CaltechTHESIS
A Caltech Library Service

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

## Citation

Gurel, Aliekber (2008) An Exact Average Formula for the Symmetric Square L-Function at the Center. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MDF7-4E10. https://resolver.caltech.edu/CaltechETD:etd-02042008-154002

## 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 → ∞.

Item Type: Thesis (Dissertation (Ph.D.)) Center; Central Point; Confluent Hypergeometric Function; Exact Formula; Fourier Coefficients; Quadratic Dirichlet L Function; Symmetric Square L-Function California Institute of Technology Physics, Mathematics and Astronomy Mathematics Scott Russell Johnson Prize for Excellence in Graduate Study in Mathematics, 2007 Public (worldwide access) Ramakrishnan, Dinakar Ramakrishnan, Dinakar (chair)Wambach, EricBrown, James L.Flach, Matthias 16 November 2007 CaltechETD:etd-02042008-154002 https://resolver.caltech.edu/CaltechETD:etd-02042008-154002 10.7907/MDF7-4E10 No commercial reproduction, distribution, display or performance rights in this work are provided. 497 CaltechTHESIS Imported from ETD-db 21 Feb 2008 03 Dec 2019 22:31

## Thesis Files  Preview
PDF (Full thesis) - Final Version
See Usage Policy.

294kB

Repository Staff Only: item control page