Vol. 286, No. 1, 2017

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 338: 1
Vol. 337: 1  2
Vol. 336: 1+2
Vol. 335: 1  2
Vol. 334: 1  2
Vol. 333: 1  2
Vol. 332: 1  2
Vol. 331: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
Elliptic curves, random matrices and orbital integrals

Jeffrey D. Achter and Julia Gordon

Appendix: S. Ali Altuğ

Vol. 286 (2017), No. 1, 1–24
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
Authors
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
Canada
S. Ali Altuğ
Massachusetts Institute of Technology
Cambridge, MA 02139-4307
United States