189-726B: Modular Forms II Assignment 7

Due: Wednesday, March 12

Note: Because this assignment is given over a two-week break, it will be longer than the usual assignments, and designed to make you learn new things. Allow enough time to do it well! A few of the questions are designed to get you to read the relevant portions of Serre’s paper “Formes modulaires et Fonctions zeta p-adiques” (of which an English translation can be found on Cameron Franc’s web page...) I advise you to try to get as far as you can without looking at Serre’s article — you’ll learn a lot more that way. Only consult Serre for hints after you’ve been stuck for a while.

1. Write down the Hasse polynomial A(Q, R) ∈ F_{13}[Q, R] for p = 13.

2. Let f ∈ M_{k} ⊗ Q, g ∈ M_{ℓ} ⊗ Q, with (p − 1)|k and t = ord_{p}(k − ℓ) + 1. Assume that t_{0} := ord_{p}(k) + 1 < t. and a_{n}(f) − a_{n}(g) ∈ p^{t}Z_{(p)} for all n ≥ 1. In class, we sketched a proof of the fact that p^{t_{0}}(a_{n}(f) − a_{n}(g)) ∈ p^{t}Z_{(p)}. Fill in the omitted details of the argument.

3. In class we explained that, if K is a totally real field of degree r, then the special value ζ_{K}(1 − k) (k ≥ 2 even) can be written as the constant term of a modular form of weight rk on SL_{2}(Z) all of whose other Fourier coefficients are rational. Use this fact to prove that ζ_{K}(1 − k) is rational. (Hint: Do not be misled by the confusing answer I gave to a question about this in my last lecture: your proof should be short and conceptual and only use what you already know well!)

4. Let U and V be the operations on formal q-expansions defined by

U(∑ a_{n}q^{n}) = ∑ a_{np}q^{n}, V (∑ a_{n}q^{n}) = ∑ a_{n}q^{np}.

Show that, if f is a p-adic modular form of weight k, then the same is true of U(f) and V (f).

5. Let f ∈ M̄_{α} be a modular form mod p, of weight α ∈ Z/(p − 1)Z and filtration w(f) = k. What is w(V f)?

6. Show that the weight two Eistenstein series P = E_{2} is a p-adic modular form of weight 2.

7. Show that if f is a p-adic modular form of weight k ∈ Z/(p − 1)Z × Z_{p}, then θf is a p-adic modular form of weight k + 2.

8. If f = ∑_{n} a_{n}q^{n} is a p-adic modular form of weight k, show that the same holds for f̃ = ∑_{(p,n)=1} a_{n}q^{n}.

9. Read Chapter 3 of Serre’s article. Although I will not be assuming this material in the subsequent lectures, familiarity with it will provide you with motivation and clarify those lectures. So, although this will not be graded, I encourage you to take this last part of the assignment seriously!