Orders in quaternion algebras over global function fields having the cancellation property (Q1112875)

Let A be a central simple algebra over a global field K, and \({\mathbb{R}}=R_ S\) some ring of S-integers in K (\(\emptyset \neq S\) containing the archimedean places). An R-order \(\theta\) is said to have the cancellation property ``CP'', if the set \(LF_ 1(\theta)\) of left ideal classes maps bijectively to the group CL(\(\theta)\) of stable isomorphism classes of locally free \(\theta\)-modules. If Eichler's condition is satisfied: (*) ``There exists \(v\in S\) such that \(A_ v\) is not a division algebra'', all the R-orders in A do have ``CP''. The authors investigate the cancellation problem for global function fields, assuming without restriction that \((i)\quad (*)\quad is\) not satisfied; \((ii)\quad K\quad is\) in fact a rational function field \({\mathbb{F}}_ q(X) \) (otherwise, ``CP'' cannot hold by previous work). In the paper under review, a complete list is given of all (hereditary) orders in quaternion algebras A over \({\mathbb{F}}_ q(X)\) having ``CP'', where first the algebras are determined whose maximal orders satisfy ``CP''. The proof uses a numerical criterion, due in the number field case to M. F. Viguéras, that compares the size of fibers of the map \(LF_ 1(\theta)\to CL(\theta)\) with the Eichler measure of \(\theta\).