Counting abelian varieties over finite fields via Frobenius densities

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.

Added author; significantly simplified global calculation in section 5; made other, smaller improvements

References in corpus (1)

• The Discriminant of an Algebraic Torus

Cited by in corpus (3)

• Polarized superspecial simple abelian surfaces with real Weil numbers
• Class numbers of CM algebraic tori, CM abelian varieties and components of unitary Shimura varieties
• On Tamagawa numbers of CM tori