Citation
Gurel, Aliekber (2008) An Exact Average Formula for the Symmetric Square LFunction at the Center. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MDF74E10. https://resolver.caltech.edu/CaltechETD:etd02042008154002
Abstract
In this thesis, we find an exact formula for the weighted average of the symmetric square Lvalues at the center. The average is taken over a Hecke eigen basis of cusp forms of SL_{2}(Z) with a fixed weight 2k. The weights are the nth Fourier Coefficients of these functions. The terms in the formula involve quadratic Dirichlet Lvalues 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 Lfunction 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 MichelRamakrishnan's averaging technique on Shimura's formula to write the weighted average of symmetric square Lvalues 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.)) 

Subject Keywords:  Center; Central Point; Confluent Hypergeometric Function; Exact Formula; Fourier Coefficients; Quadratic Dirichlet L Function; Symmetric Square LFunction 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Awards:  Scott Russell Johnson Prize for Excellence in Graduate Study in Mathematics, 2007 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  16 November 2007 
Record Number:  CaltechETD:etd02042008154002 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd02042008154002 
DOI:  10.7907/MDF74E10 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  497 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  21 Feb 2008 
Last Modified:  03 Dec 2019 22:31 
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