2026-09-20T00:26:17Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/677912022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Benjamin Brubaker.
Lennon, Catherine (Catherine Ann)
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2011-12-19T18:51:52Z
2011-12-19T18:51:52Z
2011
2011
http://hdl.handle.net/1721.1/67791
767741973
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 97-100).
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.
by Catherine Lennon.
Ph.D.
100 p.
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
copyright. They may be viewed from this source for any purpose, but
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Arithmetic and analytic properties of finite field hypergeometric functions
Thesis
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<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.</Abstract>
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