Fuselier, Jenny G. (2007-08). Hypergeometric functions over finite fields and relations to modular forms and elliptic curves. Doctoral Dissertation. Thesis uri icon

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.
  • 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.

publication date

  • August 2007