P-adic analysis and mock modular forms

dc.contributor.advisor Guerzhoy, Pavel
dc.contributor.author Kent, Zachary A.
dc.date.accessioned 2013-02-06T20:14:18Z
dc.date.available 2013-02-06T20:14:18Z
dc.date.issued 2011
dc.description 87 leaves, bound ; 29 cm.
dc.description Thesis (Ph. D.)--University of Hawaii at Manoa, 2010.
dc.description.abstract A mock modular form f+ is the holomorphic part of a harmonic Maass form f. The non-holomorphic part of f is a period integral of a cusp form g, which we call the shadow of f+. The study of mock modular forms and mock theta functions is one of the most active areas in number theory with important works by Bringmann, Ono, Zagier, Zwegers, among many others. The theory has many wide-ranging applications: additive number theory, elliptic curves, mathematical physics, representation theory, and many others. We consider arithmetic properties of mock modular forms in three different settings: zeros of a certain family of modular forms, coupling the Fourier coefficients of mock modular forms and their shadows, and critical values of modular L-functions. For a prime p > 3, we consider j-zeros of a certain family of modular forms called Eisenstein series. When the weight of the Eisenstein series is p - 1, the j-zeros are j-invariants of elliptic curves with supersingular reduction modulo p. We lift these j-zeros to a p-adic field, and show that when the weights of two Eisenstein series are p-adically close, then there are j-zeros of both series that are p-adically close. A direct method for relating the coefficients of shadows and mock modular forms is not known. This is considered to be among the first of Ono's Fundamental Problems for mock modular forms. The fact that a shadow can be cast by infinitely many mock modular forms, and the expected transcendence of generic mock modular forms pose serious obstructions to this problem. We solve these problems when the shadow is an integer weight cusp form. Our solution is p-adic, and it relies on our definition of an algebraic regularized mock modular form. We use mock modular forms to compute generating functions for the critical values of modular L-functions. To obtain this result we derive an Eichler-Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes an "Eichler-Shimura isomorphism", a "multiplicity two" Hecke theory, a correspondence between mock modular periods and classical periods, and a "Haberland-type" formula which expresses Petersson's inner product and a related antisymmetric inner product on M!k in terms of periods.
dc.format.extent 87 leaves
dc.identifier.isbn 9781124298634
dc.identifier.uri http://hdl.handle.net/10125/25934
dc.language en-US
dc.publisher University of Hawaii at Manoa
dc.relation Theses for the degree of Doctor of Philosophy (University of Hawaii at Manoa). Mathematics; no. ????
dc.rights All UHM dissertations and theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission from the copyright owner.
dc.title P-adic analysis and mock modular forms
dc.type Thesis
dc.type.dcmi Text
local.thesis.degreelevel PhD
local.thesis.department Mathematics
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