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Description

There is a classical but not very well-known connection between counting objects that are in some sense `in the same class' but not isomorphic, and volume computations. I will start by recalling the analytic class number formula, and the Minkowski-Siegel mass formula for the "number" of quadratic forms in a genus, as well as Tamagawa's reformulation of these results as a volume computation. Then I will discuss a similar formula for the number of elliptic curves in an isogeny class, and we will see that it can again appear in two versions: one is due to Gekeler (2003) and comes from probabilistic and equidistribution considerations, and the other is due to Langlands and Kottwitz and is based on a volume computation. Finally, I will talk about our recent generalization of Gekeler's result to counting principally polarized Abelian varieties, by `reverse engineering' the Langlands-Kottwitz formula.