An Exact Average Formula for the Symmetric Square L-Function at the Center

Abstract

In this thesis, we find an exact formula for the weighted average of the symmetric square L-values at the center. The average is taken over a Hecke eigen basis of cusp forms of SL2(Z) with a fixed weight 2k. The weights are the n-th Fourier Coefficients of these functions. The terms in the formula involve quadratic Dirichlet L-values at the center, Confluent Hypergeometric functions, and some arithmetic functions. The main ingredient, and the starting point, is a formula due Shimura, which relates the symmetric square L-function of a Hecke eigen form f to the inner product of f with the product of the theta function, θ; and a real analytic Eisenstein series of half integral weight, E. We apply Michel-Ramakrishnan's averaging technique on Shimura's formula to write the weighted average of symmetric square L-values in terms of the Fourier coefficients of the Eisenstein series. There are two complications. First, the levels of θ x E and f are different. Second, E is not holomorphic. That is why we first take trace of θ x E, and then we take the holomorphic projection. Computing the Fourier coefficients of the resulting function gives us the exact formula desired. Finally, we deduce the asymptotic behavior of these formulas as k → ∞.