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

TL;DR

This paper explicitly describes the relative Brauer group of function fields of certain cubic curves over fields with characteristic not 2 or 3, highlighting differences between diagonalizable and non-diagonalizable cases.

Contribution

It provides a detailed analysis of the structure of the relative Brauer group for these curves, including explicit examples and the use of cyclic twists and recent theorems.

Findings

For diagonalizable forms, all Brauer group elements are cyclic algebras from rational points.

For non-diagonalizable forms, Brauer group elements are cup products, not cyclic.

Explicit examples over Q and Q(ω) illustrate the theoretical results.

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.