8,350 research outputs found
XXL type Artin groups are CAT(0) and acylindrically hyperbolic
We describe a simple locally CAT(0) classifying space for extra extra large
type Artin groups (with all labels at least 5). Furthermore, when the Artin
group is not dihedral, we describe a rank 1 periodic geodesic, thus proving
that extra large type Artin groups are acylindrically hyperbolic. Together with
Property RD proved by Ciabonu, Holt and Rees, the CAT(0) property implies the
Baum-Connes conjecture for all XXL type Artin groups.Comment: 12 pages, 5 figures. To appear in Annales de l'Institut Fourie
On the computation of the Picard group for surfaces
We construct examples of surfaces of geometric Picard rank . Our
method is a refinement of that of R. van Luijk. It is based on an analysis of
the Galois module structure on \'etale cohomology. This allows to abandon the
original limitation to cases of Picard rank after reduction modulo .
Furthermore, the use of Galois data enables us to construct examples which
require significantly less computation time
On a BSD-type formula for L-values of Artin twists of elliptic curves
This is an investigation into the possible existence and consequences of a
Birch-Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted
by Artin representations. We translate expected properties of L-functions into
purely arithmetic predictions for elliptic curves, and show that these force
some peculiar properties of the Tate-Shafarevich group, which do not appear to
be tractable by traditional Selmer group techniques. In particular we exhibit
settings where the different p-primary components of the Tate-Shafarevich group
do not behave independently of one another. We also give examples of
"arithmetically identical" settings for elliptic curves twisted by Artin
representations, where the associated L-values can nonetheless differ, in
contrast to the classical Birch-Swinnerton-Dyer conjecture.Comment: 27 pages, new versio
Supersingular K3 Surfaces are Unirational
We show that supersingular K3 surfaces in characteristic are related
by purely inseparable isogenies. This implies that they are unirational, which
proves conjectures of Artin, Rudakov, Shafarevich, and Shioda. As a byproduct,
we exhibit the moduli space of rigidified K3 crystals as an iterated
-bundle over . To complete the picture, we also
establish Shioda-Inose type isogeny theorems for K3 surfaces with Picard rank
in positive characteristic.Comment: 31 pages; many details added, final versio
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
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