385 research outputs found

    Varieties

    Full text link
    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

    Full text link
    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

    Full text link
    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

    Full text link
    If CC is a smooth curve over an algebraically closed field kk of characteristic pp, then the structure of the maximal prime to pp 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-pp 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

    Full text link
    Let GG be a profinite group. A strongly admissible smooth representation ρ\rho of GG over C\mathbb{C} decomposes as a direct sum ρπIrr(G)mπ(ρ)π\rho \cong \bigoplus_{\pi \in \mathrm{Irr}(G)} m_\pi(\rho) \, \pi of irreducible representations with finite multiplicities mπ(ρ)m_\pi(\rho) such that for every positive integer nn the number rn(ρ)r_n(\rho) of irreducible constituents of dimension nn 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 ζρ(s)=n=1rn(ρ)ns=πIrr(G)mπ(ρ)(dimπ)s \zeta_\rho (s) = \sum_{n=1}^\infty r_n(\rho) n^{-s} = \sum_{\pi \in \mathrm{Irr}(G)} \frac{m_\pi(\rho)}{(\dim \pi)^s} associated to such a representation ρ\rho. Our primary focus is on representations ρ=IndHG(σ)\rho = \mathrm{Ind}_H^G(\sigma) of compact pp-adic Lie groups GG that arise from finite dimensional representations σ\sigma of closed subgroups HH via the induction functor. In addition to a series of foundational results - including a description in terms of pp-adic integrals - we establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro-pp 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 pp-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 pp-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

    Full text link
    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 33, and there are 3232 such groups for depth 44.Comment: 11 page

    A Hermite-Minkowski type theorem of varieties over finite fields

    Full text link
    As an application of P. Delgine's theorem (Esnault and Kerz in Acta Math. Vietnam. 37:531-562, 2012) on a finiteness of ll-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 pp

    Full text link
    We introduce a spreading out technique to deduce finiteness results for \'etale fundamental groups of complex varieties by characteristic pp methods, and apply this to recover a finiteness result proven recently for local fundamental groups in characteristic 00 using birational geometry.Comment: 15 pages, comments welcom

    On the congruence kernel for simple algebraic groups

    Full text link
    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 GG is K-isotropic then C^(S)(G) as a normal subgroup of \widehat{G}^(S) is almost generated by a single element

    Moduli of \ell-adic pro-\'etale local systems for smooth non-proper schemes

    Full text link
    Let XX be a smooth scheme over an algebraically closed field. When XX is proper, it was proved in \cite{me1} that the moduli of \ell-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 \ell is invertible in XX, 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 XX 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
    corecore