P -adic Modular Forms by Kevin Buzzard Lecture 2

• We know modular forms exist because over C there exists an explicit example (according to the classical definition), namely the Eisenstein series: E_{k} : τ ↦ ∑\begin{matrix}m,n∈Z \\ not both 0 \end{matrix} \frac{1}{(mτ+n)^{k}} , k ≥ 4. This is a level 1 modular form of

weight k. A known fact is that E_{p−1} ≡ 1 (mod p), where p is prime and p ≥ 5. So in the ring Z_{p}[[q]], E_{p−1} “looks” invertible becuase it has a power series where the constant term is a unit. However, no classical forms are of negative weight so 1/E_{p−1} is not a classical form. In addition, for example, E_{4}(πi/3) = E_{4}(i) = 0 so 1/E_{4} is not a holomorphic function on the upper half plane.

• The following is the Deligne/Katz approach to defining a p-adic modular form.

We have E/R, where E is an elliptic curve over R, an F_{p}-algebra. Now let ω ∈ H^{0}(E, Ω_{E/R}^{1}) and η ∈ H^{1}(E, O_{E}) be its dual. Consider the absolute Frobenius map, F_{abs} : O_{E} → O_{E}, f ↦ f^{p}, an additive homomorphism of sheaves of abelian groups. Now define A(E/R, ω) ∈ R, (which is actually the Hasse invariant) by setting F_{abs}^{∗}(η) = A(E/R, ω) · η which gives us that A(E/R, λω) = λ^{1−p}A(E/R, ω), λ ∈ R^{×}.

So A is a modular form of weight p − 1. Note that the boundedness condition for a modular form is satisfied by looking at A(Tate(q), ω_{can}) since the restriction of a plane curve over F_{p}[[q]] is the Tate curve over F_{p}((q)). Now A(Tate(q), ω_{can}) = 1 so if p ≥ 5, then E_{p−1} ≡ A (mod p). Therefore A = E_{p−1} (mod p) by the q-expansion principle, which says that two modular forms of level 1 and the same weight are equal if they have the same qexpansion.

We want a p-adic theory of modular forms that strongly identifies a modular form with its q-expansion so that what “looks” invertible, as in E_{p−1} (mod p), is invertible. So because the Hasse invariant A(E/R, ω) = 0 if and only if E is supersingular, we want to somehow throw away elliptic curves which are supersingular or have supersingular reduction.

• Katz’s definition of a p-adic modular form: Let p ≥ 5, R_{0} be the ring of integers in a finite extension of Q_{p}, and R an R_{0}-algebra in which p is nilpotent. A test object is

1. an elliptic curve E/R

2. a nowhere-vanishing differential ω ∈ H^{0}(E, Ω_{E/R}^{1})

3. an element Y ∈ R such that Y · E_{p−1}(E/R, ω) = 1 ∈ R.

A p-adic modular form of level 1 of weight k defined over R_{0} is a rule f sending (E/R, ω, Y) to an element of R such that

a) f(E/R, λω, λ^{p−1}Y) = λ^{−k}f(E/R, ω, Y)

b) f(Tate curve over R_{0}/p^{n}R_{0}((q)), ω_{can}, 1) is in R_{0}/p^{n}R_{0}[[q]] for all n ≥ 1. and f depends only the isomorphism class of data and behaves well under pullback.

Note that classical modular forms over R_{0} are already p-adic modular forms.

• The class of p-adic modular forms is too large to work with, but we can create subtlety through the following definition: Let R_{0} be the ring of integers in a finite extension of Q_{p} and choose ρ ∈ R_{0}\0, and R an R_{0}-algebra in which p is nilpotent. A ρ-overconvergent test object is

1. an elliptic curve E/R

2. a nowhere-vanishing differential ω ∈ H^{0}(E, Ω_{E/R}^{1})

3. an element Y ∈ R such that Y · E_{p−1}(E/R, ω) = ρ.

A ρ-overconvergent modular form is a rule on these test objects satisfying the conditions (a) and (b) in the definition of a p-adic modular form.

Note that if |ρ| < 1 then some of these test objects might have supersingular geometric fibers.

• We don’t want to throw away too much so the idea is to define E, an elliptic curve, as having “very supersingular reduction” if A(E(mod p) / R/pR) = 0 ∈ R/pR. Now if R is the ring of integers in a highly ramified extension of Q_{p} then R/pR could be huge so maybe there are a lot of elliptic curve whose reductions are supersingular but not very supersingular.