Brauer groups of curves and unramified central simple algebras over their function fields
The main object of the paper under review is the Brauer group \(\text{Br}(X)\) of a smooth geometrically connected projective curve \(X\) defined over a local non-archimedean field \(k\). The authors' goal is to describe the 2-torsion of this group (assuming \(k\) to be non-dyadic) by representing the elements of \({}_2\text{Br}(X)\) by quaternion algebras over \(k(X)\). The focus is on the case where \(X\) is a hyperelliptic curve. The results heavily depend on the reduction of \(X\). If \(X\) is an elliptic curve, theorems 7-10 give a full description of \({}_2\text{Br}(X)\). Moreover, in that case theorems 11-12 (respectively theorems 13-16) give a representation of the elements of \({}_{2^n}\text{Br}(X)\) and \({}_{3^n}\text{Br}(X)\) (respectively \({}_{p^{\infty }}\text{Br}(X)\)) by unramified central cyclic algebras. If \(X\) is a hyperelliptic curve with good reduction, theorem 18 also gives a full description of \({}_2\text{Br}(X)\). In the case of bad reductiom, such classification is available for hyperelliptic quintics (theorems 20-50). Finally, the authors consider the case where \(k\) is an \(m\)-dimensional local field. Their basic assumption here is that the \(n\)-torsion of the Jacobian of \(X\) is \(k\)-rational. Theorems 51-52 state that any class of algebras in \({}_n\text{Br}(X)\) can be represented as the tensor product of some constant algebra and \(\ell\) cyclic algebras and provide an explicit upper bound for \(\ell\).
scientific article; zbMATH DE number 1551879
scientific article