Weighted equidistribution theoreMS in the theory of modular forMS

Abstract

We give three results concerning the distribution of eigenvalues of Hecke operators acting on spaces of modular cusp forms of weight k with respect to Γ_0(N) by attaching some weights to them. These results extend some classical results. In the 1960s, M. Sato and J. Tate made a conjecture regarding the distribution laws for the Fourier coefficients at primes of a fixed Hecke eigenform. In 1997, J-P Serre considered a vertical analogue of the Sato-Tate conjecture: he fixed a prime p and considered the set of p-th Fourier coefficients of all Hecke eigenforms of weight k with respect to Γ_0(N). He then derived a distribution law for such families as N + k → ∞. Serre’s theorem was made effective by M. R. Murty and K. Sinha, who found explicit error terms in Serre’s theorem. His theorem was also generalized by C. C. Li in 2004 to derive an equidistribution law for Serre’s families by attaching some suitable weights to the elements. In our first theorem, we extend the work of Murty and Sinha and find the error term in Li’s weighted equidistribution theorem. In 2006, H. Nagoshi proved two theorems. In his first theorem, he showed that by varying the primes p and the weights k, the Sato-Tate distribution law holds and in his second theorem, he proves a type of central limit theorem for the Fourier coefficients at primes of Hecke eigenforms with respect to Γ_0(1) and weights k → ∞. Our second and third results are the weighted analogues of Nagoshi’s first and second theorems respectively, with the weights as defined by Li.