An Explicit Abelian Surface with Maximal Galois Action

@article{Greicius2018AnEA,
title={An Explicit Abelian Surface with Maximal Galois Action},
author={Quinn Greicius and Aaron Landesman},
journal={arXiv: Number Theory},
year={2018}
}
• Published 30 March 2018
• Mathematics
• arXiv: Number Theory
We construct an explicit example of a genus $2$ curve $C$ over a number field $K$ such that the adelic Galois representation arising from the action of $\operatorname{Gal}(\overline{K}/K)$ on the Jacobian of $C$ has image $\operatorname{GSp}_4(\widehat{\mathbb Z})$.

References

SHOWING 1-10 OF 26 REFERENCES
Hyperelliptic curves with maximal Galois action on the torsion points of their Jacobians
• Mathematics
• 2017
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
An explicit Jacobian of dimension 3 with maximal Galois action
We gives an explicit genus 3 curve over Q such that the Galois action on the torsion points of its Jacobian is a large as possible. That such curves exist is a consequence of a theorem of D.
Surjectivity of Galois representations in rational families of abelian varieties
• Mathematics
Algebra & Number Theory
• 2019
In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as
Elliptic Curves with Surjective Adelic Galois Representations
An analysis of maximal closed subgroups of GL2() results in an example of a number field K and an elliptic curve E/K that admits a surjective adelic Galois representation.
Explicit Determination of the Images of the Galois Representations Attached to Abelian Surfaces with End(A) = Z
We give an effective version of a resuIt reported by Serre asserting that the images of the Galois representations attached to an abelian surface with End(A) = Z are as large as possible for almost
Lifting subgroups of symplectic groups over Z /
For a positive integer g, let Sp2g(R) denote the group of 2g × 2g symplectic matrices over a ring R. Assume g ≥ 2. For a prime number , we give a self-contained proof that any closed subgroup of
Big symplectic or orthogonal monodromy modulo $\ell$
Lifting Subgroups of Symplectic Groups over $\mathbb{Z} / \ell \mathbb{Z}$
For a positive integer $g$, let $\mathrm{Sp}_{2g}(R)$ denote the group of $2g \times 2g$ symplectic matrices over a ring $R$. Assume $g \ge 2$. For a prime number $\ell$, we show that any closed