On an algebra associated to a ternary cubic curve.
Let \(f(x_1,\dots,x_n)\) be an \(n\)-ary form of degree \(d\) over a field \(F\). The `Clifford algebra' of \(f\) is defined to be the \(F\)-algebra \(C_f=F\{x_1,\dots,x_n\}/I\) where \(I\) is the two-sided ideal generated by elements of the form \((\alpha_1x_1+\cdots+\alpha_nx_n)^d-f(\alpha_1,\dots,\alpha_n)\) for \(\alpha_1,\dots,\alpha_n\in F\). When \(d=2\), this is the classical Clifford algebra. When \(d>2\), the Clifford algebra \(C_f\) has been studied by various authors. For \(d>3\) or \(d=3\) and \(n=2\), \(C_f\) contains a free \(F\)-algebra on two generators and hence it is not finitely generated over its center and therefore Azumaya. \textit{D.~E.~Haile} proved [in Am. J. Math. 106, 1269-1280 (1984; Zbl 0585.13001)] that the Clifford algebra of a binary cubic form is Azumaya over its center of rank 9. He also described this center as the affine coordinate ring of an elliptic curve related to the given form. In the current paper, the author generalizes those results. Starting with a binary cubic form \(f(x,y)=ax^3+3bx^2y+3cxy^2+dy^3\), she defines a cubic plane curve \(C\) by the equation \(z^3-exyz-f(x,y)=0\) where \(e\) is a parameter. She then defines a (non-usual) Clifford algebra \(A_C\) associated to this curve as \(A_C=F\{x,y\}/I\) where \(I\) is the two-sided ideal generated by elements of the form \((\alpha x+\beta y)^3-e\alpha\beta(\alpha x+\beta y)-f(\alpha,\beta)\). She proves that the center of \(A_C\) is isomorphic to the affine coordinate ring of the Jacobian of the curve \(C\). She defines a group homomorphism \(\Phi\colon E(F)\to\text{Br}(F)\) in the obvious manner, and she proves that the image is the relative Brauer group \(\text{Br}(F(C)/F)\). Finally, she describes \(\Phi\) explicitly in the diagonal case, i.e. when \(b=c=0\). It must be noted that for \(e=0\), these results give Haile's results mentioned above.
scientific article; zbMATH DE number 5931637
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