International Mathematics Research Papers

Journal Article Minorations des hauteurs normalisées des sous-variétés des puissances des courbes elliptiques Get access Sinnou David, Sinnou David Institut de Mathématiques de Jussieu UMR 7586 du C. N. R. S. — UFR 921, Théorie des Nombres & Géométrie et Dynamique, 4, Place Jussieu, 75252 Paris cedex 05 Correspondence to be sent to: Sinnou David. e-mail: david@math.jussieu.fr Search for other works by this author on: Oxford Academic Google Scholar Patrice Philippon Patrice Philippon Institut de Mathématiques de Jussieu UMR 7586 du C. N. R. S. — UFR 921, Théorie des Nombres & Géométrie et Dynamique, 4, Place Jussieu, 75252 Paris cedex 05 Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Papers, Volume 2007, 2007, rpm006, https://doi.org/10.1093/imrp/rpm006 Published: 01 January 2007 Article history Received: 15 December 2006 Published: 01 January 2007 Revision received: 07 May 2007 Accepted: 20 May 2007

Let C be the union of two general connected, smooth, nonrational curves X and Y intersecting transversally at a point P. Assume that P is a general point of X or of Y. Our main result describes all limits on C of special Weierstrass points along smooth curves degenerating to C. As an application, we recover in a unified and conceptually simpler way the computations made by Diaz and Cukierman of divisor classes of curves with special Weierstrass points in the moduli space of stable curves. In our approach there are no multiplicity issues, an usual nuisance of the method of test curves.

The author, and independently De Concini, conjectured that the monodromy of the Casimir connection of a simple Lie algebra g is described by the quantum Weyl group operators of the quantum group U_h(g). The aim of this paper, and of its sequel [TL4], is to prove this conjecture. The proof relies upon the use of quasi-Coxeter algebras, which are to generalised braid groups what Drinfeld's quasitriangular quasibialgebras are to the Artin braid groups B_n. Using an appropriate deformation cohomology, we reduce the conjecture to the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra Ug which interpolates between the quasi-Coxeter structure underlying the Casimir connection and the quasitriangular quasibialgebra underlying the KZ equations. The existence of this structure will be proved in [TL4].

We show that, under fairly general conditions, many elements of a p-adic group can be well approximated by a product whose factors have properties that are helpful in performing explicit character computations.

We relate the representations of the rational Cherednik algebras associated with the complex reflection group G(m,1,n) to sheaves on Nakajima quiver varieties associated with extended Dynkin gaphs via a Z-algebra construction. As the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. We interpret the ordering on category O geometrically using this relationship; we also relate the geometry to the a-function for Hecke algebras with unequal parameters.

Let $X$ be a compact metric space and let $\Lambda$ be a $\Z^k$ ($k\ge 1$) action on $X.$ We give a solution to a version of Voiculescu's problem of AF-embedding: The crossed product $C(X)\rtimes_{\Lambda}\Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $\Lambda$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $\Lambda: G\to Aut(C)$ be a monomorphism. We show that $C\rtimes_{\Lambda} G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $\Lambda$-invariant tracial state.

This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by ( the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal representations of finite groups of Lie type are distinguished relative to particular quadratic characters. As an application of the main results, we obtain necessary and sufficient conditions for equivalence of two of Yu's supercuspidal representations associated to distinct G-data.

We consider the energy-critical non-linear focusing Schrödinger equation in dimension N = 3, 4, 5. An explicit stationary solution, W, of this equation is known. In [KeM], the energy E(W) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article, we study the dynamics at the critical level E(u) = E(W) and classify the corresponding solutions. This gives in particular a dynamical characterization of W.

We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.

Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called a multiplicative preprojective algebra. In this paper, we study a moduli space of ( semi) stable representations of such an algebra ( the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there is a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety ( which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is star-shaped, the multiplicative quiver variety parameterizes Simpson's ( poly) stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight, and associated graded local systems around each puncture.

We construct the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). This construction is only valid over Spec C, but a special case is a GIT presentation of the moduli space of stable curves of genus g with n marked points, \bar M_g,n; this is valid over Spec Z. Our method follows that used in the case n=0 by Gieseker to construct \bar M_g, though our proof that the semistable set is nonempty is entirely different.

We study quotients of quasi-affine schemes by unipotent groups over fields of characteristic 0. To do this, we introduce a notion of stability which allows us to characterize exactly when a principal bundle quotient exists and, together with a cohomological vanishing criterion, to characterize whether or not the resulting quasi-affine quotient scheme is affine. We completely analyze the case of G_a-invariant hypersurfaces in a linear G_a-representation W; here the above characterizations admit simple geometric and algebraic interpretations. As an application, we produce arbitrary dimensional families of non-isomorphic smooth quasi-affine but not affine n-dimensional varieties (n \geq 6) that are contractible in the sense of A^1-homotopy theory. Indeed, existence follows without any computation; yet explicit defining equations for the varieties depend only on knowing some linear G_a- and SL_2- invariants, which, for a sufficiently large class, we provide. Similarly, we produce infinitely many non-isomorphic examples in dimensions 4 and 5. Over C, the analytic spaces underlying these varieties are non-isomorphic, non-Stein, topologically contractible and often diffeomorphic to C^n.

The purpose of this article is to present a practical approach to computing zeta functions of smooth projective hypersurfaces over finite fields which is based on relative rigid cohomology. The method presented is a reformulation of a computational idea due to Lauder, which in turn was inspired by a beautiful paper by Dwork. The algorithm of Lauder, though a theoretical breakthrough, does not appear to be of immediate practical use; in particular, the precision-loss estimates appear too coarse to be useful. We derive a practical algorithm by recasting the computational idea in the setting of rigid cohomology and undertaking a delicate analysis of the precision loss, which is essential to improve the run-time behaviour. This also clarifies the relation to earlier point counting methods based on Monsky-Washnitzer cohomology, and shows how the methods can be gainfully combined. We have implemented our algorithm in the programing language Magma and present some examples which we have computed.

We consider the totally asymmetric simple exclusion process (TASEP) in discrete time with sequential update. The joint distribution of the positions of selected particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. We focus on periodic initial conditions where particles occupy dZ, d>=2. In the proper large time scaling limit, the fluctuations of particle positions are described by the Airy_1 process. Interpreted as a growth model, this confirms universality of fluctuations with flat initial conditions for a discrete set of slopes.

The resonance relations are identities between coordinates of functions with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Baker-Akhiezer type, satisfying the resonance relations. We give an alternative representation-theoretic construction of this space, using the traces of regularized intertwining operators for the quantum group, and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equations. We define the quantum conformal blocks as distinguished Weyl anti-invariant hypergeometric qKZB solutions with values in a tensor product of finite-dimensional modules. We prove that for generic q the dimension of the space of quantum conformal blocks equals the dimension of the quantum group invariants, and is computed by the Verlinde algebra when q is a root of unity.

We generalize the motivic incarnation morphism from the theory of arithmetic integration to the relative case, where we work over a base variety S over a field k of characteristic zero. We develop a theory of constructible effective Chow motives over S, and we show how to associate a motive to any S-variety. We give a geometric proof of relative quantifier elimination for pseudo-finite fields, and we construct a morphism from the Grothendieck ring of the theory of pseudo-finite fields over S, to the tensor product of Q with the Grothendieck ring of constructible effective Chow motives. This morphism yields a motivic realization of parameterized arithmetic integrals. Finally, we define relative arc and jet spaces, and the three relative motivic Poincare series.

We give a streamlined proof of a quantitative version of a result from P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) which is crucial for the proof of universality in the bulk P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) and also at the edge P. Deift and D. Gioev, {Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices. Comm. Pure Appl. Math. (in press) for orthogonal and symplectic ensembles of random matrices. As a byproduct, this result gives asymptotic information on a certain ratio of the β=1,2,4 partition functions for log gases.

We present a conjecture on the lower bound of essential minima of projective subvarieties of polarized abelian varieties and we give a proof of this conjecture when the abelian variety is isogeneous to a power of an elliptic curve endowed with its canonical product polarisation. We deduce estimates for the number of points of small height on subvarieties of such powers of elliptic curves, depending only on the geometric parameters involved. Combining with a theorem of G. Remond, we deduce a new uniform version of the Mordell-Lang statement in this case. All the results are entirely explicit, they enable us to produce in a separate text, written in collaboration with M. Nakamaye, examples of families of curves satisfying a strong uniform bound on the number of their rational points, as proposed by B. Mazur.

We show that the operator Hilbert space OH introduced by Pisier embeds into the predual of the hyerfinite III1 factor. The main new tool is a Khintchine type inequality for the generators of the CAR algebra with respect to a quasi-free state. Our approach yields a Khintchine type inequality for the q-gaussian variables for all values q between -1 and 1. These results are closely related to recent results of Pisier and Shlyakhtenko in the free case.

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since in practice all known cases, e.g. hyperelliptic, superelliptic and Cab curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over Fpn , the expected running time is e O(ng+ng), whereas the space complexity amounts to e O(ng), assuming p is fixed.