Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Relating Tate-Shafarevich group of an elliptic curve with class group   The paper formulates a precise relationship between the Tate-Shafarevich
group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the
classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) =
GL_2({\mathbb Z}/p)$ operates by its standard 2 dimensional representation over
${\mathbb Z}/p$. We establish such a relationship in most cases.

22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph