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Genus one curves from division algebras of degree 3 Saltman, David J

Description

If $D/F$ is a division algebra of degree 3, then the Severi-Brauer variety of $D$, call it $X$, is a form of the projective plane. The line bundle $O(3)$ is defined on $X$, which says it makes sense to talk about cubic curves on $X$. Since $X$ has no rational points, these are genus one curve and not elliptic curves. However, they are principle homogeneous spaces over their Jacobians $E$, which are elliptic curves. Which ones occur?

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