E/R, ω) ∈ R such that (a) f(E/R, ω) only depends on the isomorphism class of the data

(b) if E/R_{1} and E′/R_{2} are elliptic curves, and we have β : Spec(R_{2}) → Spec(R_{1}), then we can form the pullback β^{∗}E of E to Spec(R_{2}) along β, and if we have an isomorphism from E′ to β^{∗}E then the data of the f’s should match up too.

(c) f(T ate(q), ω_{can}) ∈ R_{0}[[q]] (boundedness conditions)

(d) f(E/R, λω) = λ^{−k}f(E/R, ω) for all λ ∈ R^{∗}

Explanation of (c): There is an elliptic curve called T ate(q) defined over the p-adic completion of R_{0}((q)), with a canonical nowhere-vanishing differential ω_{can}, and by definition f(T ate(q), ω_{can}) will be in the p-adic completion of R_{0}((q)), and the assertion is that it has to be in the much smaller ring R_{0}[[q]] (the point is that we want to rule out poles at infinity).

Non-computable definition: a modular form is a well behaved rule on (Elliptic curves, differentials).

David Whitehouse

California Institute of Technology March 26, 2001