Curves C that are Cyclic Twists of Y^2 = X^3+c and the Relative Brauer Groups Br(k(C)/k

Abstract

Let k be a field with char(k) not 2 or 3. Let C_f be the projective curve of a binary cubic form f, and k(C_f) the function field of C_f. In this paper we explicitly describe the relative Brauer group Br(k(C_f)/k) of k(C_f) over k. When f is diagonalizable we show that every algebra in Br(k(C_f)/k) is a cyclic algebra obtainable using the y-coordinate of a k-rational point on the Jacobian E of C_f. But when f is not diagonalizable, the algebras in Br(k(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=Q, the rationals, and for k=Q(omega) where omega is a primitive third root of unity. The approach is to realize C_f as a cyclic twist of its Jacobian E, an elliptic curve, and then apply a recent theorem of Ciperiani and Krashen.