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.     

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     

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.     

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     

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.     

Space-time decay of Navier–Stokes flows invariant under rotations

We show that the solutions to the non-stationary Navier–Stokes equations in (d=2,3) which are left invariant under the action of discrete subgroups of the orthogonal group O(d) decay much faster as than in generic case and we compute, for each subgroup, the precise decay rates in space-time of the velocity field.     

Order and algebra isomorphisms of spaces of regular operators

If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:(E) →(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU⁻¹ for all T ∈ (E).     

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     

Sectional curvatures of Kähler moduli

If X is a compact Kähler manifold of dimension n, we let denote the cone of Kähler classes, and the level set given by classes D with Dⁿ=1. This space is naturally a Riemannian manifold and is isometric to the manifold of Kähler forms with ⁿ some fixed volume form, equipped with the Hodge metric, as studied previously by Huybrechts. We study these spaces further, in particular their geodesics and sectional curvatures. Conjecturally, at least for Calabi–Yau manifolds and probably rather more generally, these sectional curvatures should be bounded between and zero. We find simple formulae for the sectional curvatures, and prove both the bounds hold for various classes of varieties, developing along the way a mirror to the Weil–Petersson theory of complex moduli. In the case of threefolds with h^1,1=3, we produce an explicit formula for this curvature in terms of the invariants of the cubic form. This enables us to check the bounds by computer for a wide range of examples. Finally, we explore the implications of the non-positivity of these curvatures.     

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      

On the elliptic genus of generalised Kummer varieties

Borisov and Libgober ([2]) recently proved a conjecture of Dijkgraaf, Moore, Verlinde, and Verlinde (see [6]) on the elliptic genus of a Hilbert scheme of points on a surface. We show how their result can be used together with our work on complex genera of generalised Kummer varieties [17] to deduce the following formula, conjectured by Kawai and Yoshioka ([15]), on the elliptic genus of a generalised Kummer variety A^[[n>^]] of dimension 2(n–1): Here is the weak Jacobi form of weight –1 and index and V(n) is the Hecke operator sending Jacobi forms of index r to Jacobi forms of index nr (see [7]).The author was supported by the Deutsche Forschungsgemeinschaft.Revised version: 14 November 2003     

An immersion theorem for Vaisman manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering , with monodromy acting on by Kähler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on . We prove that any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold (ⁿ0). As an application, we obtain that any Sasakian manifold has a contact immersion to an odd-dimensional sphere.Mathematics Subject Classification (2000): 53C55, 14E25, 53C25Liviu Ornea is member of EDGE, Research Training Network HRPN-CT-2000-00101, supported by the European Human Potential Programme.Misha Verbitsky is an EPSRC advanced fellow supported by CRDF grant RM1-2354-MO02 and EPSRC grant GR/R77773/01.Both authors acknowledge financial support from Ecole Polytechnique (Palaiseau).     

On the zero-distribution of Epstein zeta-functions

We prove that the mean value of the real parts of the nontrivial zeros of the Epstein zeta-function associated with a positive definite quadratic form in n variables is equal to . Furthermore, we show that Epstein zeta-functions in general have an asymmetric zero-distribution with respect to the critical line Re .     

On the dynamics near infinity of some polynomial mappings in

We construct the Green current for a random iteration of horizontal-like mappings in . This is applied to the study of a polynomial map with the following properties: i. infinity is f-attracting; ii. f contracts the line at infinity to a point not in the indeterminacy set. We study for such mappings the escape rates near infinity, i.e. the set of possible values of the function We show in particular that the set of possible values can contain an interval. On the other hand the Green current T of f can be decomposed into pieces associated to an itinerary defined by the indeterminacy points. This allows us to prove that exists ||T||-a.e. and we give its value in terms of explicit quantities depending on f.     

Integral varieties of the canonical cone structure on <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">P</Emphasis>

The canonical cone structure on a compact Hermitian symmetric space G/P is the fiber bundle where is the cone of the highest weight vectors under the action of the reductive part of P. It is known that the cone coincides with the cone of the vectors tangent to the lines in G/P passing through x, when we consider G/P as a projective variety under its homogeneous embedding into the projective space of the irreducible representation space V of G with highest weight associated to P. A subvariety X of G/P is said to be an integral variety of at all smooth points xG/P. Equivalently, an integral variety of is a subvariety of G/P whose embedded projective tangent space at each smooth point is a linear space We prove a kind of rigidity of the integral varieties under some dimension condition. After making a uniform setting to study the problem, we apply the theory of Lie algebra cohomology as a main tool. Finally we show that the dimension condition is necessary by constructing counterexamples.     

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].     

Birational geometry of quartic 3-folds II: The importance of being -factorial

The paper explores the birational geometry of terminal quartic 3-folds. In doing this I develop a new approach to study maximal singularities with positive dimensional centers. This allows to determine the pliability of a -factorial quartic with ordinary double points, and it shows the importance of -factoriality in the context of birational geometry of uniruled 3-folds.Mathematics Subject Classification (1991): 14E07, 14J30, 14E30Partially supported by EAGER and Geometria sulle Varietà Algebriche (MIUR).Revised version: 2 January 2004     

On Cross-intersecting Families of Sets

A family of -element subsets and a family of k-element subsets of an n-element set are cross-intersecting if every set from has a nonempty intersection with every set from . We compare two previously established inequalities each related to the maximization of the product , and give a new and short proof for one of them. We also determine the maximum of for arbitrary positive weights ,_k.     

Isomorphisms of algebras associated with directed graphs

Given countable directed graphs G and G, we show that the associated tensor algebras (G) and (G) are isomorphic as Banach algebras if and only if the graphs G are G are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for algebraic isomorphisms. We also show that given countable directed graphs G, G, the free semigroupoid algebras and are isomorphic as dual algebras if and only if the graphs G are G are isomorphic. In particular, spatially isomorphic free semigroupoid algebras are unitarily isomorphic. For free semigroupoid algebras associated with locally finite directed graphs with no sinks, the graph forms an invariant for algebraic isomorphisms as well.Mathematics Subject Classification (2000): 47L80, 47L55, 47L40Acknowledgments. We would like to thank the referee for several constructive suggestions on the initial draft and for bringing to our attention the work in [8,9]. The first author was partially supported by a research grant from ECU and the second author by an NSERC research grant and start up funds from the University of Guelph. We thank David Pitts for enlightening conversations and Alex Kumjian for helpful comments on the literature.