Vol. 286, No. 1, 2017

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Elliptic curves, random matrices and orbital integrals

Jeffrey D. Achter and Julia Gordon

Appendix: S. Ali Altuğ

Vol. 286 (2017), No. 1, 1–24

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.

elliptic curve, orbital integral, isogeny class
Mathematical Subject Classification 2010
Primary: 11G20, 22E35
Secondary: 14G15
Received: 15 July 2016
Accepted: 19 July 2016
Published: 9 December 2016
Jeffrey D. Achter
Department of Mathematics
Colorado State University
Weber Building
Fort Collins, CO 80523-1874
United States
Julia Gordon
University of British Columbia
Vancouver, BC V6T 1Z2
S. Ali Altuğ
Massachusetts Institute of Technology
Cambridge, MA 02139-4307
United States