2010/04/05 by Darrell Haile, Darrell E. Haile, Ilseop Han +4
Mathematics · #14H52 #16K50 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #math.RA #msc:14H52 #msc:16K50
paper · pdf · doi:10.48550/arxiv.1004.0714
arxiv created 2010/04/05 · openalex publication_date 2010/04/05 · arxiv updated 2010/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a field with char(k) not 2 or 3. Let Cf be the projective curve of a binary cubic form f, and k(Cf) the function field of Cf. In this paper we explicitly describe the relative Brauer group Br(k(Cf)/k) of k(Cf) over k. When f is diagonalizable we show that every algebra in Br(k(Cf)/k) is a cyclic algebra obtainable using the y-coordinate of a k-rational point on the Jacobian E of Cf. But when f is not diagonalizable, the algebras in Br(k(Cf)/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 Cf as a cyclic twist of its Jacobian E, an elliptic curve, and then apply a recent theorem of Ciperiani and Krashen.