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Fully maximal and fully minimal abelian varieties and curves

Banff International Research Station for Mathematical Innovation and Discovery

We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists over its minimal field of definition. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic. For supersingular curves of genus 3 in characteristic 2, we use a parametrization of a moduli space of such curves by Viana and Rodriguez to determine the L-polynomial and the type of each. This is joint work with Valentijn Karemaker.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: Colorado State University

2017-11-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63723

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/