Large 2-adic Galois image and non-existence of certain abelian surfaces over Q

TL;DR

This paper investigates the structure of Galois representations associated with abelian surfaces and hyperelliptic curves, providing bounds and non-existence results for certain cases, especially involving 2-adic Galois images.

Contribution

It offers a complete description of Honda systems for specific p-divisible groups, establishes bounds on abelian conductors, and proves non-existence of certain favorable abelian surfaces.

Findings

Galois group G is as large as possible under certain conditions.

Non-existence results for specific favorable abelian surfaces with large N.

Characterization of Galois images for Jacobians of hyperelliptic curves.

Abstract

Motivated by our arithmetic applications, we required some tools that might be of independent interest. Let $\mathcal E$ be an absolutely irreducible group scheme of rank $p^4$ over $\mathbb Z_p$. We provide a complete description of the Honda systems of $p$-divisible groups $\mathcal G$ such that $\mathcal G[p^{n+1}]/\mathcal G[p^n] \simeq \mathcal E$ for all $n$. Then we find a bound for the abelian conductor of the second layer $\mathbb Q_p(\mathcal G[p^2])/\mathbb Q_p(\mathcal G[p])$, stronger in our case than can be deduced from Fontaine's bound. Let $\pi\!: \, {\rm Sp}_{2g}(\mathbb Z_p) \to {\rm Sp}_{2g}(\mathbb F_p)$ be the reduction map and let $G$ be a closed subgroup of ${\rm Sp}_{2g}(\mathbb Z_p)$ with $\overline{G} = \pi(G)$ irreducible and generated by transvections. We fill a gap in the literature by showing that if $p=2$ and $G$ contains a transvection, then $G$ is as large as possible in ${\rm Sp}_{2g}(\mathbb Z_p)$ with given reduction $\overline{G}$, i.e. $G = \pi^{-1}(\overline{G})$. One simple application arises when $A = J(C)$ is the Jacobian of a hyperelliptic curve $C\!: \, y^2 + Q(x)y = P(x)$, where $Q(x)^2 + 4P(x)$ is irreducible in $\mathbb Z[x]$ of degree $m=2g+1$ or $2g+2$, with Galois group $\mathcal S_m \subset {\rm Sp}_{2g}(\mathbb F_2)$. If the Igusa discriminant $I_{10}$ of $C$ is odd and some prime $q$ exactly divides $I_{10}$, then $G = {\operatorname{Gal}}(\mathbb Q(A[2^\infty])/\mathbb Q)$ is $\tilde{\pi}^{-1}(\mathcal S_m)$, where $\tilde{\pi}\!: \, {\rm GSp}_{2g}(\mathbb Z_p) \to {\rm Sp}_{2g}(\mathbb F_p)$. When $m = 5$, $Q(x) = 1$ and $I_{10} = N$ is a prime, $A = J(C)$ is an example of a $\textit{favorable}$ abelian surface. We use the machinery above to obtain non-existence results for certain favorable abelian surfaces, even for large $N$.