TY - JOUR

T1 - The trace of frobenius of elliptic curves and the p-adic gamma function

AU - McCarthy, Dermot

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013

Y1 - 2013

N2 - 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).

AB - 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).

KW - Elliptic curves

KW - Hypergeometric functions

KW - Modular forms

KW - P-adic gamma function

KW - Trace of Frobenius

UR - http://www.scopus.com/inward/record.url?scp=84878841782&partnerID=8YFLogxK

U2 - 10.2140/pjm.2013.261.219

DO - 10.2140/pjm.2013.261.219

M3 - Article

AN - SCOPUS:84878841782

VL - 261

SP - 219

EP - 236

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -