We study the local topological zeta function associated to a complex function that is holomorphic at the origin of $\mathbb{C}^2$ (respectively $\mathbb{C}^3$). We determine all possible poles less than −1/2 (respectively −1). On $\mathbb{C}^2$ our result is a generalization of the fact that the log canonical threshold is never in ]5/6,1[. Similar statements are true for the motivic zeta function.

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra $S$ in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adjoin inverses to some of the arrows in the quiver, we show that the path algebra of the new quiver with relations is $M_n(S)$ where $n$ is the number of vertices in our quiver. The slogan would be that every finitely presented algebra is Morita equivalent to a universal localization of a finite dimensional algebra.

We prove a portion of a conjecture of Conrad, Diamond, and Taylor, yielding some new cases of the Fontaine–Mazur conjectures, specifically, the modularity of certain potentially Barsotti–Tate Galois representations. The proof follows the template of Wiles, Taylor–Wiles, and Breuil–Conrad–Diamond–Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the ring of integers of a local field. This makes crucial use of the filtered $\phi_1$-modules of Breuil.

We give estimates for exponential sums of the form $\sum_{n \leq N}\Lambda(n)\exp(2 \pi i a g^n/m)$, where m is a positive integer, a and g are integers relatively prime to m, and $\Lambda$ is the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form $\sum_{p \leq N}\exp(2 \pi i a M_p/m)$, where Mp is the Mersenne number; $M_p=2^p-1$ for any prime p. We also estimate some closely related sums, including $\sum_{n \leq N}\mu(n)\exp(2 \pi i a g^n/m)$ and $\sum_{n \leq N}\mu^2(n)\exp(2 \pi i a g^n/m)$, where $\mu$ is the Möbius function.

The multifractal formalism for functions has been proved to be valid for a large class of selfsimilar functions. All the functions that have been studied are (or turned out to be) associated with a family of contractions which satisfies some separation conditions. In this paper, we extend the validity in the presence of overlaps involved by the well-known $n$-scale dilation family. Our method of proof is based on wavelet analysis and some interesting properties of this family.

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. Moduli spaces of these objects are obtained with fixed generic polar parts at each singularity, which amounts to fixing a coadjoint orbit of the group $GL_r(\mathbb{C}[z]/z^n)$. We prove that they carry complete hyper-Kähler metrics.

Let X/S be a smooth morphism of schemes in characteristic p and let $(E,\nabla)$ be a sheaf of $\mathcal{O}_{X}$-modules with integrable connection on X. We give a formula for the cohomology sheaves of the de Rham complex of $(E,\nabla)$ in terms of a Higgs complex constructed from the p-curvature of $(E,\nabla)$. This formula generalizes the classical Cartier isomorphism, with which it agrees when $(E,\nabla)$ is the constant connection.

Nous appelons polynôme quasi-ordinaire de Laurent un polynôme unitaire $f(Y)$ dont les coefficients sont des séries de Laurent à plusieurs variables et tel que son discriminant soit le produit d’un monôme de Laurent et d’une série entière de terme constant non-nul. Si la dérivée $\partial f/\partial Y$ rendue unitaire est encore quasi-ordinaire de Laurent—ce qui peut être toujours obtenu par changement de base—nous montrons que l’on peut mesurer le contact de ses facteurs avec ceux de $f$ en fonction d’invariants discrets de $f$ qui mesurent le contact entre ses racines, codés sous la forme de l’arbre d’Eggers–Wall. Tous les calculs sont faits en termes de chaînes et de cochaînes supportées par cet arbre. Ce travail constitue une généralisation de résultats connus pour les germes de courbes planes.

AMS 2000 Mathematics subject classification: Primary 32S25. Secondary 14M25

We study the fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups. We use this additional structure to analyse geometric/topological properties of stacks. We give an explicit formula for the fundamental group of the coarse moduli space. As an application, we find an explicit formula for the fundamental group of the geometric quotient of an arbitrary algebraic group action. Also, we use these additional structures to give a necessary and sufficient for an algebraic stack to be uniformizable (i.e. quotient of an algebraic space by a finite group action).

AMS 2000 Mathematics subject classification: Primary 14A20. Secondary 14F35; 14L30

We prove a conjecture made by Frank Peterson on the global structure of the Dickson algebras arising as odd primary general linear group invariants. The Dickson algebra $W_{n}$ of invariants in a rank $n$ polynomial algebra over $\mathbb{F}_{p}$ is an unstable algebra over the mod $p$ Steenrod algebra. We prove that $W_{n}$ is a free unstable algebra on a certain cyclic module, modulo just one additional relation. The result is both similar to and different from the corresponding result we previously obtained with Frank Peterson at the prime 2. We also extend our characterization to the algebras of invariants under the special linear groups.

Following the analogy between primes and knots, we introduce the refined Milnor invariants for prime numbers and establish their connection with certain Massey products in Galois cohomology. This generalizes the well-known relation between the power residue symbol and cup product and gives a cohomological interpretation of L. Rédei's triple symbol.

We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in $\mathbb{C}^*$ (or, more generally, with coefficients in the complex points of an algebraic torus over $\mathbb{C}$) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse stating that the admissible homomorphisms of a Weil group to the Langlands dual group of a reductive group can be lifted to an extension of the Langlands dual group by a torus.

We analyse the non-commutative space underlying the quantum group $\textrm{SU}_q(2)$ from the spectral point of view, which is the basis of non-commutative geometry, and show how the general theory developed in our joint work with Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudo-differential calculus, the Wodzciki-type residue, and the local cyclic co-cycle giving the index formula. The co-chain whose co-boundary is the difference between the original Chern character and the local one is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. This specific example allows us to illustrate the general notion of locality in non-commutative geometry. The formulae computing the residue are ‘local’. Locality by stripping all the expressions from irrelevant details makes them computable. The key feature of this spectral triple is its equivariance, i.e. the $\textrm{SU}_q(2)$-symmetry. We shall explain how this naturally leads to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries.

AMS 2000 Mathematics subject classification: Primary 81R50; 19K33; 46L; 58B34

We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves.

In this paper we apply the techniques and results from the theory of multifractal divergence points to give a systematic and detailed account of the Hausdorff dimensions of sets of numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their $N$-adic expansion. Using earlier methods and results we investigate and compute the Hausdorff dimension of several new sets of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of numbers for which the limiting frequencies of the digits in their $N$-adic expansion do not exist. Such sets have only very rarely been studied. In addition, our techniques provide simple proofs of (substantial generalizations of) known results, by Cajar and Drobot and Turner and others, on the Hausdorff dimension of sets of normal and non-normal numbers.

The annihilator $J^{\perp}$ of a weak*-closed inner ideal $J$ in the JBW*-triple $A$ consists of elements $b$ of $A$ for which $\{J\,b\,A\}$ is equal to zero, and the kernel $\mathrm{Ker}(J)$ of $J$ consists of those elements $b$ in $A$ for which $\{J\,b\,J\}$ is equal to zero. The annihilator $J^{\perp}$ is also a weak*-closed inner ideal in $A$, and $A$ enjoys the Peirce decomposition \begin{eqnarray*} A &=& J \oplus_M J^{\perp} \oplus J_1\\ &=& J_2 \oplus_M J_0 \oplus J_1, \end{eqnarray*} where $J_1$ is the intersection of the kernels of $J$ and $J^{\perp}$. When all of the usual Peirce arithmetical relations hold, $J$ is said to be a Peirce inner ideal. A pair $(J,K)$ of weak*-closed inner ideals is said to be compatible when \[A = \bigoplus_{j,k = 0}^2 J_j \cap K_k.\] By analysing the biannihilator $J^{\perp\perp}$ of a Peirce inner ideal $J$ it is shown that, if $J$ is a Peirce inner ideal, then $J^{\perp}$ is a Peirce inner ideal. Furthermore, if $K$ is a weak*-closed inner ideal in $A$ such that $(J,K)$ is a compatible pair, then $(J^{\perp},K)$ is a compatible pair.

Existence varieties (briefly e-varieties) of regular semigroups were introduced in [1] as classes of regular semigroups closed under the formation of homomorphic images, regular subsemigroups and direct products. Existence varieties of orthodox semigroups, which are simply the sub-e-varieties of the e-variety $\hbox{\sp O}$ of all orthodox semigroups, were independently introduced in [4] under the name of bivarieties. Moreover, in [4], the notions of bifree objects, biidentities and biinvariant congruences were introduced within $\hbox{\sp O}$ in such a way that a theory properly generalizing the theory of varieties of inverse semigroups arose. Existence varieties of orthodox semigroups include all varieties of orthodox completely regular semigroups, and hence, in particular, all varieties of idempotent semigroups, that is, all subvarieties of the variety $\hbox{\sp B}$ of all bands. These varieties form a countable lattice whose full description is well known.

As the first step of research on functional equations for multiple zeta-functions, we present a candidate of the functional equation for a class of two variable double zeta-functions of the Hurwitz–Lerch type, which includes the classical Euler sum as a special case.

We extend a result of M. Heins by showing that for any sequence of points $(z_n)$ in the unit disk ${\Bbb D}$ tending to the boundary, there is a Blaschke product $B$ which is universal for noneuclidian translates in the sense that the set $\{B((z\,{+}\,z_n)/(1\,{+}\,\overline{z}_nz))\,{:} n\,{\in}\,{\Bbb N}\}$ is locally uniformly dense in the set of all holomorphic functions bounded by one on ${\Bbb D}$. From this, we conclude that for every countable set ${\sp L}$ of hyperbolic/parabolic automorphisms of the unit disk there exists a Blaschke product which is a common cyclic vector in $H^2$ for the composition operators associated with the elements in ${\sp L}$. These results are obtained by transferring the associated approximation problems to interpolation problems on the corona of $H^\infty$.

Using the Fiedler–Polyak–Viro Gauß diagram formulae we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of a given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots. In particular, we prove that there are no achiral almost positive knots and classify all almost positive diagrams of the unknot. We give an application to contact geometry (Legendrian knots) and property P.