28 research outputs found
Mild pro-p-groups with 4 generators
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.Comment: 12 pages. No figures. LaTe
Mild pro-2-groups and 2-extensions of Q with restricted ramification
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 . 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
Demuskin groups, Galois modules, and the elementary type conjecture
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).Comment: v2 (20 pages); added theorem characterizing decompositions into free
and trivial modules; to appear in J. Algebr
Remark on fundamental groups and effective Diophantine methods for hyperbolic curves
In a letter from Grothendieck to Faltings, it was suggested that a positive answer to the section conjecture should imply finiteness of points on hyperbolic curves over number fields. In this paper, we point out instead the analogy between the section conjecture and the finiteness conjecture for the Tate-Shafarevich group of elliptic curves. That is, the section conjecture should provide a terminating algorithm for finding all rational points on a hyperbolic curve equipped with a rational point