Curves $\mathcal {C}$ that are cyclic twists of $Y^{2}=X^{3}+c$ and the relative Brauer groups $Br(k(\mathcal {C})/k)$

Abstract:

Let $k$ be a field with $\operatorname {char}(k) \neq 2,3$. Let $\mathcal {C}_{f}$ be the projective curve of a binary cubic form $f$, and $k(\mathcal {C}_{f})$ the function field of $\mathcal {C}_{f}$. In this paper we explicitly describe the relative Brauer group $\operatorname {Br}(k(\mathcal {C}_{f})/k)$ of $k(\mathcal {C}_{f})$ over $k$. When $f$ is diagonalizable we show that every algebra in $\operatorname {Br}(k(\mathcal {C}_{f})/k)$ is a cyclic algebra obtainable using the $y$-coordinate of a $k$-rational point on the Jacobian $\mathcal {E}$ of $\mathcal {C}_{f}$. But when $f$ is not diagonalizable, the algebras in $\operatorname {Br}(k(\mathcal {C}_{f})/k)$ are presented as cup products of cohomology classes, but not as cyclic algebras. In particular, we provide several specific examples of relative Brauer groups for $k=\mathbb {Q}$, the rationals, and for $k=\mathbb {Q}(\omega )$, where $\omega$ is a primitive third root of unity. The approach is to realize $\mathcal {C}_{f}$ as a cyclic twist of its Jacobian $\mathcal {E}$, an elliptic curve, and then apply a recent theorem of Ciperiani and Krashen.