Dissertations, Theses, and Capstone Projects

Date of Degree

6-2020

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Brooke Feigon

Committee Members

Krzysztof Klosin

Carlos Moreno

Subject Categories

Number Theory

Keywords

triple product L-function, central L-value, nonvanishing, relative trace formula, quaternion algebra, period integral

Abstract

Harris and Kudla (2004) proved a conjecture of Jacquet, that the central value of a triple product L-function does not vanish if and only if there exists a quaternion algebra over which a period integral of three corresponding automorphic forms does not vanish. Moreover, Gross and Kudla (1992) established an explicit identity relating central L-values and period integrals (which are finite sums in their case), when the cusp forms are of prime levels and weight 2. Böcherer, Schulze-Pillot (1996) and Watson (2002) generalized this identity to more general levels and weights, and Ichino (2008) proved an adelic period formula which would work for all the cases. In this thesis we use Ichino's period formula combined with a relative trace formula to show exact averages of certain families of triple product L-functions. We also present some applications of the average formulas to the nonvanishing problem.

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