Biorthogonal systems and quotient spaces via baire category methods

We show that every Banach space X of density smaller that the Baire category number admits a quotient with a long Schauder basis that can be taken of length ω₁ if X is not separable. So, assuming that the Baire category number does not take its minimal possible value, a Banach space X is separable if and only if all biorthogonal systems of X are countable.     

Poisson transform for the Heisenberg group and eigenfunctions of the sublaplacian

We define an analogue of Poisson transform on the Heisenberg group and use it to characterise joint eigenfunctions of the sublaplacian and T=i∂_t in terms of certain analytic functionals.     

Global Solutions to the Two Dimensional Quasi-Geostrophic Equation with Critical or Super-Critical Dissipation

The two dimensional quasi-geostrophic (2D QG) equation with critical and super-critical dissipation is studied in Sobolev space H^s(ℝ²). For critical case (α=), existence of global (large) solutions in H^s is proved for s≥ when is small. This generalizes and improves the results of Constantin, D. Cordoba and Wu [4] for s = 1, 2 and the result of A. Cordoba and D. Cordoba [8] for s=. For s≥1, these solutions are also unique. The improvement for pushing s down from 1 to is somewhat surprising and unexpected. For super-critical case (α ∈ (0,)), existence and uniqueness of global (large) solution in H^s is proved when the product is small for suitable s≥2−2α, p ∈ [1,∞] and β ∈ (0,1].     

On Resolvable Multipartite G-Designs II

In this paper, it has been proved that and are factorable, where ⊗ denotes wreath product of graphs. As a consequence, a resolvable (k,n,k,2λ) multipartite –design exists for even k. These results generalize the results of Ushio on –factorizations of complete tripartite graphs.     

Existence of singular solutions of a degenerate equation in R 2

Let a₁,a₂, . . . ,a_m ∈ ℝ², 2≤f ∈ C([0,∞)), g_i ∈ C([0,∞)) be such that 0≤g_i(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation u_t=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−a_i|→−g_i(t) as |x−a_i|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u₀ where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ² with prescribed singularities at a finite number of points in the domain.     

Quelques résultats sur les déformations équivariantes des courbes stables

Let G be a finite group, let g≥2 and g ′ ≥ 0 be integers. We introduce the algebraic stack classifying the stable curves of genus g endowed with an action of G faithful in each geometric fiber and such that the quotient of each fiber is a semi-stable curve of genus g′. We study the completion of the local rings of this algebraic stack. They are closely related to universal equivariant deformation rings R_C,G of stable curves endowed with a faithful action of G. A useful tool for this purpose is a local-global principle generalizing the one obtained by Bertin and Mézard in [BM00]. We then use the results we already proved in [Mau03b] and [Mau03a] to describe some properties of the space (purity, dimension).     

Integral structures in automorphic line bundles on the p-adic upper half plane

Given an automorphic line bundle of weight k on the Drinfeld upper half plane X over a local field K, we construct a GL₂(K)-equivariant integral lattice in as a coherent sheaf on the formal model underlying Here is ramified of degree 2. This generalizes a construction of Teitelbaum from the case of even weight k to arbitrary integer weight k. We compute and obtain applications to the de Rham cohomology H_dR¹( X, Sym_K^k(St)) with coefficients in the k-th symmetric power of the standard representation of SL₂(K) (where k0) of projective curves X uniformized by X: namely, we prove the degeneration of a certain reduced Hodge spectral sequence computing H_dR¹( X, Sym_K^k(St)), we re-prove the Hodge decomposition of H_dR¹( X, Sym_K^k(St)) and show that the monodromy operator on H_dR¹( X, Sym_K^k(St)) respects integral de Rham structures and is induced by a universal monodromy operator defined on , i.e. before passing to the -quotient.Mathematics Subject Classification (2000): 11F33, 11F12, 11G09, 11G18I wish to thank Peter Schneider and Jeremy Teitelbaum for generously providing me with some helpful private notes on their own work, and for their interest. I am also grateful to Matthias Strauch for useful discussions on odd weight modular forms. I thank Christophe Breuil for his interest and his insisting on lattices for the entire G-action. Finally I thank the referee for his suggestions concerning the presentation of several technical constructions.     

Canonical height functions for affine plane automorphisms

Let be a polynomial automorphism of dynamical degree δ≥2 over a number field K. We construct height functions defined on that transform well relative to f, which we call canonical height functions for f. These functions satisfy the Northcott finiteness property, and a -valued point on is f-periodic if and only if its height is zero. As an application, we give an estimate on the number of points with bounded height in an infinite f-orbit.     

Pro-<Emphasis Type="Italic">p</Emphasis>-Iwahori Hecke ring and supersingular -representations

The motivation of this paper is the search for a Langlands correspondence modulo p. We show that the pro-p-Iwahori Hecke ring of a split reductive p-adic group G over a local field F of finite residue field F _q with q elements, admits an Iwahori-Matsumoto presentation and a Bernstein Z-basis, and we determine its centre. We prove that the ring is finitely generated as a module over its centre. These results are proved in [11] only for the Iwahori Hecke ring. Let p be the prime number dividing q and let k be an algebraically closed field of characteristic p. A character from the centre of to k which is “as null as possible” will be called null. The simple -modules with a null central character are called supersingular. When G=GL(n), we show that each simple -module of dimension n containing a character of the affine subring is supersingular, using the minimal expressions of Haines generalized to , and that the number of such modules is equal to the number of irreducible k-representations of the Weil group W _F of dimension n (when the action of an uniformizer p _F in the Hecke algebra side and of the determinant of a Frobenius Fr _F in the Galois side are fixed), i.e. the number N _n (q) of unitary irreducible polynomials in F _q [X] of degree n. One knows that the converse is true by explicit computations when n=2 [10], and when n=3 (Rachel Ollivier). An erratum to this article can be found at     

Topological classification of holomorphic, semi-hyperbolic germs, in ``Quasi-Absence' of resonances

We give the classification, under topological conjugacy, of invertible holomorphic germs f:, with λ₁, . . . ,λ_n eigenvalues of d f₀, and |λ_i|≠1 for i=2, . . . ,n while λ₁ is a root of the unity, in the suitable hypothesis of ``quasi-absence' of resonances (i.e., assuming that for r_i≥0 and i=2, . . . ,n, with ).     

A Remark on the Linearized Stability of Positive Solutions for Systems Involving the p-Laplacian

The aim of this paper is to prove some stability result for nonlinear elliptic systems of the form where Δ_p denotes the p-Laplacian operator defined by Δ_pz = div(|∇ z|^p-2∇ z); p > 2, Ω is a bounded domain in R^N (N > 1) with smooth boundary where with h = 1 when α = 1, λ is a positive parameter and f,g are C² functin on [0,∞) × [0,∞). We prove stability and instability results of positive stationary solutions under various choices of f and g.     

Hyponormality of Toeplitz operators with rational symbols

In this article we introduce a notion of `division' for rational functions and then give a criterion for hyponormality of (f, g are rational functions) in the cases where g divides f. Furthermore we show that we may assume, without loss of generality, that g divides f when we consider the hyponormality of . Supported in part by a grant from Faculty Research Fund, Sungkyunkwan University, 2004. Supported in part by a grant (R14-2003-006-01000-0) from the Korea Research Foundation.     

A class of <Emphasis Type="Italic">p</Emphasis>-adic Galois representations arising from abelian varieties over

Let V be a p-adic representation of the absolute Galois group G of that becomes crystalline over a finite tame extension, and assume p2. We provide necessary and sufficient conditions for V to be isomorphic to the p-adic Tate module V_p() of an abelian variety defined over . These conditions are stated on the filtered (,G)-module attached to V.Mathematics Subject Classification (2000): 14F30, 11G10, 11F80, 14G20, 14F20     

Minimal Connected τ-Critical Hypergraphs

A hypergraph is τ-critical if τ(−{E})<τ() for every edge E ∈ , where τ() denotes the transversal number of . We show that if is a connected τ-critical hypergraph, then −{E} can be partitioned into τ()−1 stars of size at least two, for every edge E ∈ . An immediate corollary is that a connected τ-critical hypergraph has at least 2τ()−1 edges. This extends, in a very natural way, a classical theorem of Gallai on colour-critical graphs, and is equivalent to a theorem of Füredi on t-stable hypergraphs. We deduce a lower bound on the size of τ-critical hypergraphs of minimum degree at least two.     

Iwasawa invariants of galois deformations

Fix a residual ordinary representation :G_F→GL_n(k) of the absolute Galois group of a number field F. Generalizing work of Greenberg–Vatsal and Emerton–Pollack–Weston, we show that the Iwasawa invariants of Selmer groups of deformations of depends only on and the ramification of the deformation.     

Some q-convexity properties of coverings of complex manifolds

Let C be a nowhere dense compact analytic subset of a 2-dimensional complex manifold X. A result of Napier is the construction of a neighborhood V of C such that for any covering space in which is holomorphically convex, there is a C ^∞ exhaustion function φ on which is strictly plurisubharmonic on π^-1(V) away from the compact irreducible components of . Colţoiu and Vajaitu obtained a q-convex version of this result for C a q-dimensional fiber of a proper surjective submersion X→Y. The goal of this paper is a similar version for Cprojective.     

The Maximum Size of 3-Wise Intersecting and 3-Wise Union Families

Let be an n-uniform hypergraph on 2n vertices. Suppose that and holds for all F₁,F₂,F₃ ∈ . We prove that the size of is at most . The second author was supported by MEXT Grant-in-Aid for Scientific Research (B) 16340027     

On topological Kadec norms

We present a topological analogue of the classic Kadec Renorming Theorem, as follows. Let be two separable metric topologies on the same set X. We prove that every point in X has an -neighbourhood basis consisting of sets that are -closed if and only if there exists a function φ: X→ℝ that is -lower semi-continuous and such that is the weakest topology on X that contains and that makes φ continuous. An immediate corollary is that the class of almost n-dimensional spaces consists precisely of the graphs of lower semi-continuous functions with at most n-dimensional domains.     

Rational curves of minimal degree and characterizations of projective spaces

In this paper we investigate complex uniruled varieties X whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point x ∈ X form a linear subspace of T_xX. As a first application of our main result, we give a unified geometric proof of Mori's, Wahl's, Campana-Peternell's and Andreatta-Wiśniewski's characterizations of . We also give a characterization of products of projective spaces in terms of the geometry of their families of rational curves of minimal degree.     

Simplicial complexes lying equivariantly over the affine building of GL(<Emphasis Type="Italic">N</Emphasis>)

Let G=GL(N,K), K a non-archimedean local field and X be the semisimple affine building of G over K. We construct a projective tower of G-sets with X(0)=X. They are obtained by using a minor modification in Bruhat and Tits original construction (an idea due to P. Schneider). Using the structure of X as an abstract building, we construct a projective tower of simplicial G-complexes such that, for each r, X_(r) is canonically a geometrical realization of X_r. In the case N=2, X_r has a natural two-sheeted covering _r and we show that the supercuspidal part of the cohomology space is characterized by a nice property.Mathematics Subject Classification (2000): 14R25, 20E42, 20G25, 55U10, 57S25