Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counterexamples to the local-global divisibility over elliptic curves   Let $p \geq 5$ be a prime number. We find all the possible subgroups $G$ of
${\rm GL}_2 ( \mathbb{Z} / p \mathbb{Z} )$ such that there exists a number
field $k$ and an elliptic curve ${\mathcal{E}}$ defined over $k$ such that the
${\rm Gal} ( k ( {\mathcal{E}}[p] ) / k )$-module ${\mathcal{E}}[p]$ is
isomorphic to the $G$-module $( \mathbb{Z} / p \mathbb{Z} )^2$ and there exists
$n \in \mathbb{N}$ such that the local-global divisibility by $p^n$ does not
hold over ${\mathcal{E}} ( k )$.

11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
1
The categories of this paper are: cs