Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counterexamples to the local-global principle for non-singular plane
  curves and a cubic analogue of Ankeny-Artin-Chowla-Mordell conjecture   In this article, we introduce a systematic and uniform construction of
non-singular plane curves of odd degrees $n \geq 5$ which violate the
local-global principle. Our construction works unconditionally for $n$
divisible by $p^2$ for some odd prime number $p$. Moreover, our construction
also works for $n$ divisible by some $p \geq 5$ which satisfies a conjecture on
$p$-adic properties of the fundamental units of $\mathbb{Q}(p^{1/3})$ and
$\mathbb{Q}((2p)^{1/3})$. This conjecture is a natural cubic analogue of the
classical Ankeny-Artin-Chowla-Mordell conjecture for $\mathbb{Q}(p^{1/2})$ and
easily verified numerically.

27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin