On the Fourier coefficients of 2-dimensional vector-valued modular forms

Fourier Coefficients of Modular Forms

These notes describe some conjectures and results related to the distribution of Fourier coefficients of modular forms. This is a rough draft and these notes should forever be considered incomplete.

Oscillations of Fourier Coefficients of Modular Forms

a(p) = 2p~ @ ) cos 0(p). Since we know the truth of the Ramanujan-Petersson conjecture, it follows that the 0(p)'s are real. Inspired by the Sato-Tate conjecture for elliptic curves, Serre [14] conjectured that the 0(p)'s are uniformly distributed in the interval [0, rc] with respect to the 1 measure -sin2OdO. Following Serre, we shall refer to this as the Sato-Tate r~ conjecture, there being n...

Divisors of Fourier coefficients of modular forms

Let d(n) denote the number of divisors of n. In this paper, we study the average value of d(a(p)), where p is a prime and a(p) is the p-th Fourier coefficient of a normalized Hecke eigenform of weight k ≥ 2 for Γ0(N) having rational integer Fourier coefficients.

Some vector valued Siegel modular forms of genus 2

is a module over the ring of all modular forms with respect to the group Γ2[4, 8]. We are interested in its structure. By Igusa, the ring of modular forms is generated by the ten classical theta constants θ[m]. The module M contains a submodule N which is generated by 45 Cohen-Rankin brackets {θ[m], θ[n]}. We determine defining relations for this submodule and compute its Hilbert function (Theo...

Vector valued hermitian and quaternionic modular forms