Let $K$ be a field of odd characteristic $p$, let $f\left(x\right)$ be an irreducible separable polynomial of degree $n\ge 5$ with big Galois group (the symmetric group or the alternating group). Let $C$ be the hyperelliptic curve ${y}^{2}=f\left(x\right)$ and $J\left(C\right)$ its jacobian. We prove that $J\left(C\right)$ does not have nontrivial endomorphisms over an algebraic closure of $K$ if either $n\ge 7$ or $p\ne 3$.

The main result of this paper implies that if an abelian variety over a field $F$ has a maximal isotropic subgroup of $n$-torsion points all of which are defined over $F$, and $n\ge 5$, then the abelian variety has semistable reduction away from $n$. This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its $n$-torsion points are defined over a field $F$ and $n\ge 3$, then the abelian variety has semistable reduction away from $n$. We also give information about the Néron models...

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