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     

<Emphasis Type="Italic">p</Emphasis>-adic properties of division polynomials and elliptic divisibility sequences

For a fixed rational point P E (K) on an elliptic curve, we consider the sequence of values (F_n (P))_n1 of the division polynomials of E at P. For a finite field we prove that the sequence is periodic. For a local field we prove (under certain hypotheses) that there is a power q=p^e so that for all m1, the limit of exists in K and is algebraic over We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.Mathematics Subject Classification (1991): 11G07, 11D61, 14G20, 14H52The authors research supported by NSA grant H98230-04-1-0064.     

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     

Any nilpotence degree occurs in mod-<Emphasis Type="Italic">p</Emphasis> cohomology rings of <Emphasis Type="Italic">p</Emphasis>-groups

Let p be an odd prime number and let n be an arbitrary positive integer. We prove that there exists a p-group whose mod-p cohomology ring has a nilpotent element H²() satisfying ⁿ0,^n+p–1=0. As a corollary, we exhibit a p-group whose mod-p cohomology ring contains an element of nilpotency degree n+1.Mathematical Subject Classification (2000): 20J06, 20D15, 55R40To Phuong and Nin     

Modified Ariki-Koike algebras and cyclotomic <Emphasis Type="Italic">q</Emphasis>-Schur algebras

The modified Ariki-Koike algebra is a variation of the original Ariki-Koike algebra over an integral domain R. When R is a rational function field over the independent parameters, But for general R, is not isomorphic to , and has a simpler structure than . In this paper, we construct a cellular basis of which has a similar property as the cellular basis of introduced by Dipper-James-Mathas. By comparing these two cellular bases, we obtain some estimate on the decomposition numbers of in terms of the decomposition numbers of . We also prove the integral form of the Schur-Weyl reciprocity between a certain quantum algebra U_q and on the tensor space      

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     

On the commutativity of <Emphasis Type="Italic">C</Emphasis>* -algebras

Let be a C* -algebra. Let f be a non-constant complex-valued continuous function defined on a closed interval I. We shall show that f densely spans As an application, is commutative if f(x)f(y)=f(y)f(x) for all self-adjoint elements x and y in with spectrums contained in I.Mathematics Subject Classification (1991):Primary 46L05     

Surfaces of general type with <Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">g</Emphasis></Subscript>=0, <Emphasis Type="Italic">K</Emphasis><Superscript>2</Superscript>=6 and non birational bicanonical map

Let S be a minimal surface of general type with p_g=0 and K²=6, such that its bicanonical map is not birational. The map is a morphism of degree 4 onto a surface. The case of deg = 4 is completely classified in [Topology, 40 (5) (2001), 977–991] and the present paper completes the characterization of these surfaces. It is proven that the degree of cannot be equal to 3, and the geometry of surfaces with deg = 2 is analysed in detail. The last section contains three examples of such surfaces, two of which appear to be new.Mathematics Subject Classification (2000): 14J29     

Hardy algebras, <Emphasis Type="Italic">W</Emphasis>*-correspondences and interpolation theory

Given a von Neumann algebra M and a W*-correspondence E over M, we construct an algebra H(E) that we call the Hardy algebra of E. When M= =E, H(E) is the classical Hardy space H of bounded analytic functions on the unit disc. When M= and E= H(E) is the free semigroup algebra studied by Popescu, Davidson and Pitts and many others. We show that given any faithful normal representation of M on a Hilbert space H there is a natural correspondence E over the commutant (M), called the -dual of E, and that H(E) can be realized in terms of (B(H)-valued) functions on the open unit ball ((E)*) in the space of adjoints of elements in E. We prove analogues of the Nevanlinna-Pick theorem in this setting and discover other aspects of the value distribution theory for elements in H(E). We also analyze the boundary behavior of elements in H(E) and obtain generalizations of the Sz.-Nagy–Foia functional calculus and the functional calculus of Popescu for c.n.c. row contractions. The correspondence E has a dual that is naturally isomorphic to E and the commutants of certain, so-called induced representations of H(E) can be viewed as induced representations of H(E). For these induced representations a double commutant theorem is proved.Supported in part by grants from the National Science Foundation and from the U.S.-Israel Binational Science Foundation.Supported in part by the U.S.-Israel Binational Science Foundation and by the Fund for the Promotion of Research at the Technion.Revised version: 11 March 2004     

On Resolvable Multipartite <Emphasis Type="Italic">G</Emphasis>-Designs

A necessary and sufficient condition for the existence of a _km–factorization of the complete symmetric k–partite multi-digraph K*(n₁,n₂,...,n_k) is obtained for odd k. As a consequence, a resolvable (k,n,km,) multipartite _km–design exists for odd k if and only if m|n. This deduces a result of Ushio when m=1 and k=3. Further, a necessary and sufficient condition for the existence of a _km–factorization of is established for even k, where denotes the wreath product of graphs. Finally, a simple and short proof for the non-existence of a _k–factorization of is obtained for odd k.Acknowledgments.The author thanks Dr. P. Paulraja for his useful ideas in writing this paper and the Department of Science and Technology, New Delhi, for its support (Project Grant No. DST/MS/103/99).Final version received: November 17, 2003     

The classification of <Emphasis Type="Italic">p</Emphasis>-local finite groups over the extraspecial group of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript> and exponent <Emphasis Type="Italic">p</Emphasis>

In this paper we classify the p-local finite groups over p¹⁺²₊, the extraspecial group of order p³ and exponent p for odd p. This study reduces to the classification of the saturated fusion systems over p¹⁺²₊, which will be characterized by the outer automorphism group, the number of -radical subgroups and the automorphism group of each nontrivial -radical subgroup. As part of this classification, we obtain three new exotic 7-local finite groups.Partially supported by MCYT grant BFM2001-2035.Partially supported by MCYT grant BFM2001-1825.Both authors have been supported by the EU grant nr HPRN-CT-1999-00119.in final form: 1 October 2003     

Ordinary elliptic curves of high rank over with constant <Emphasis Type="Italic">j</Emphasis>-invariant

We show that under the assumption of Artins Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over with arbitrarily high rank and constant j-invariant. For odd primes p, this result follows from a theorem we prove which states that whenever p is a generator of (/)*/–1 ( an odd prime) there exists a hyperelliptic curve over whose Jacobian is isogenous to a power of one ordinary elliptic curve.Mathematics Subject Classification (2000): Primary: 11G05; Secondary: 11G20, 14H40, 14H52Revised version: 3 February 2004Acknowledgements. The research of Jasper Scholten is funded by the AREHCC project of the European Commission (Fifth Framework Program IST - 2001).     

Positive Projective Space of a <Emphasis Type="Italic">C</Emphasis>*-Algebra with a Tracial Conditional Expectation

Let be a C*-algebra, a subalgebra of its center and Φ: → a tracial faithful conditional expectation. We define the positive projective space as the quotient where G⁺ is the space of positive invertible elements of , and if there exists g invertible in such that a′ = |g|²a. When is abelian, this space is a set of representatives for probability densities equivalent to a given one. The aim of this paper is to endow ℙ⁺ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements in ℙ⁺, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space ℙ⁺ with the given geodesic distance is non positively curved.     

Commutativity of Almost <Emphasis Type="Italic">F</Emphasis>-algebras

Let A be an Archimedean vector lattice, let be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ ₀:A × A→ B is a positive orthosymmetric bimorphism, then there exists a positive bimorphism extension Ψ of Ψ ₀ to × in B which is orthosymmetric. This leads to a new and short proof of the commutativity of the almost f-algebras multiplications.     

Bounded trace <Emphasis Type="Italic">C</Emphasis>*-algebras and integrable actions

Let G be a second countable group, A be a separable C*-algebra with bounded trace and a strongly continuous action of G on A. Suppose that the action of G on induced by is free and the G-orbits are locally closed. We show that the crossed product A×G has bounded trace if and only if G acts integrably (in the sense of Rieffel and an Huef) on . In the course of this, we show that the extent of non-properness of an integrable action gives rise to a lower bound for the size of the (finite) upper multiplicities of the irreducible representations of the crossed product.Mathemactics Subject Classification (1991): 46L55     

Path regularity for Feller semigroups via Gaussian kernel estimates and generalizations to arbitrary semigroups on <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>

Given γ ∈ (−1,1), we present a dyadic growth condition on the finite dimensional distributions of operator semigroups on C₀(E which - for γ>0 and Feller semigroups - assures that the corresponding Feller process has paths in local Hölder spaces and in weighted Besov spaces of order γ. We show that, for operator semigroups satisfying Gaussian kernel estimates of order m>1, condition holds for all and even for all in the case of Feller semigroups. Such Gaussian kernel estimates are typical for Feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on ℝ^D of order m≥D.     

Some remarks on the Hardy-Littlewood maximal function on variable <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class (ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝⁿ).     

On the Achromatic Index of for a Prime <Emphasis Type="Italic">q</Emphasis>

The achromatic index of a graph G is the maximum number of colours that can be assigned to edges of G in such a way that the colouring is proper and any two colours meet on two adjacent edges. To compute the achromatic index of complete graphs seems to be a very difficult task. Indeed, that knowledge would yield all odd projective plane orders. The aim of the paper is to establish a nontrivial lower bound of the achromatic index of for a prime q7.Acknowledgments The first author gratefully acknowledges a support of the Slovak grant VEGA 1/7467/20. The authors wish to thank to an anonymous referee for an easy proof of Lemma 8.Final version received: November 10, 2003     

On the logarithmic plurigenera of complements of plane curves

Let B be a (not necessarily irreducible) plane curve in ². In the present article, we prove that if and only if Moreover, we determine the curve B when and Mathematics Subject Classification (2000): 14R05, 14H50, 14J26     

Partition of <Emphasis Type="Italic">C</Emphasis><Subscript>4</Subscript>-Designs into Minimum and Maximum Number of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-Designs

Let be a C₄-design of order n and index , on the vertex set V, |V|=n. If V₁V_m=V is a partition of the vertex set, such that the intersections of the with V_i form a P₃-design of order |V_i| and the same index , for each 1im, then 2m log₃(2n+1). The minimum bound is best possible for every . The maximum bound is best possible for =2, and hence also for every even .Supported by MIUR, Italy and CNR-GNSAGAAlso affiliated with the Department of Computer Science, University of Veszprém, Hungary; supported in part by the Hungarian Scientific Research Fund, grant OTKA T-32969AMS classification: 05B05