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

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 $N$.