Faddeev invariants for central simple algebras over rational function fields

• Published in: Manuscripta mathematica Vol. 145; no. 1-2; pp. 71 - 88
• Main Author: Sivatski, A. S.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2014
• Subjects: Algebraic Geometry; Calculus of Variations and Optimal Control; Optimization; Geometry; Lie Groups; Mathematics; Mathematics and Statistics; Number Theory; Topological Groups; 11E04; 16H05; 11E81
• ISSN: 0025-2611; 1432-1785
• DOI: 10.1007/s00229-014-0667-9

Let p be a prime, k a field, containing a primitive p th root of unity, char k ≠  p . We give an upper bound for the Faddeev index of a central simple algebra of exponent p over the rational function field k ( t ) in the case where the ramification set of the algebra consists of rational points. This bound depends only on the number of ramification points and in certain cases turns out to be strict. In the case where p =  2 and the ramification set in A k 1 consists of three rational points we compute the Faddeev index, using the language of quadratic forms. Let X be a smooth geometrically irreducible complete curve over k . We show that there exist algebras of exponent p over k ( X ) with the prescribed Faddeev index, provided there are algebras of exponent p and arbitrarily large index over k . In the last section of the paper we consider another invariant of a central simple algebra of prime exponent p over k ( t ), the so called Faddeev cyclic length. In certain cases we compute this invariant, using triviality of the divided power operations on central simple cyclic algebras of exponent p .