Convergence and Correlations of Coefficients of Cusp Forms

Abstract

In this thesis, we discuss aspects of lower-order statistical behavior of coefficients of GL2 automorphic forms. First, we establish two cases of recently observed correlation phenomena, referred to as “murmurations,” between root numbers and L-function coefficients. The first is for the family of weight k modular cuspidal newforms. In that case, we show that averages of P-th Fourier coefficients correlated against the root number in a family of forms of conductor ∼ N converge to a function of P/N. In the second case (from joint work with Booker, Lee, Lowry-Duda, and Seymour-Howell), we prove an analogous result for the family of weight 0 level 1 Maass forms. In this case, additional averaging on P is required, and the answer is given by a measure evaluated on the interval of P-averaging. Finally, in joint work with Sarnak, we give new rates of convergence to the Plancherel measure for coefficients of holomorphic forms of weight 2 and bound the number of d-dimensional factors of the Jacobian of the modular curve.