Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $N$.

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The categories of this paper are: cs
19
The categories of this paper are: eess
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th