Open Problems on Central Simple Algebras

Abstract

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

v2 has some small revisions to the text. Some items are re-numbered, compared to v1

References in corpus (4)

• Twisted sheaves and the period-index problem
• Applications of patching to quadratic forms and central simple algebras
• Weakly commensurable arithmetic groups, lengths of closed geodesics and isospectral locally symmetric spaces
• Quaternion algebras with the same subfields

Cited by in corpus (10)

• Period-index and u-invariant questions for function fields over complete discretely valued fields
• Word equations in simple groups and polynomial equations in simple algebras
• Refinements to patching and applications to field invariants
• On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions
• Essential dimension of algebraic groups, including bad characteristic
• On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions
• Surjectivity of the total Clifford invariant and Brauer dimension
• On the Brauer $p$-dimension of Henselian discrete valued fields of residual characteristic $p>0$
• On the Noether Problem for torsion subgroups of tori
• Period-index bounds for arithmetic threefolds