189-726B: Modular Forms II Assignment 11

Due: Wednesday, April 9

1. Let f be a p-adic modular form of weight k ∈ Z with fourier coefficients in Z_{p}. Show that f can be written (not necessarily uniquely) as

f = ∑_{ν=0^{∞}} p^{a(ν)}f_{ν}E_{p−1}^{−ν},

where the f_{ν} are classical modular forms of weight k + (p − 1)ν with fourier coefficients in Z_{p}, and a(ν)⟶∞ as ν⟶∞.

More precisely, show that for any ordinary test object (A, ω)_{/R} defined over a p-adic ring R (meaning that the natural map R⟶ lim(R/p^{n}R) is an isomorphism), we have

f(A, ω)_{/R} = ∑_{ν=0^{∞}} p^{a(ν)}f_{ν}(A, ω)E_{p−1}(A, ω)^{−ν}.

2. Let X be a non-singular projective algebraic curve over C and let η be a differential form of the second kind on X over C. Given subsets U ⊂ V in X(C), with U closed and V open for the usual topology, recall that there exists a smooth function f_{U,V} on X(C) which is identically 1 on U and vanishes outside V. Use this fact to give an explicit formula for a smooth differential form η^{an} on X(C) representing the same class as η in the de Rham cohomology of X(C). Your answer should involve the singularities P_{1}, …, P_{r} of η, mutually disjoint open discs U_{j} containing P_{j}, and the local primitives F_{j} of η on U_{j}.

3. Let ω and η be differential forms of the first and second kind respectively on X over C. Use the expression for η^{an} obtained in part 2 to prove the assertion made in class that:

⟨ω, η⟩ = \frac{1}{2πi} ∫_{X(C)} ω^{an} ∧ η^{an},

where the pairing on the left is the pairing on algebraic deRham cohomology defined in terms of residues of differential forms.