Arithmetic and analytic properties of finite field hypergeometric functions
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2011
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| Online Access: | http://hdl.handle.net/1721.1/67791 |
| _version_ | 1826188282521714688 |
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| author | Lennon, Catherine (Catherine Ann) |
| author2 | Benjamin Brubaker. |
| author_facet | Benjamin Brubaker. Lennon, Catherine (Catherine Ann) |
| author_sort | Lennon, Catherine (Catherine Ann) |
| collection | MIT |
| description | Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. |
| first_indexed | 2024-09-23T07:57:16Z |
| format | Thesis |
| id | mit-1721.1/67791 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T07:57:16Z |
| publishDate | 2011 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/677912022-01-13T07:54:35Z Arithmetic and analytic properties of finite field hypergeometric functions Lennon, Catherine (Catherine Ann) Benjamin Brubaker. Massachusetts Institute of Technology. Dept. of Mathematics. Massachusetts Institute of Technology. Department of Mathematics Mathematics. 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. 2011-12-19T18:51:52Z 2011-12-19T18:51:52Z 2011 2011 Thesis http://hdl.handle.net/1721.1/67791 767741973 eng 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. http://dspace.mit.edu/handle/1721.1/7582 100 p. application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Lennon, Catherine (Catherine Ann) Arithmetic and analytic properties of finite field hypergeometric functions |
| title | Arithmetic and analytic properties of finite field hypergeometric functions |
| title_full | Arithmetic and analytic properties of finite field hypergeometric functions |
| title_fullStr | Arithmetic and analytic properties of finite field hypergeometric functions |
| title_full_unstemmed | Arithmetic and analytic properties of finite field hypergeometric functions |
| title_short | Arithmetic and analytic properties of finite field hypergeometric functions |
| title_sort | arithmetic and analytic properties of finite field hypergeometric functions |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/67791 |
| work_keys_str_mv | AT lennoncatherinecatherineann arithmeticandanalyticpropertiesoffinitefieldhypergeometricfunctions |