Two-dimensional families of hyperelliptic Jacobians with big monodromy

Zarhin, Yuri

doi:10.1090/tran/6579

Two-dimensional families of hyperelliptic Jacobians with big monodromy
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Abstract:

Let $K$ be a global field of characteristic different from $2$ and $u(x)\in K[x]$ be an irreducible polynomial of even degree $2g\ge 6$ whose Galois group over $K$ is either the full symmetric group $\mathbf {S}_{2g}$ or the alternating group $\mathbf {A}_{2g}$. We describe explicitly how to choose (infinitely many) pairs of distinct $t_1, t_2 \in K$ such that the $g$-dimensional Jacobian of a hyperelliptic curve $y^2=$ $(x-t_1)(x-t_2))u(x)$ has no nontrivial endomorphisms over an algebraic closure of $K$ and has big monodromy.

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