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arXiv · 1905.11603

Counting abelian varieties over finite fields via Frobenius densities

Jeff Achter·Salim Ali Altug·Luis Garcia·Julia Gordon·Wen-Wei Li·Thomas Rüd
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Abstract

Let $[X,\lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor $\nu_v([X,\lambda])$ for each place $v$ of $\mathbb Q$, and show that the product of these factors essentially computes the size of the isogeny class of $[X,\lambda]$. The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes, and in particular does not rely on a calculation of class numbers.

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Jeff Achter, Salim Ali Altug, Luis Garcia, Julia Gordon, Wen-Wei Li, Thomas Rüd. 2019-05-28. Counting abelian varieties over finite fields via Frobenius densities. https://doi.org/10.2140/ant.2023.17.1239

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