Local Heuristics and an Exact Formula for Abelian Surfaces Over Finite Fields
Jeffrey D. Achter, Cassandra Williams · Canadian Mathematical Bulletin · 2015
Abstract Consider a quartic q-Weil polynomial ƒ. Motivated by equidistribution considerations, we define, for each prime ℓ, a local factor that measures the relative frequency with which ƒ mod ℓ occurs as the characteristic polynomial of a symplectic similitude over 𝔽ℓ. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over 𝔽q with Weil polynomial ƒ.