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The trace of
Frobenius of elliptic curves and the p-adic gamma function
Dermot McCarthy |
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Vol. 261 (2013), No. 1, 219–236
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Abstract |
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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). |
Keywords
elliptic curves, p-adic gamma
function, trace of Frobenius, hypergeometric functions,
modular forms
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Mathematical Subject Classification 2010
Primary: 11G20, 33E50
Secondary: 33C99, 11S80
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Milestones
Received: 21 May 2012
Revised: 6 September 2012
Accepted: 11 September 2012
Published: 28 February 2013
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Authors |
| Dermot McCarthy | |
| Deptartment of Mathematics Texas A&M University Mailstop 3368 College Station, TX 77843 United States |
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