Rumely’s local global principle for algebraic ${\mathrm P}{\mathcal S}{\mathrm C}$ fields over rings

Skolem Density Problems over Algebraic Psc Fields over Rings

A Local-global Principle for Weak Approximation on Varieties over Function Fields

We present a new perspective on the weak approximation conjecture of Hassett and Tschinkel: formal sections of a rationally connected fibration over a curve can be approximated to arbitrary order by regular sections. The new approach involves the study of how ideal sheaves pullback to Cartier divisors. 1. Statement of results Disclaimer. This is the first draft of an article which was circulate...

Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains

— Let R be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let L and k be its fraction field and residue field respectively. Let ΩR be the set of rank 1 discrete valuations of L corresponding to codimension 1 points of regular proper models of Spec R. We prove that a quadratic form q over L satisfies the local-global principle with respect to ΩR in the follo...

Haagerup Property for Algebraic Groups over Local Fields

We classify, among the linear algebraic groups over a local field of characteristic zero, those that have the Haagerup property (also called a-(T)-menability). Our method relies essentially on a discussion on the existence of a subgroup isomorphic, up to a finite covering, to the semidirect product of SL2 by an irreducible representation, or a one-dimensional central extension of an even-dimens...

Curves over Every Global Field Violating the Local-global Principle

There is an algorithm that takes as input a global field k and produces a curve over k violating the local-global principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n.