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Arithmetic and analytic properties of finite field hypergeometric functions

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dc.contributor.advisor Benjamin Brubaker. en_US
dc.contributor.author Lennon, Catherine (Catherine Ann) en_US
dc.contributor.other Massachusetts Institute of Technology. Dept. of Mathematics. en_US
dc.date.accessioned 2011-12-19T18:51:52Z
dc.date.available 2011-12-19T18:51:52Z
dc.date.copyright 2011 en_US
dc.date.issued 2011 en_US
dc.identifier.uri http://hdl.handle.net/1721.1/67791
dc.description Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. en_US
dc.description Cataloged from PDF version of thesis. en_US
dc.description Includes bibliographical references (p. 97-100). en_US
dc.description.abstract The intent of this thesis is to provide a detailed study of the arithmetic and analytic properties of Gaussian (finite field) hypergeometric series. We present expressions for the number of F,-points on certain families of varieties as special values of these functions. We also present "hypergeometric trace formulas" for the traces of Hecke operators on spaces of cusp forms of levels 3 and 9. These formulas lead to a simple expression for the Fourier coefficients of r(3z)', the unique normalized cusp form of weight 4 and level 9. We then use this to show that a certain threefold is "modular" in the sense that the number of its F,-points is expressible in terms of these coefficients. In this way, we use Gaussian hypergeometric series as a tool for connecting arithmetic and analytic objects. We also discuss congruence relations between Gaussian and truncated classical hypergeometric series. In particular, we use hypergeometric transformation identities to express the pth Fourier coefficient of the unique newform of level 16 and weight 4 as a special value of a Gaussian hypergeometric series, when p =1 (mod 4). We then use this to prove a special case of Rodriguez-Villegas' supercongruence conjectures. en_US
dc.description.statementofresponsibility by Catherine Lennon. en_US
dc.format.extent 100 p. en_US
dc.language.iso eng en_US
dc.publisher Massachusetts Institute of Technology en_US
dc.rights M.I.T. 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. See provided URL for inquiries about permission. en_US
dc.rights.uri http://dspace.mit.edu/handle/1721.1/7582 en_US
dc.subject Mathematics. en_US
dc.title Arithmetic and analytic properties of finite field hypergeometric functions en_US
dc.type Thesis en_US
dc.description.degree Ph.D. en_US
dc.contributor.department Massachusetts Institute of Technology. Dept. of Mathematics. en_US
dc.identifier.oclc 767741973 en_US


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