Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A positive proportion of cubic fields are not monogenic yet have no
  local obstruction to being so   We show that a positive proportion of cubic fields are not monogenic, despite
having no local obstruction to being monogenic. Our proof involves the
comparison of $2$-descent and $3$-descent in a certain family of Mordell curves
$E_k \colon y^2 = x^3 + k$. As a by-product of our methods, we show that, for
every $r \geq 0$, a positive proportion of curves $E_k$ have Tate--Shafarevich
group with $3$-rank at least $r$.

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The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
3
The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th