Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Genus 1 Curves in Severi--Brauer Surfaces   In a talk at the Banff International Research Station in 2015 Asher Auel
asked questions about genus one curves in Severi-Brauer varieties $SB(A)$. More
specifically he asked about the smooth cubic curves in Severi-Brauer surfaces,
that is in $SB(D)$ where $D/F$ is a degree three division algebra. Even more
specifically, he asked about the Jacobian, $E$, of these curves. In this paper
we give a version of an answer to both these questions, describing the
surprising connection between these curves and properties of the algebra $A$.
Let $F$ contain $\rho$, a primitive third root of one. Since $D/F$ is cyclic,
it is generated over $F$ by $x,y$ such that $xy = \rho{yx}$ and we call $x,y$ a
skew commuting pairs. The connection mentioned above is between the Galois
structure of the three torsion points $E[3]$ and the Galois structure of skew
commuting pairs in extensions $D \otimes_F K$. Given a description of which $E$
arise, we then describe, via Galois cohomology, which $C$ arise.

23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin