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

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 $\mathbf{F}_{\mathfrak{p}}$-rational points for almost all primes $\mathfrak{p}$ 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 $\geq 4$ such that $\char93 A_{\p}(\mathbf{F}_{\mathfrak{p}})$ is even for almost all primes $\mathfrak{p}$ of $K$, but there does not exist a $K$-isogenous $A'$ such that $\char93 A'(K)_{tors}$ is even.