A Radical Characterization of Abelian Varieties

TL;DR

This paper demonstrates that the prime factors of the number of points on an abelian variety over finite fields, for a density one set of primes, uniquely determine the variety up to isogeny, extending Faltings' theorem.

Contribution

It introduces a new characterization of abelian varieties using prime factor data of point counts, providing an explicit finite extension for determination.

Findings

Prime factors of point counts determine abelian varieties up to isogeny.

The proof involves analysis of $\, ext{l}$-adic monodromy groups and Weyl group actions.

Extends Faltings' theorem by using prime factor data instead of Frobenius polynomials.

Abstract

Let $A$ be a square-free abelian variety defined over a number field $K$. Let $S$ be a density one set of prime ideals $\mathfrak{p}$ of $\mathcal{O}_K$. A famous theorem of Faltings says that the Frobenius polynomials $P_{A,\mathfrak{p}}(x)$ for $\mathfrak{p}\in S$ determine $A$ up to isogeny. We show that the prime factors of $|A(\mathbb{F}_{\mathfrak{p}})|=P_{A,\mathfrak{p}}(1)$ for $\mathfrak{p}\in S$ also determine $A$ up to isogeny over an explicit finite extension of $K$. The proof relies on understanding the $\ell$-adic monodromy groups which come from the $\ell$-adic Galois representations of $A$, and the absolute Weyl group action on their weights.