29 research outputs found
Commuting-Liftable Subgroups of Galois Groups II
Let denote either a positive integer or , let be a fixed
prime and let be a field of characteristic different from . In the
presence of sufficiently many roots of unity in , we show how to recover
some of the inertia/decomposition structure of valuations inside the maximal
-abelian Galois group of using the maximal
-abelian-by-central Galois group of , whenever is sufficiently
large relative to .Comment: 62 pages; final version; NOTE: numbering has changed from previous
version
The Galois action on geometric lattices and the mod- I/OM
This paper studies the Galois action on a special lattice of geometric
origin, which is related to mod- abelian-by-central quotients of
geometric fundamental groups of varieties. As a consequence, we formulate and
prove the mod- abelian-by-central variant/strengthening of a conjecture
due to Ihara/Oda-Matsumoto.Comment: Final version. Minor changes/corrections, introduction expanded. Will
appear in Inventione
Almost-Commuting-Liftable Subgroups of Galois Groups
Let K be a field and \ell be a prime such that char K \neq \ell. In the
presence of sufficiently many roots of unity in K, we show how to recover some
of the inertia/decomposition structure of valuations inside the maximal
(\Z/\ell)-abelian resp. pro-\ell-abelian Galois group of K using its
(Z/\ell)-central resp. pro-\ell-central extensions.Comment: Version 2: updated two references, added a few words to the argument
in Theorem 3, fixed a few typos. All results and arguments are the same. 38
page
Four-fold Massey products in Galois cohomology
In this paper, we develop a new necessary and sufficient condition for the
vanishing of 4-Massey products of elements in the mod-2 Galois cohomology of a
field. This new description allows us to define a splitting variety for
4-Massey products, which is shown in the Appendix to satisfy a local-to-global
principle over number fields. As a consequence, we prove that, for a number
field, all such 4-Massey products vanish whenever they are defined. This
provides new explicit restrictions on the structure of absolute Galois groups
of number fields.Comment: Final version: several corrections made throughout the paper; some
sections reorganized; will appear in Compositio Mathematic
Abstraction boundaries and spec driven development in pure mathematics
In this article we discuss how abstraction boundaries can help tame
complexity in mathematical research, with the help of an interactive theorem
prover. While many of the ideas we present here have been used implicitly by
mathematicians for some time, we argue that the use of an interactive theorem
prover introduces additional qualitative benefits in the implementation of
these ideas.Comment: To appear in a special volume of the Bull. Amer. Math. Soc.; 14 pg.;
feedback welcome
Galois Module Structure of \Z/\ell^n-th Classes of Fields
In this paper we use the Merkurjev-Suslin theorem to explore the structure of
arithmetically significant Galois modules that arise from Kummer theory. Let K
be a field of characteristic different from a prime \ell, n a positive integer,
and suppose that K contains the (\ell^n)^th roots of unity. Let L be the
maximal \Z/\ell^n-elementary abelian extension of K, and set G = \Gal(L|K). We
consider the G-module J = L^\times/\ell^n and denote its socle series by J_m.
We provide a precise condition, in terms of a map to H^3(G,\Z/\ell^n),
determining which submodules of J_{m-1} embed in cyclic modules generated by
elements of J_m. This generalizes a theorem of Adem, Gao, Karaguezian, and
Minac which deals with the case m=\ell^n=2. This description of J_m/J_{m-1} can
be viewed as an analogue of the classical Hilbert's Theorem 90 and it is
helpful for understanding the G-module J.Comment: Final version: to appear in Bull. of the London Math So