32 research outputs found
Cohomological Hasse principle and motivic cohomology for arithmetic schemes
In 1985 Kazuya Kato formulated a fascinating framework of conjectures which
generalizes the Hasse principle for the Brauer group of a global field to the
so-called cohomological Hasse principle for an arithmetic scheme. In this paper
we prove the prime-to-characteristic part of the cohomological Hasse principle.
We also explain its implications on finiteness of motivic cohomology and
special values of zeta functions.Comment: 47 pages, final versio
Galois sections for abelianized fundamental groups
Given a smooth projective curve of genus at least 2 over a number field
, Grothendieck's Section Conjecture predicts that the canonical projection
from the \'etale fundamental group of onto the absolute Galois group of
has a section if and only if the curve has a rational point. We show that there
exist curves where the above map has a section over each completion of but
not over . In the appendix Victor Flynn gives explicit examples in genus 2.
Our result is a consequence of a more general investigation of the existence
of sections for the projection of the \'etale fundamental group `with
abelianized geometric part' onto the Galois group. We give a criterion for the
existence of sections in arbitrary dimension and over arbitrary perfect fields,
and then study the case of curves over local and global fields more closely. We
also point out the relation to the elementary obstruction of
Colliot-Th\'el\`ene and Sansuc.Comment: This is the published version, except for a characteristic 0
assumption added in Section 5 which was unfortunately omitted there. Thanks
to O. Wittenberg for noticing i
On Selmer groups of abelian varieties over â„“-adic Lie extensions of global function fields
The six operations for sheaves on Artin stacks II: Adic Coefficients
International audienceIn this paper we develop a theory of Grothendieck's six operations for adic constructible sheaves on Artin stacks continuing the study of the finite coefficients case in math.AG/0512097