David Harbater

Local-global principles for torsors over arithmetic curves

We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for groups with rational components, we compute it explicitly and show that it is finite. This yields necessary and sufficient conditions for local-global principles to hold. Our results rely on first obtaining a Mayer-Vietoris sequence for Galois cohomology and then showing that torsors can be patched. We also give new applications to quadratic forms and central simple algebras.

Patching subfields of division algebras

Given a field F, one may ask which finite groups are Galois groups of field extensions E/F such that E is a maximal subfield of a division algebra with center F. This question was originally posed by Schacher, who gave partial results over the field of rational numbers. Using patching, we give a complete characterization of such groups in the case that F is the function field of a curve over a complete discretely valued field with algebraically closed residue field of characteristic zero, as well as results in related cases.

Local-global principles for Galois cohomology

This paper proves local-global principles for Galois cohomology groups over function fields

Refinements to Patching and Applications to Field Invariants

F

Differential Galois groups over Laurent series fields

of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for

EMBEDDING PROBLEMS AND OPEN SUBGROUPS

H^n(F, Z/mZ(n-1))

Weierstrass preparation and algebraic invariants

, for all

The local lifting problem for actions of finite groups on curves

n>1

Global Oort groups

. This is motivated by work of Kato and others, where such principles were shown in related cases for

Abhyankar’s conjectures in Galois theory: Current status and future directions

n=3