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A Radical Characterization of Abelian Varieties

2017

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

doi:10.7298/x41v5c30
fatcat:pysdlrw2mrccdjiwl6ffgvahx4