Hypergeometric functions over finite fields and relations to modular forms and elliptic curves

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Title: Hypergeometric functions over finite fields and relations to modular forms and elliptic curves
Author: Fuselier, Jenny G.
Abstract: The theory of hypergeometric functions over finite fields was developed in the mid - 1980s by Greene . Since that time , connections between these functions and elliptic curves and modular forms have been investigated by mathematicians such as Ahlgren , Frechette , Koike , Ono , and Papanikolas . In this dissertation , we begin by giving a survey of these results and introducing hypergeometric functions over finite fields . We then focus on a particular family of elliptic curves whose j -invariant gives an automorphism of P1 . We present an explicit relationship between the number of points on this family over Fp and the values of a particular hypergeometric function over Fp . Then , we use the same family of elliptic curves to construct a formula for the traces of Hecke operators on cusp forms in level 1 , utilizing results of Hijikata and Schoof . This leads to formulas for Ramanujan ?s -function in terms of hypergeometric functions .
URI: http : / /hdl .handle .net /1969 .1 /ETD -TAMU -1547
Date: 2009-05-15

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Hypergeometric functions over finite fields and relations to modular forms and elliptic curves. Available electronically from http : / /hdl .handle .net /1969 .1 /ETD -TAMU -1547 .

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