Two-dimensional families of hyperelliptic jacobians with big monodromy

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 $S_{2g}$ or the alternating group $A_{2g}$. We describe explicitly how to choose (infinitely many) pairs of distinct elements $t_1,t_2$ of $K$ 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 $\ell$-adic monodromy.

22 pages. arXiv admin note: text overlap with arXiv:0804.4264

References in corpus (4)

• An open image theorem for a general class of abelian varieties
• Very simple 2-adic representations and hyperelliptic jacobians
• Surjectivity Criteria for p-adic Representations, Part I
• Homomorphisms of abelian varieties