56:08
published on April 25, 2024
Motives and (super-)representation theory: principles and case studies
By Yves André
Local densities compute isogeny classes
Appears in collection : Arithmetic, Geometry, Cryptography and Coding Theory / Arithmétique, géométrie, cryptographie et théorie des codes
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 actually computes an adelic orbital integral which visibly counts the desired cardinality. Moreover, the new perspective allows a natural generalization to higher-dimensional abelian varieties. This is joint work with Julia Gordon and S. Ali Altug.
