Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A parametrized set of explicit elements of $\Sha(E/\Q)[3]$   In this paper, we give explicit equations for homogeneous spaces
corresponding to a rational isogeny of degree $3$. An explicit set of elliptic
curves with elements of order $3$ in their Tate-Shafarevich group is
constructed. Combining this gives explicit examples of plane cubics over $\Q$
that have a point everywhere locally, but not globally.

35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
13
The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg