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Chris Hall : Hilbert irreducibility for abelian varieties

If $K$ is the rational function field $K=\mathbb{Q}(t)$, then a polynomial $f$ in $K[x]$ can be regarded as a one-parameter family of polynomials over $\mathbb{Q}$. If $f$ is irreducible, then a basic form of Hilbert's irreducibility theorem states that there are infinitely many $t$ in $\mathbb{Q}$ for which the specialized polynomial $f_t$ is irreducible over $\mathbb{Q}$. 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 $\mathbb{Q}$ such that the abelian variety $A_t$ is not simple over $\mathbb{Q}$.

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