On vector-valued modular forms and their Fourier coefficients

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.

Vector valued hermitian and quaternionic modular forms

Construction of Vector Valued Modular Forms from Jacobi Forms

We give a geometrical construction of the canonical automorphic factor for the Jacobi group and construct new vector valued modular forms from Jacobi forms by differentiating them with respect to toroidal variables and then evaluating at zero.