### Abstract

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p > 3, that the trace of Frobenius of any elliptic curve over F_{p}, whose j -invariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over F_{p} when p = 1 (mod 12).

Original language | English |
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Pages (from-to) | 219-236 |

Number of pages | 18 |

Journal | Pacific Journal of Mathematics |

Volume | 261 |

Issue number | 1 |

DOIs | |

State | Published - Jun 17 2013 |

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### Keywords

- Elliptic curves
- Hypergeometric functions
- Modular forms
- P-adic gamma function
- Trace of Frobenius