Algebraic elements in division algebras over function fields of curves
Let \(C\) be a curve over the field \(k\) with function field \(K\) and \(D\) a division algebra with center \(K\). Consider a maximal order \({\mathcal O}_{\Lambda}\) in \(D\), i.e. a sub-\({\mathcal O}_ C\)-sheaf of maximal orders of the constant sheaf \(\mathbb D\). As in the commutative case, one may consider the cohomology spaces of coherent \({\mathcal O}_{\Lambda}\)-modules, and relate them by an analogue of the Riemann-Roch theorem. One has two different notions of genus for \(D\), \(g_ D\) and \(g_{\Lambda}\), where \(g_{\Lambda}\) depends on the choice of the maximal order \({\mathcal O}_{\Lambda}\). Some natural questions arise: -- Under which circumstances does \(g_{\Lambda}\) depend only on \(D\)? -- Does \(D\) contain \(k\)-algebraic sub-division algebras? Several partial answers to these questions are given, assuming additional conditions on \(k\) or on the genus of \(K\) over \(k\). Furthermore, some very instructive examples and counterexamples are provided.
scientific article; zbMATH DE number 3959573