Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians

Abstract

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb{Q}$ whose Jacobians have such maximal adelic Galois representations.

arXiv comment25 pages

References in corpus (6)

• Elliptic curves with maximal Galois action on their torsion points
• Hilbert's irreducibility theorem and the larger sieve
• Surjectivity of Galois Representations in Rational Families of Abelian Varieties
• An explicit Jacobian of dimension 3 with maximal Galois action
• Residual Representations of Semistable Principally Polarized Abelian Varieties
• Lifting Subgroups of Symplectic Groups over $\mathbb{Z}/\ell \mathbb{Z}$

Cited by in corpus (2)

• Surjectivity of Galois Representations in Rational Families of Abelian Varieties
• Lifting Subgroups of Symplectic Groups over $\mathbb{Z}/\ell \mathbb{Z}$