Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local-Global principles for certain images of Galois representations   Let $K$ be a number field and let $E/K$ be an elliptic curve whose mod $\ell$
Galois representation locally has image contained in a group $G$, up to
conjugacy. We classify the possible images for the global Galois representation
in the case where $G$ is a Cartan subgroup or the normalizer of a Cartan
subgroup. When $K = \mathbf{Q}$, we deduce a counterexample to the local-global
principle in the case where $G$ is the normalizer of a split Cartan and $\ell =
13$. In particular, there are at least three elliptic curves (up to twist) over
$\mathbf{Q}$ whose mod $13$ image of Galois is locally contained in the
normalizer of a split Cartan, but whose global image is not.

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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci