Vol. 261, No. 1, 2013

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Vol. 286: 1  2
Vol. 285: 1  2
Vol. 284: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
The trace of Frobenius of elliptic curves and the p-adic gamma function

Dermot McCarthy

Vol. 261 (2013), No. 1, 219–236
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).

Keywords
elliptic curves, p-adic gamma function, trace of Frobenius, hypergeometric functions, modular forms
Mathematical Subject Classification 2010
Primary: 11G20, 33E50
Secondary: 33C99, 11S80
Milestones
Received: 21 May 2012
Revised: 6 September 2012
Accepted: 11 September 2012
Published: 28 February 2013
Authors
Dermot McCarthy
Deptartment of Mathematics
Texas A&M University
Mailstop 3368
College Station, TX 77843
United States