The trace of frobenius of elliptic curves and the p-adic gamma function

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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 Fp, 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 Fp when p = 1 (mod 12).

Original languageEnglish
Pages (from-to)219-236
Number of pages18
JournalPacific Journal of Mathematics
Volume261
Issue number1
DOIs
StatePublished - Jun 17 2013

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Keywords

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

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