Number Theory Seminar
In 2003, E.-U. Gekeler gave a formula for the number of elliptic curves in an isogeny class (over a finite field), based on probabilistic and equidistribution considerations. This formula is in a sense similar to Siegel's formula expressing the size of a genus of a quadratic form as a product of local densities. It is well-known that Siegel's formula is equivalent to the computation of the Tamagawa volume of the orthogonal group. In the same spirit, there is a formula, due to Langlands and Kottwitz, expressing the cardinality of an isogeny class of principally polarized abelian varieties as an orbital integral. It turns out that by carefully comparing different natural measures on the orbits, one can see a direct connection between Gekeler's formula and the formula of Langlands and Kottwitz, and therefore generalize Gekeler-style formula to higher dimension. In the process we encounter some questions that appear classical but seem to be surprisingly difficult to answer, which I am hoping to discuss. This is a joint project with Jeffrey Achter, S. Ali Altug, and Luis Garcia.