Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Explicit Kummer Theory for Elliptic Curves   Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in
E(K)$ be a point of infinite order, and let $N^{-1}\alpha$ be the set of
$N$-division points of $\alpha$ in $E(\bar{K})$. We prove strong effective and
uniform results for the degrees of the Kummer extensions $[K(E[N],N^{-1}\alpha)
: K(E[N])]$. When $K=\mathbb{Q}$, and under a minimal assumption on $\alpha$,
we show that the inequality $[\mathbb{Q}(E[N],N^{-1}\alpha) : \mathbb{Q}(E[N])]
\geq cN^2$ holds with a constant $c$ independent of both $E$ and $\alpha$.

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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph