Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mod-p reducibility, the torsion subgroup, and the Shafarevich-Tate group   Let $E$ be an optimal elliptic curve over $\Q$ of prime conductor $N$. We
show that if for an odd prime $p$, the mod $p$ representation associated to $E$
is reducible (in particular, if $p$ divides the order of the torsion subgroup
of $E(\Q)$), then the $p$-primary component of the Shafarevich-Tate group of
$E$ is trivial. We also state a related result for more general abelian
subvarieties of $J_0(N)$ and mention what to expect if $N$ is not prime.

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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: q-bio
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The categories of this paper are: cs
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph