189-726B: Modular Forms II Assignment 1

Due: Wednesday, January 16

1. Let F be a number field and let

ζ(F, s) = ∑_{a}(Na)^{−s}

be its Dedekind zeta-function, where the sum is taken over all integral ideals of F.

If λ is an integral ideal, define N(λ) to be the index of λ in O_{F}, and extend this definition to fractional ideals in the obvious way. To any fractional ideal λ we can associate the zeta function

ζ_{λ}(s) = ∑_{x∈λ/O_{F}^{×}} |Nx|^{−s},

and to an ideal class C, the zeta-function

ζ(C, s) = (Nλ)^{−s}ζ_{λ}(s),

where λ is any representative ideal in C. Show that

ζ(F, s) = ∑_{C} ζ(C, s),

where the sum is taken over the distinct ideal classes of F.

2. Fill in the details that are needed to prove the integral representation for the L-function of a cusp form f of weight k on Γ_{0}(N):

Λ(f, s) = ∫_{1/\sqrt{N}}^{∞} f(it)t^{s} \frac{dt}{t} + i^{k}N^{k/2−s} ∫_{1/\sqrt{N}}^{∞} g(it)t^{k−s} \frac{dt}{t},

where g := f|_{k}S_{N}.

3. Use the modular transformation formula for the Riemann theta-function

θ(τ) = ∑_{n=1}^{∞} e^{2πin^{2}τ}

which you saw last semester to derive the functional equation for the Riemann zeta-function.