Applications of patching to quadratic forms and central simple algebras

Abstract

This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the u-invariant of p-adic function fields, for p odd. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.

48 pages; connectivity now required in the definition of rational group; beginning of Section 4 reorganized; other minor changes

References in corpus (1)

• Period and index in the Brauer group of an arithmetic surface (with an appendix by Daniel Krashen)

Cited by in corpus (39)

• Period-index and u-invariant questions for function fields over complete discretely valued fields
• Open Problems on Central Simple Algebras
• Refinements to patching and applications to field invariants
• Local-global principle for reduced norms over function fields of p-adic curves
• Dualité et principe local-global pour les tores sur une courbe au-dessus de C((t))
• The Pythagoras number and the $u$-invariant of Laurent series fields in several variables
• Hasse Principle for Simply Connected Groups over Function Fields of Surfaces
• Local-global principles for tori over arithmetic curves
• On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions
• Patching over Berkovich Curves and Quadratic Forms
• On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions
• Differential Galois Groups over Laurent Series Fields
• Division Algebras and Quadratic Forms over Fraction Fields of Two-dimensional Henselian Domains
• Patching and local-global principles for homogeneous spaces over function fields of p-adic curves
• Indecomposable and Noncrossed Product Division Algebras over Curves over Complete Discrete Valuation Rings
• Local-global principles for norm one tori over semi-global fields
• Diagonal forms of higher degree over function fields of $p$-adic curves
• Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields
• Local-Global Principle for Unitary Groups Over Function Fields of p-adic Curves
• Local-global Galois theory of arithmetic function fields
• A Tannakian approach to patching
• Hasse principle for hermitian spaces over semi-global fields
• Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
• A cohomological Hasse principle over two-dimensional local rings
• On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3
• Weierstrass preparation and algebraic invariants
• Generalized period-index problem with an application to quadratic forms
• Local-global principles for homogeneous spaces over some two-dimensional geometric global fields
• Reduced norms of division algebras over complete discrete valuation fields of local-global type
• Cyclicity and indecomposability in the Brauer group of a $p$-adic curve
• Patching subfields of division algebras
• Period-index bounds for arithmetic threefolds
• Symbol length in the Brauer group of a field
• Linkage of Pfister forms over semi-global fields
• Embedding problems of division algebras
• Gerbe patching and a Mayer-Vietoris sequence over arithmetic curves
• On the u-Invariant of Function Fields of Curves Over Complete Discretely Valued Fields
• Torsion in Chow groups of zero cycles of homogeneous projective varieties
• Local triviality for G-torsors