Instruct: Identify the main and secondary category of Arxiv papers based on the titles
Query: Ideal class groups of number fields and Bloch-Kato's Tate-Shafarevich
  groups for symmetric powers of elliptic curves
18
The categories of this paper are: econ
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom