Elliptic curves, random matrices and orbital integrals

TL;DR

This paper provides a new proof that the size of an isogeny class of elliptic curves over a finite field can be computed using an adelic orbital integral, linking algebraic and analytic perspectives.

Contribution

It offers a transparent proof connecting Gekeler's product formula to adelic orbital integrals, answering a question by N. Katz.

Findings

The product formula equals an adelic orbital integral.

The proof clarifies the relationship between elliptic curve isogeny classes and orbital integrals.

It provides a new perspective on counting elliptic curves over finite fields.

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. 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.