Ure C. Prime Torsion in the Brauer Group of an Elliptic Curve. Transactions of the American Mathematical Society (under revision). Submitted.

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## Abstract

We determine generators and relations in the d-torsion of the Brauer group of an elliptic curve, provided that the d-torsion of the elliptic curve itself is rational over the base field, for any integer d≥2. For q any odd prime, we further give an algorithm to explicitly calculate generators and relations of the q-torsion of the Brauer group of an elliptic curve over any base field of characteristic different from two, three, and q, containing a primitive q-th root of unity. These generators are symbol algebras and the relations arise as their tensor products. We deduce an upper bound on the symbol length of the prime torsion of Br(E)/Br(k).

*Last updated on 11/07/2022*