385 research outputs found
Varieties
This text is devoted to the theory of varieties, which provides an important
tool, based in universal algebra, for the classification of regular languages.
In the introductory section, we present a number of examples that illustrate
and motivate the fundamental concepts. We do this for the most part without
proofs, and often without precise definitions, leaving these to the formal
development of the theory that begins in Section 2. Our presentation of the
theory draws heavily on the work of Gehrke, Grigorieff and Pin (2008) on the
equational theory of lattices of regular languages. In the subsequent sections
we consider in more detail aspects of varieties that were only briefly evoked
in the introduction: Decidability, operations on languages, and
characterizations in formal logic.Comment: This is a chapter in an upcoming Handbook of Automata Theor
Some aspects of profinite group theory
A survey of recent results about profinite groups, and results about infinite
and finite groups where the theory of profinite groups plays a leading role
Semi-galois Categories I: The Classical Eilenberg Variety Theory
This paper is an extended version of our proceedings paper announced at
LICS'16; in order to complement it, this version is written from a different
viewpoint including topos-theoretic aspect on our work. Technically, this paper
introduces and studies the class of semi-galois categories, which extend galois
categories and are dual to profinite monoids in the same way as galois
categories are dual to profinite groups; the study on this class of categories
is aimed at providing an axiomatic reformulation of Eilenberg's theory of
varieties of regular languages--- a branch in formal language theory that has
been developed since the mid 1960's and particularly concerns systematic
classification of regular languages, finite monoids, and deterministic finite
automata. In this paper, detailed proofs of our central results announced at
LICS'16 are presented, together with topos-theoretic considerations. The main
results include (I) a proof of the duality theorem between profinite monoids
and semi-galois categories, extending the duality theorem between profinite
groups and galois categories; based on this results on semi-galois categories,
we then discuss (II) a reinterpretation of Eilenberg's theory from a viewpoint
of duality theorem; in relation with this reinterpretation of the theory, (III)
we also give a purely topos-theoretic characterization of classifying topoi BM
of profinite monoids M among general coherent topoi, which is a topos-theoretic
application of (I). This characterization states that a topos E is equivalent
to the classifying topos BM of some profinite monoid M if and only if E is (i)
coherent, (ii) noetherian, and (iii) has a surjective coherent point. This
topos-theoretic consideration is related to the logical and geometric problems
concerning Eilenberg's theory that we addressed at LICS'16, which remain open
in this paper.Comment: Updated some part of our proceedings paper published in Proc. of
LICS1
Generators and relations for the etale fundamental group
If is a smooth curve over an algebraically closed field of
characteristic , then the structure of the maximal prime to quotient of
the \'etale fundamental group is known by analytic methods. In this paper, we
discuss the properties of the fundamental group that can be deduced by purely
algebraic techniques. We describe a general reduction from an arbitrary curve
to the projective line minus three points, and show what can be proven
unconditionally about the maximal pro-nilpotent and pro-solvable quotients of
the prime-to- fundamental group. Included is an appendix which treats the
tame fundamental group from a stack-theoretic perspective.Comment: 26 pages. Significant revision; errors corrected, various points
clarifie
Zeta functions associated to admissible representations of compact p-adic Lie groups
Let be a profinite group. A strongly admissible smooth representation
of over decomposes as a direct sum of irreducible
representations with finite multiplicities such that for every
positive integer the number of irreducible constituents of
dimension is finite. Examples arise naturally in the representation theory
of reductive groups over non-archimedean local fields. In this article we
initiate an investigation of the Dirichlet generating function associated to such a representation .
Our primary focus is on representations of
compact -adic Lie groups that arise from finite dimensional
representations of closed subgroups via the induction functor. In
addition to a series of foundational results - including a description in terms
of -adic integrals - we establish rationality results and functional
equations for zeta functions of globally defined families of induced
representations of potent pro- groups. A key ingredient of our proof is
Hironaka's resolution of singularities, which yields formulae of Denef-type for
the relevant zeta functions.
In some detail, we consider representations of open compact subgroups of
reductive -adic groups that are induced from parabolic subgroups. Explicit
computations are carried out by means of complementing techniques: (i)
geometric methods that are applicable via distance-transitive actions on
spherically homogeneous rooted trees and (ii) the -adic Kirillov orbit
method. Approach (i) is closely related to the notion of Gelfand pairs and
works equally well in positive defining characteristic.Comment: 61 pages; small changes, contains an abridged Section 5.3. Final
version to be published in Trans. Amer. Math. Soc. (arXiv version contains an
additional footnote in Section 5.3
On finite generation of self-similar groups of finite type
A self-similar group of finite type is the profinite group of all
automorphisms of a regular rooted tree that locally around every vertex act as
elements of a given finite group of allowed actions. We provide criteria for
determining when a self-similar group of finite type is finite,
level-transitive, or topologically finitely generated. Using these criteria and
GAP computations we show that for the binary alphabet there is no infinite
topologically finitely generated self-similar group given by patterns of depth
, and there are such groups for depth .Comment: 11 page
A Hermite-Minkowski type theorem of varieties over finite fields
As an application of P. Delgine's theorem (Esnault and Kerz in Acta Math.
Vietnam. 37:531-562, 2012) on a finiteness of -adic sheaves on a variety
over a finite field, we show the finiteness of \'etale coverings of such a
variety with given degree whose ramification bounded along an effective Cartier
divisor. This can be thought of a higher dimensional analogue of the classical
Hermite-Minkowski theorem
Finiteness of \'etale fundamental groups by reduction modulo
We introduce a spreading out technique to deduce finiteness results for
\'etale fundamental groups of complex varieties by characteristic methods,
and apply this to recover a finiteness result proven recently for local
fundamental groups in characteristic using birational geometry.Comment: 15 pages, comments welcom
On the congruence kernel for simple algebraic groups
This paper contains several results about the structure of the congruence
kernel C^(S)(G) of an absolutely almost simple simply connected algebraic group
G over a global field K with respect to a set of places S of K. In particular,
we show that C^(S)(G) is always trivial if S contains a generalized arithmetic
progression. We also give a criterion for the centrality of C^(S)(G) in the
general situation in terms of the existence of commuting lifts of the groups
G(K_v) for v \notin S in the S-arithmetic completion \widehat{G}^(S). This
result enables one to give simple proofs of the centrality in a number of
cases. Finally, we show that if K is a number field and is K-isotropic then
C^(S)(G) as a normal subgroup of \widehat{G}^(S) is almost generated by a
single element
Moduli of -adic pro-\'etale local systems for smooth non-proper schemes
Let be a smooth scheme over an algebraically closed field. When is
proper, it was proved in \cite{me1} that the moduli of -adic continuous
representations of \pi_1^\et(X), \LocSys(X), is representable by a
(derived) \Ql-analytic space. However, in the non-proper case one cannot
expect that the results of \cite{me1} hold mutatis mutandis. Instead, assuming
is invertible in , one has to bound the ramification at infinity of
those considered continuous representations. The main goal of the current text
is to give a proof of such representability statements in the open case. We
also extend the representability results of \cite{me1}. More specifically,
assuming is assumed to be proper, we show that \LocSys(X) admits a
canonical shifted symplectic form and we give some applications of such
existence result
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