19,192 research outputs found
Mumford dendrograms and discrete p-adic symmetries
In this article, we present an effective encoding of dendrograms by embedding
them into the Bruhat-Tits trees associated to -adic number fields. As an
application, we show how strings over a finite alphabet can be encoded in
cyclotomic extensions of and discuss -adic DNA encoding. The
application leads to fast -adic agglomerative hierarchic algorithms similar
to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint
of -adic geometry, to encode a dendrogram in a -adic field means
to fix a set of -rational punctures on the -adic projective line
. To is associated in a natural way a
subtree inside the Bruhat-Tits tree which recovers , a method first used by
F. Kato in 1999 in the classification of discrete subgroups of
.
Next, we show how the -adic moduli space of
with punctures can be applied to the study of time series of
dendrograms and those symmetries arising from hyperbolic actions on
. In this way, we can associate to certain classes of dynamical
systems a Mumford curve, i.e. a -adic algebraic curve with totally
degenerate reduction modulo .
Finally, we indicate some of our results in the study of general discrete
actions on , and their relation to -adic Hurwitz spaces.Comment: 14 pages, 6 figure
A Satake isomorphism in characteristic p
Suppose that G is a connected reductive group over a p-adic field F, that K
is a hyperspecial maximal compact subgroup of G(F), and that V is an
irreducible representation of K over the algebraic closure of the residue field
of F. We establish an analogue of the Satake isomorphism for the Hecke algebra
of compactly supported, K-biequivariant functions f: G(F) \to End V. These
Hecke algebras were first considered by Barthel-Livne for GL_2. They play a
role in the recent mod p and p-adic Langlands correspondences for GL_2(Q_p), in
generalisations of Serre's conjecture on the modularity of mod p Galois
representations, and in the classification of irreducible mod p representations
of unramified p-adic reductive groups.Comment: 24 pages, revise
The classification of p-compact groups and homotopical group theory
We survey some recent advances in the homotopy theory of classifying spaces,
and homotopical group theory. We focus on the classification of p-compact
groups in terms of root data over the p-adic integers, and discuss some of its
consequences e.g. for finite loop spaces and polynomial cohomology rings.Comment: To appear in Proceedings of the ICM 2010
The classification of 2-compact groups
We prove that any connected 2-compact group is classified by its 2-adic root
datum, and in particular the exotic 2-compact group DI(4), constructed by
Dwyer-Wilkerson, is the only simple 2-compact group not arising as the
2-completion of a compact connected Lie group. Combined with our earlier work
with Moeller and Viruel for p odd, this establishes the full classification of
p-compact groups, stating that, up to isomorphism, there is a one-to-one
correspondence between connected p-compact groups and root data over the p-adic
integers. As a consequence we prove the maximal torus conjecture, giving a
one-to-one correspondence between compact Lie groups and finite loop spaces
admitting a maximal torus. Our proof is a general induction on the dimension of
the group, which works for all primes. It refines the
Andersen-Grodal-Moeller-Viruel methods to incorporate the theory of root data
over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and
we show that certain occurring obstructions vanish, by relating them to
obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s.Comment: 47 page
On a classification of irreducible admissible modulo representations of a -adic split reductive group
We give a classification of irreducible admissible modulo representations
of a split -adic reductive group in terms of supersingular representations.
This is a generalization of a theorem of Herzig.Comment: 25 page
Structure, classifcation, and conformal symmetry, of elementary particles over non-archimedean space-time
It is known that no length or time measurements are possible in sub-Planckian
regions of spacetime. The Volovich hypothesis postulates that the
micro-geometry of spacetime may therefore be assumed to be non-archimedean. In
this letter, the consequences of this hypothesis for the structure,
classification, and conformal symmetry of elementary particles, when spacetime
is a flat space over a non-archimedean field such as the -adic numbers, is
explored. Both the Poincar\'e and Galilean groups are treated. The results are
based on a new variant of the Mackey machine for projective unitary
representations of semidirect product groups which are locally compact and
second countable. Conformal spacetime is constructed over -adic fields and
the impossibility of conformal symmetry of massive and eventually massive
particles is proved
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