Local heuristics and an exact formula for abelian surfaces over finite fields

Abstract

Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution considerations we define, for each prime $\ell$, a local factor which measures the relative frequency with which $f\mathrm{mod}\ell$ occurs as the characteristic polynomial of a symplectic similitude over ${\mathbb{F}}_{\ell }$. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over ${\mathbb{F}}_q$ with Weil polynomial $f$.

15 pages, section 6.4 added, minor revisions. Final version; to appear in Canadian Mathematical Bulletin

Cited by in corpus (3)

• Polarizations of abelian varieties over finite fields via canonical liftings
• Counting abelian varieties over finite fields via Frobenius densities
• Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields