On the classification of 3-dimensional non-associative division algebras over \(p\)-adic fields
In the paper [Contemp. Math. 463, 29--44 (2008; Zbl 1163.17005)], the authors studied not necessarily associative division algebras of dimension 3 over a field \(K\). Given such an algebra \(A\), define a ternary form \(f:A\to K\) by \(f(x)=\det(\text{L}_x)\), where \(\text{L}_x:A\to A,\:a\mapsto xa\). Since \(A\) is a division algebra, \(f\) is anisotropic. Let \(C_f\) be the set of zeros of \(f\) over \(\bar{K}\), viewed as a subset of the projective space \(\mathbb{P}^2(\bar{K})\). Then \(C_f\) is either a union of three lines, or an absolute irreducible curve of genus one. The article under review features a more detailed analysis of the second case, when \(K\) is a finite extension of \(\mathbb{Q}_p\), the field of \(p\)-adic numbers, for some prime \(p\). With this assumption on \(K\), using a result from [loc. cit.], the authors prove that every anisotropic, absolutely irreducible ternary cubic form \(f\) for which \(C_f\) is a genus one curve can be written as \(f(x)=\det(\text{L}_x)\), where \(\text{L}_x\) is left multiplication in some division algebra \(A\) over \(K\). Next, in what is the main part of the article, they construct all such cubics \(f\). The problem of finding all division algebras \(A\) for which the form \(x\mapsto\det(\text{L}_x)\) is equivalent to a given \(f\) was treated in [loc. cit.]. This, together with the results in the present article, means that every \(3\)-dimensional division algebra over \(K\) for which \(C_f\) is an absolutely irreducible curve of genus one can be constructed.
scientific article; zbMATH DE number 5955269
scientific article