Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the vanishing of almost all primary components of the
  Shafarevich-Tate group of elliptic curves over the rationals   The Shafarevich-Tate and Selmer groups arise in the context of Kummer theory
for elliptic curves. The finiteness of the Shafarevich-Tate group of an
elliptic curve $E$ over the field of rational numbers is included in the Birch
and Swinnerton-Dyer conjectures, and is still an open question.
  We present an overview of the Shafarevich-Tate and Selmer groups of an
elliptic curve in the framework of Galois cohomology. Known results on the
finiteness of the Shafarevich-Tate group are mentioned, including results of
Coates and Wiles, Rubin, Gross and Zagier, and Kolyvagin.
  We then prove the vanishing of the $\ell$-primary component of the
Shafarevich-Tate group for almost all primes $\ell$, for any elliptic curve $E$
over the rationals without complex multiplication.

37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs