Honda-Tate theory for log abelian varieties over finite fields

TL;DR

This paper extends Honda-Tate theory to log abelian varieties over finite fields, providing classifications of isogeny classes using Weil numbers and rational points in simplices, generalizing classical results.

Contribution

It generalizes Honda-Tate theory to log abelian varieties over finite fields, describing isogeny classes via Weil numbers and rational points in generalized simplices.

Findings

Complete classification of isogeny classes over standard log points.

Description of simple isogeny classes via rational points.

Extension of classical Honda-Tate theory to log abelian varieties.

Abstract

In this article we study the Honda-Tate theory for log abelian varieties over an fs log point $S=(\mathrm{Spec}(\mathbf{k}),M_S)$ for $\mathbf{k}=\mathbb{F}_q$ a finite field, generalizing the classical Honda-Tate theory for abelian varieties over $\mathbf{k}$. For the standard log point $S$, we give a complete description of the isogeny classes of such log abelian varieties using Weil $q$-numbers of weight 0,1, and 2. In the general case where $M_S$ admits a global chart $P\to\mathbf{k}$ with $P=\mathbb{N}^k$, we also give a complete description of simple isogeny classes of log abelian varieties over $S$ in terms of rational points in generalized simplices.