Periods of modular forms and central values of L-functions

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Title: Periods of modular forms and central values of L-functions
Author: Hopkins, Kimberly Michele
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 .
URI: http : / /hdl .handle .net /2152 /ETD -UT -2010 -05 -1423
Date: 2010-10-21

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Periods of modular forms and central values of L-functions. Doctoral dissertation, University of Texas at Austin. Available electronically from http : / /hdl .handle .net /2152 /ETD -UT -2010 -05 -1423 .

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