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.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-1547 |
| Date | 15 May 2009 |
| Creators | Fuselier, Jenny G. |
| Contributors | Papanikolas, Matthew |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
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