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.