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
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by Jenny G. Fuselier

Proc. Amer. Math. Soc. 138 (2010), 109-123

DOI: https://doi.org/10.1090/S0002-9939-09-10068-0

Published electronically: August 28, 2009

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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 $j$-invariant $\frac {1728}{t}$ and values of a particular $_2F_1$-hypergeometric function over ${\mathbb {F}_p}$. We also give a formula for traces of Hecke operators on spaces of cusp forms of weight $k$ 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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