Periods of modular forms and central values of L-functions

Date
2010-05
Authors
Hopkins, Kimberly Michele
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Abstract

This thesis is comprised of three problems in number theory. The introduction is Chapter 1. The first problem is to partially generalize the main theorem of Gross, Kohnen and Zagier to higher weight modular forms. In Chapter 2, we present two conjectures which do this and some partial results towards their proofs as well as numerical examples. This work provides a new method to compute coefficients of weight k+1/2 modular forms for k>1 and to compute the square roots of central values of L-functions of weight 2k>2 modular forms. Chapter 3 presents four different interpretations of the main construction in Chapter 2. In particular we prove our conjectures are consistent with those of Beilinson and Bloch. The second problem in this thesis is to find an arithmetic formula for the central value of a certain Hecke L-series in the spirit of Waldspurger's results. This is done in Chapter 4 by using a correspondence between special points in Siegel space and maximal orders in quaternion algebras. The third problem is to find a lower bound for the cardinality of the principal genus group of binary quadratic forms of a fixed discriminant. Chapter 5 is joint work with Jeffrey Stopple and gives two such bounds.

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