arXiv subjects
Publications and source records attributed to John Labute.
In this article we trace the genesis of a theorem that gives for the first time examples of Galois group $G_S$ of the maximal $p$-extension of $\mathbb{Q}$, unramified outside a finite set of primes not containing $p$, that are of cohomological dimension $2$. The pro-$p$-group $G_S$ is a fab pro-$p$-group which means that all its derived factors are finite.
We show that certain cyclically pinched one-relator groups are residually torsion-free nilpotent.
Let p be an odd prime, let S be a finite set of primes q congruent to 1 mod p but not mod p^2 and let G_S be the Galois group of the maximal p-extension of Q un-ramified outside of S. If r is a continuous homomorphism of G_S into GL_2(Z_p) then under certain conditions on the linking numbers of S we show that r=1 if its reduction mod p is 1. We also show that the reduction of r mod p is 1 if r can be put in triangular form mod p^3.
We investigate the relations in Galois groups of maximal p-extensions of fields, the structure of their natural filtrations, and their relationship with the Bloch-Kato conjecture proved by Rost and Voevodsky with Weibel's patch. Our main focus is on the third degree, but we provide examples for all degrees.
Using the mixed Lie algebras of Lazard, we extend the results of the first author on mild groups to the case p=2. In particular, we show that for any finite set S_0 of odd rational primes we can find a finite set S of odd rational primes containing S_0 such that the Galois group of the maximal 2-extension of Q unramified outside S is mild. We thus produce a projective system of such Galois groups which converge to the maximal pro-2-quotient of the absolute Galois group of $\Q$ unramified at 2 and $\infty$. Our results also allow results of Alexander Schmidt on pro-p-fundamental groups of marked arithmetic curves to be extended to the case p=2 over a global field which is either a function field of odd characteristic or a totally imaginary number field.
Let p be an odd prime and S a finite set of primes = 1 mod p. We give an effective criterion for determining when the Galois group G=G_S(p) of the maximal p-extension of Q unramified outside of S is mild when |S|=4 and the cup product H^1(G,Z/pZ) \otimes H^1(G,Z/pZ) --> H^2(G,Z/pZ) is surjective.
Let p be a prime and F(p) the maximal p-extension of a field F containing a primitive p-th root of unity. We give a new characterization of Demuskin groups among Galois groups Gal(F(p)/F) when p=2, and, assuming the Elementary Type Conjecture, when p>2 as well. This characterization is in terms of the structure, as Galois modules, of the Galois cohomology of index p subgroups of Gal(F(p)/F).
Let p be a prime and F a field containing a primitive pth root of unity. Then for n in N, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is <=n if and only if the corestriction maps H^n(H,Fp) -> H^n(G,Fp) are surjective for all open subgroups H of index p. Using this result we derive a surprising generalization to dim_Fp H^n(H,Fp) of Schreier's formula for dim_Fp H^1(H,Fp).