A Radical Characterization of Abelian Varieties

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 $\mathrm{\mid }A({\mathbb{F}}_{\mathfrak{p}})\mathrm{\mid }=P_{A,\mathfrak{p}}(1)$ for $\mathfrak{p}\in S$ also determine $A$ up to isogeny over an explicit finite extension of $K$. The proof relies on understanding the $\mathrm{\ell }$-adic monodromy groups which come from the $\mathrm{\ell }$-adic Galois representations of $A$, and the absolute Weyl group action on their weights.