Consider an ordinary isogeny class of elliptic curves over a finite, prime field. Inspired by a random matrix heuristic (which is so strong it’s false), Gekeler defines a local factor for each rational prime. Using the analytic class number formula, he shows that the associated infinite product computes the size of the isogeny class.

I’ll explain a transparent proof of this formula; it turns out that this product actu- ally computes an adelic orbital integral which visibly counts the desired cardinality. Moreover, it the new perspective allows a natural generalization to higher-dimensional abelian varieties.

This is joint work with Julia Gordon and S. Ali Altug.