A Radical Characterization of Abelian Varieties

Heung Shan Theodore Hui
Let A be a square-free abelian variety defined over a number field K. Let S be a density one set of prime ideals p of O K . A famous theorem of Faltings says that the Frobenius polynomials P A,p (x) for p ∈ S determine A up to isogeny. We show that the prime factors of |A(F p )| = P A,p (1) for p ∈ S also determine A up to isogeny over an explicit finite extension of K. The proof relies on understanding the -adic monodromy groups which come from the -adic Galois representations of A, and the
more » ... olute Weyl group action on their weights. We also show that there exists an explicit integer e ≥ 1 such that after replacing K by a suitable finite extension, the Frobenius polynomials of A at p must equal to the e-th power of a separable polynomial for a density one set of prime ideals p ⊆ O K .
doi:10.7298/x41v5c30 fatcat:pysdlrw2mrccdjiwl6ffgvahx4