Elliptic curves, random matrices and orbital integrals

Jeffrey D. Achter and Julia Gordon

Vol. 286 (2017), No. 1, 1–24

DOI: 10.2140/pjm.2017.286.1

Abstract

An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a product formula, in terms of congruence considerations involving that polynomial, for the size of such an isogeny class (over a finite prime field). In this paper we give a new transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality. This answers a question posed by N. Katz and extends Gekeler’s work to ordinary elliptic curves over arbitrary finite fields.

Keywords

elliptic curve, orbital integral, isogeny class

Mathematical Subject Classification 2010

Primary: 11G20, 22E35

Secondary: 14G15

Milestones

Received: 15 July 2016

Accepted: 19 July 2016

Published: 9 December 2016