On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I (original) (raw)

Extensions of unipotent groups, Massey products and Galois cohomology

arXiv (Cornell University), 2017

We study the vanishing of four-fold Massey products in mod p Galois cohomology. First, we describe a sufficient condition, which is simply expressed by the vanishing of some cup-products, in direct analogy with the work of Guillot, Mináč and Topaz for p = 2. For local fields with enough roots of unity, we prove that this sufficient condition is also necessary, and we ask whether this is a general fact. We provide a simple splitting variety, that is, a variety which has a rational point if and only if our sufficient condition is satisfied. It has rational points over local fields, and so, if it satisfies a local-global principle, then the Massey Vanishing conjecture holds for number fields with enough roots of unity. At the heart of the paper is the construction of a finite group U 5 (F p), which has U 5 (F p) as a quotient. Here U n (F p) is the group of unipotent n × n-matrices with entries in the field F p with p elements; it is classical that U n+1 (F p) is intimately related to n-fold Massey products. Although U 5 (F p) is much larger than U 5 (F p), its definition is very natural, and for our purposes, it is easier to study.

Local-global principles for Galois cohomology

Commentarii Mathematici Helvetici, 2014

This paper proves local-global principles for Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for H n pF, Z{mZpn´1qq, for all n ą 1. This is motivated by work of Kato and others, where such principles were shown in related cases for n " 3. Using our results in combination with cohomological invariants, we obtain localglobal principles for torsors and related algebraic structures over F . Our arguments rely on ideas from patching as well as the Bloch-Kato conjecture.

Field theory and the cohomology of some Galois groups

Arxiv preprint math/0009011, 2000

Abstract. We prove that two arithmetically significant extensions of a field F coincide if and only if the Witt ring WF is a group ring Z/n[G]. Furthermore, working modulo squares with Galois groups which are 2-groups, we establish a theorem analogous to Hilbert's theorem 90 and show ...

On the topology of relative orbits for actions of algebraic groups over complete fields

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2010

We investigate the problem of equipping a topology on cohomology groups (sets) in its relation with the problem of closedness of (relative) orbits for the action of algebraic groups on affine varieties defined over complete, especially p-adic fields and give some applications.

On cohomological systems of Galois representations

Banach Center Publications, 2016

The paper contains an expanded version of the talk delivered by the first author during the conference ALANT3 in Będlewo in June 2014. We survey recent results on independence of systems of Galois representations attached to-adic cohomology of schemes. Some other topics ranging from the Mumford-Tate conjecture and the Geyer-Jarden conjecture to applications of geometric class field theory are also considered. In addition, we have highlighted a variety of open questions which can lead to interesting research in near future.

On the Hasse principle for finite group schemes over global function fields

Mathematical Research Letters, 2012

Let K be a global function field of characteristic p > 0 and let M be a (commutative) finite and flat K-group scheme. We show that the kernel of the canonical localization map H 1 (K, M) −→ all v H 1 (K v , M) in flat (fppf) cohomology can be computed solely in terms of Galois cohomology. We then give applications to the case where M is the kernel of multiplication by p m on an abelian variety defined over K.