Hilbert irreducibility for abelian varieties

If $K$ is the rational function field $K=Q(t)$, then a polynomial $f$ in $K[x]$ can be regarded as a one-parameter family of polynomials over $Q$. If $f$ is irreducible, then a basic form of Hilbert’s irreducibility theorem states that there are infinitely many $t$ in $Q$ for which the specialized polynomial $f_t$ is irreducible over $Q$. In this talk we will discuss analogous theorems for an abelian variety $A/K$ regarded as a one-parameter family of abelian varieties over $K$. For example, we will exhibit $A$ which are simple over $K$ and for which there are only finitely many $t$ in $Q$ such that the abelian variety $A_t$ is not simple over $Q$.