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

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 \(f\bmod \ell\) occurs as the characteristic polynomial of a symplectic similitude over \(\mathbb{F}_\ell\)...

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Bibliographic Details
Published in:arXiv.org 2015-05
Main Authors: Achter, Jeff, Williams, Cassie
Format: Article
Language:English
Subjects:
Fields (mathematics)
Polynomials
Online Access:Get full text
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Summary: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 \(f\bmod \ell\) 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 \(f\).
ISSN:2331-8422
DOI:10.48550/arxiv.1403.3037