Vol. 2, No. 3, 2009

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ISSN: 1944-4184 (e-only)
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Numerical evidence on the uniform distribution of power residues for elliptic curves

Jeffrey Hatley and Amanda Hittson

Vol. 2 (2009), No. 3, 305–321
Abstract

Elliptic curves are fascinating mathematical objects which occupy the intersection of number theory, algebra, and geometry. An elliptic curve is an algebraic variety upon which an abelian group structure can be imposed. By considering the ring of endomorphisms of an elliptic curve, a property called complex multiplication may be defined, which some elliptic curves possess while others do not. Given an elliptic curve E and a prime p, denote by Np the number of points on E over the finite field Fp. It has been conjectured that given an elliptic curve E without complex multiplication and any modulus M, the primes for which Np is a square modulo p are uniformly distributed among the residue classes modulo M. This paper offers numerical evidence in support of this conjecture.

Keywords
elliptic curve, power residue, uniform distribution, computational algebraic geometry
Mathematical Subject Classification 2000
Primary: 11Y99
Milestones
Received: 21 September 2008
Revised: 1 April 2009
Accepted: 11 April 2009
Published: 3 October 2009

Communicated by Nigel Boston
Authors
Jeffrey Hatley
Department of Mathematics
University of Massachusetts
Amherst, MA 01003-9305
United States
Amanda Hittson
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706-1388
United States