226 research outputs found
The Weil-\'etale fundamental group of a number field I
Lichtenbaum has conjectured the existence of a Grothendieck topology for an
arithmetic scheme such that the Euler characteristic of the cohomology
groups of the constant sheaf with compact support at infinity
gives, up to sign, the leading term of the zeta-function at .
In this paper we consider the category of sheaves on this
conjectural site for the spectrum of a number ring. We
show that has, under natural topological assumptions, a well
defined fundamental group whose abelianization is isomorphic, as a topological
group, to the Arakelov Picard group of . This leads us to give a list of
topological properties that should be satisfied by . These
properties can be seen as a global version of the axioms for the Weil group.
Finally, we show that any topos satisfying these properties gives rise to
complexes of \'etale sheaves computing the expected Lichtenbaum cohomology.Comment: 40 pages. To appear in Kyushu Journal of Mathematic
Zeta functions of regular arithmetic schemes at s=0
Lichtenbaum conjectured the existence of a Weil-\'etale cohomology in order
to describe the vanishing order and the special value of the Zeta function of
an arithmetic scheme at in terms of Euler-Poincar\'e
characteristics. Assuming the (conjectured) finite generation of some \'etale
motivic cohomology groups we construct such a cohomology theory for regular
schemes proper over . In particular, we obtain
(unconditionally) the right Weil-\'etale cohomology for geometrically cellular
schemes over number rings. We state a conjecture expressing the vanishing order
and the special value up to sign of the Zeta function at
in terms of a perfect complex of abelian groups
. Then we relate this conjecture to
Soul\'e's conjecture and to the Tamagawa number conjecture of Bloch-Kato, and
deduce its validity in simple cases.Comment: 53 pages. To appear in Duke Math.
The Weil-\'etale fundamental group of a number field II
We define the fundamental group underlying to Lichtenbaum's Weil-\'etale
cohomology for number rings. To this aim, we define the Weil-\'etale topos as a
refinement of the Weil-\'etale sites introduced in \cite{Lichtenbaum}. We show
that the (small) Weil-\'etale topos of a smooth projective curve defined in
this paper is equivalent to the natural definition given in
\cite{Lichtenbaum-finite-field}. Then we compute the Weil-\'etale fundamental
group of an open subscheme of the spectrum of a number ring. Our fundamental
group is a projective system of locally compact topological groups, which
represents first degree cohomology with coefficients in locally compact abelian
groups. We apply this result to compute the Weil-\'etale cohomology in low
degrees and to prove that the Weil-\'etale topos of a number ring satisfies the
expected properties of the conjectural Lichtenbaum topos.Comment: 59 pages. To appear in Selecta Mathematic
On the Weil-étale cohomology of number fields
We give a direct description of the category of sheaves on Lichtenbaum's Weil-étale site of a number ring. Then we apply this result to define a spectral sequence relating Weil-étale cohomology to Artin-Verdier étale cohomology. Finally we construct complexes of étale sheaves computing the expected Weil-étale cohomology
Collective Motion with Anticipation: Flocking, Spinning, and Swarming
We investigate the collective dynamics of self-propelled particles able to
probe and anticipate the orientation of their neighbors. We show that a simple
anticipation strategy hinders the emergence of homogeneous flocking patterns.
Yet, anticipation promotes two other forms of self-organization: collective
spinning and swarming. In the spinning phase, all particles follow synchronous
circular orbits, while in the swarming phase, the population condensates into a
single compact swarm that cruises coherently without requiring any cohesive
interactions. We quantitatively characterize and rationalize these phases of
polar active matter and discuss potential applications to the design of
swarming robots.Comment: 6 pages, 4 figure
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