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The class number of hereditary orders in non-Eichler algebras over global function fields. (English) Zbl 0627.16003
Let A be a central simple algebra over a global function field K, R a ring of “integers” in K. A is supposed not to satisfy the Eichler condition with respect to R (i.e. A is a non-Eichler division algebra over R). For hereditary R-orders \(\Theta\) in A the class numbers of the isomorphism classes of locally free left \(\Theta\)-ideals are studied. In the case where \(K=F_ q(t)\), \(R=F_ q[t]\) and A a non-Eichler \((F_ q[t])\)-algebra of prime index over K, an explicit class number formula is obtained. The main ingredients are Eichler’s formula for the stably free measure and the study of the different embeddings of rings of integers in splitting fields of A. In the final section it is indicated how the formulae can be extended to the general case.

MSC:
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
11R58 Arithmetic theory of algebraic function fields
11R52 Quaternion and other division algebras: arithmetic, zeta functions
16P10 Finite rings and finite-dimensional associative algebras
16Kxx Division rings and semisimple Artin rings
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References:
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