Hypergeometric functions over 𝔽_{𝕡} and relations to elliptic curves and modular forms

Fuselier, Jenny

doi:10.1090/S0002-9939-09-10068-0

Hypergeometric functions over ${\mathbb{F}_p}$ and relations to elliptic curves and modular forms

Author: Jenny G. Fuselier
Journal: Proc. Amer. Math. Soc. 138 (2010), 109-123
MSC (2000): Primary 11F30; Secondary 11T24, 11G20, 33C99
DOI: https://doi.org/10.1090/S0002-9939-09-10068-0
Published electronically: August 28, 2009
MathSciNet review: 2550175
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Abstract: For primes $p\equiv 1 \pmod{12}$ , we present an explicit relation between the traces of Frobenius on a family of elliptic curves with -invariant $\frac{1728}{t}$ and values of a particular -hypergeometric function over ${\mathbb{F}_p}$ . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's $\tau$ -function in terms of hypergeometric functions.

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