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Local Heuristics and an Exact Formula for Abelian Surfaces Over Finite Fields

Published online by Cambridge University Press:  20 November 2018

Jeffrey Achter  and
Cassandra Williams
Jeffrey Achter
Affiliation:
Colorado State University, Fort Collins, CO 80523-1874, USA e-mail: achter@math.colostate.edu
Cassandra Williams
Affiliation:
James Madison University, Harrisonburg, VA 22807, USA e-mail: willi5cl@jmu.edu
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Abstract

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Consider a quartic $q$ -Weil polynomial $f$ . Motivated by equidistribution considerations, we define, for each prime $\ell$ , a local factor that 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 $f$ .

Keywords

14K02abelian surfacesfinite fieldsrandom matrices

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 673 - 691
Copyright
Copyright © Canadian Mathematical Society 2015

References

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