Arithmetic of binary cubic forms
The main objects of the paper under review are an integer binary cubic form \(F\) and its ``determining'' form \(q_F\) (Hessian) introduced by Eisenstein. He proved that the correspondence \(F\mapsto q_F\) commutes with the natural action of \(\text{SL}_2(\mathbb Z)\) (and therefore takes classes of cubic forms to classes of quadratic forms) and showed that a primitive quadratic form \(q\) of discriminant \(D=4d\) with \(d\) square-free admits a cubic form \(F\) with \(q_F=q\) if and only if the class of \(q\) is a 3-torsion element in the class group of binary cubic forms of discriminant \(D\). The authors' goal is to extend Eisenstein's results to cubic forms over any integral domain \(R\) of characteristic not 2 or 3. They define a cubic form on a projective \(R\)-module \(M\) of rank 2 with values in \(R\) and associate to it a binary quadratic mapping \(q\colon M\to {\mathcal D}(M)\) where \({\mathcal D}(M):=\text{{Hom}} (\wedge ^2M\to\wedge^2M^*)\). To describe the composition law for binary quadratic forms and mappings in such a generality, they use the language of Clifford algebras, as in \textit{M.~Kneser} [J. Number Theory 15, 406-413 (1982; Zbl 0493.10029)]. The key concept is that of a cubic \(C\)-form where \(C\) is a quadratic \(R\)-algebra in the sense of Kneser [loc.cit.]; these are forms \(F(x)=T(x,x,x)\) where \(T(x,y,z)\) is a symmetric trilinear form with the property that \(T(cx,y,z)\) is symmetric for any \(c\in C\). The main results of the paper (Theorems 5.1 and 5.2) describe the \(C\)-isomorphism classes of cubic \(C\)-forms and can be viewed as rough general analogues of Eisenstein's results. If \(R\) is a PID, more precise results are proved which coincide with classical ones for \(R=\mathbb Z\). The authors also present a cohomological interpretation of their results (in terms of a Kummer exact sequence in flat cohomology) and some explicit computations.
scientific article; zbMATH DE number 1558753