Large 2-adic Galois image and non-existence of certain abelian surfaces over ${\mathbb Q}$
Volume 183 / 2018
Abstract
This paper has two themes, linked by an arithmetic application.
Let $\pi\colon {\rm Sp}_{2g}({\mathbb Z}_p) \to {\rm Sp}_{2g}({\mathbb F}_p)$ be the reduction map and let $G$ be a closed subgroup of ${\rm Sp}_{2g}({\mathbb Z}_p)$ with ${\hskip2.pt\overline{\hskip-2.pt G \hskip-.2pt}\hskip.2pt} = \pi(G)$ irreducible and generated by transvections. To fill a gap in the literature, we show that if $p=2$ and $G$ contains a transvection, then $G$ is as large as possible in ${\rm Sp}_{2g}({\mathbb Z}_2)$ with the given reduction ${\hskip2.pt\overline{\hskip-2.pt G \hskip-.2pt}\hskip.2pt}$, i.e. $G = \pi^{-1}({\hskip2.pt\overline{\hskip-2.pt G \hskip-.2pt}\hskip.2pt})$. For example, let $A$ be the Jacobian of a hyperelliptic curve $C\colon y^2 + Q(x)y = P(x)$, where $f = Q^2 + 4P$ 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 discriminant of $f$ is exactly divisible by an odd prime, then $G = \operatorname{Gal}({\mathbb Q}(A[2^\infty])/{\mathbb Q})$ is $\tilde{\pi}^{-1}({\mathcal S}_m)$, where $\tilde{\pi} \colon {\rm GSp}_{2g}({\mathbb Z}_2) \to {\rm Sp}_{2g}({\mathbb F}_2)$.
Let ${\mathcal E}$ be an absolutely irreducible commutative 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.
If $m = 5$, $Q(x) = 1$ and the Igusa discriminant $I_{10} = N$ is a prime, then the Jacobian $A = J(C)$ is an example of a favorable abelian surface. Non-existence results for certain favorable abelian surfaces follow, even for large $N$. Also, we determine the parameters for the Honda system associated to $A[4]$ over ${\mathbb Z}_2$ in terms of the coefficients of $f$ by exploiting the $x-T$ map usually used for 2-descent.
