Hypergeometric Functions Over Finite Fields and Relations to Modular Forms and Elliptic Curves

Released on by Jenny G. Fuselier
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  • Hypergeometric Functions Over Finite Fields and Relations to Modular Forms and Elliptic Curves Book Detail

  • Author : Jenny G. Fuselier
  • Release Date : 2010
  • Publisher :
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  • Pages :
  • ISBN 13 :
  • File Size : 51,51 MB
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Hypergeometric Functions Over Finite Fields and Relations to Modular Forms and Elliptic Curves by Jenny G. Fuselier PDF Summary

Book Description: 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 [pi]-function in terms of hypergeometric functions.

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Hypergeometric Functions, My Love

Hypergeometric Functions, My Love

File Size : 72,72 MB
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The classical story - of the hypergeometric functions, the configuration space of 4 points on the projective line, elliptic curves, elliptic modular functions a