A computational approach to the 2-torsion structure of abelian threefolds

Abstract

Let A be a three-dimensional abelian variety defined over a number field K. Using techniques of group theory and explicit computations with MAGMA, we show that if A has an even number of Fp-rational points for almost all primes p of K, then there exists a K-isogenous A' which has an even number of K-rational torsion points. We also show that there exist abelian varieties A of all dimensions > 4 such that #Ap(Fp) is even for almost all primes p of K, but there does not exist a K-isogenous A' such that #A'(K) tors is even.