Finite group subschemes of abelian varieties over finite fields

TL;DR

This paper classifies the finite group subschemes of abelian varieties over finite fields using Newton polygons, and applies this to classify zeta functions of Kummer surfaces.

Contribution

It provides a new classification of group schemes within isogeny classes of abelian varieties over finite fields based on Newton polygons.

Findings

Classification of $B[ ext{ell}]$ group schemes in terms of Newton polygons

Determination of zeta functions of Kummer surfaces over finite fields

Connection between Newton polygons and group scheme structures

Abstract

Let $A$ be an abelian variety over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by the Weil polynomial $f_A$. We assume that $f_A$ is separable. For a given prime number $\ell\neq\mathrm{char}\, k$ we give a classification of group schemes $B[\ell]$, where $B$ runs through the isogeny class, in terms of certain Newton polygons associated to $f_A$. As an application we classify zeta functions of Kummer surfaces over $k$.