The trace of Frobenius of elliptic curves and the \(p\)-adic gamma function

DOI10.2140/PJM.2013.261.219zbMATH Open1296.11079arXiv1205.5001OpenAlexW2140422992MaRDI QIDQ1951189

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Publication date: 29 May 2013

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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 $mathbb{F}_p$, 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 $mathbb{F}_p$ when $pequiv1pmod12$.

Full work available at URL: https://arxiv.org/abs/1205.5001

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