Show simple item record

dc.contributor.advisorBenjamin Brubaker.en_US
dc.contributor.authorLennon, Catherine (Catherine Ann)en_US
dc.contributor.otherMassachusetts Institute of Technology. Dept. of Mathematics.en_US
dc.date.accessioned2011-12-19T18:51:52Z
dc.date.available2011-12-19T18:51:52Z
dc.date.copyright2011en_US
dc.date.issued2011en_US
dc.identifier.urihttp://hdl.handle.net/1721.1/67791
dc.descriptionThesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.en_US
dc.descriptionCataloged from PDF version of thesis.en_US
dc.descriptionIncludes bibliographical references (p. 97-100).en_US
dc.description.abstractThe 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.statementofresponsibilityby Catherine Lennon.en_US
dc.format.extent100 p.en_US
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsM.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.urihttp://dspace.mit.edu/handle/1721.1/7582en_US
dc.subjectMathematics.en_US
dc.titleArithmetic and analytic properties of finite field hypergeometric functionsen_US
dc.typeThesisen_US
dc.description.degreePh.D.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematics
dc.identifier.oclc767741973en_US


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record