Period-index problems in WC-groups IV: a local transition theorem

2009/07/15 by Pete L. Clark, Clark, Pete L.
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.0907.2497

20 pages; comments welcome

arxiv created 2009/07/15 · arxiv updated 2009/12/01

Abstract

Let K be a complete discretely valued field with perfect residue field k. Assuming upper bounds on the relation between period and index for WC-groups over k, we deduce corresponding upper bounds on the relation between period and index for WC-groups over K. Up to a constant depending only on the dimension of the torsor, we recover theorems of Lichtenbaum and Milne in a "duality free" context. Our techniques include the use of LLR models of torsors under abelian varieties with good reduction and a generalization of the period-index obstruction map to flat cohomology. In an appendix, we consider some related issues of a field-arithmetic nature.

Related

• The period-index problem in WC-groups II: abelian varieties
• The period-index problem in WC-groups I: elliptic curves
• Central simple algebras, Milnor K-theory and homogeneous spaces over complete discretely valued fields of dimension 2
• Period-index in top cohomology over semiglobal fields
• The period-index problem and Hodge theory