TY - JOUR

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

AU - McCarthy, Dermot

PY - 2013/2/28

Y1 - 2013/2/28

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(mod12).

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(mod12).

U2 - 10.2140/pjm.2013.261.219

DO - 10.2140/pjm.2013.261.219

M3 - Article

SP - 219

EP - 236

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

ER -