International Mathematics Research Notices - Advance Access

Rational Curves on Smooth Cubic Hypersurfaces

Coskun, I., Starr, J. — 2009-07-02

We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in of degree 2d ≤ min (n + 4, 2n – 2) and dimension at least three is irreducible and of the expected dimension.

Symmetric Waves Are Traveling Waves

Ehrnstrom, M., Holden, H., Raynaud, X. — 2009-07-02

We show that horizontally symmetric water waves are traveling waves. The result is valid for the Euler equations, and is based on a general principle that applies to a large class of nonlinear partial differential equations, including some of the most famous model equations for water waves. A detailed analysis is given for weak solutions of the Camassa–Holm equation. In addition, we establish the existence of nonsymmetric linear rotational waves for the Euler equations.

Foliations on Hypersurfaces in Holomorphic Symplectic Manifolds

Sawon, J. — 2009-07-02

Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction |_Y of the holomorphic symplectic form induces a rank one foliation on Y. We investigate situations where this foliation has compact leaves; in such cases, we obtain a space of leaves Y/ F which has dimension 2n – 2 and admits a holomorphic symplectic form.

Moduli Spaces of Semistable Sheaves on Singular Genus 1 Curves

Ruiperez, D. H., Martin, A. C. L., Gomez, D. S., Prieto, C. T. — 2009-07-02

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure-dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of nonisomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle E_N of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank r is isomorphic to the rth symmetric product of the rational curve with one node.

An A{infty}-Structure for Lines in a Plane

Kajiura, H. — 2009-07-01

As an explicit example of an A-structure associated with geometry, we construct explicitly an A-structure for a Fukaya category of finitely many lines (Lagrangians) in , i.e. we also define nontransversal A-products. The Acategory is constructed so that it is A-homotopy equivalent to a differential graded category of DeRham type. This construction is motivated by homological mirror symmetry of (two-)tori, where is the covering space of a two-torus. The strategy is based on an algebraic reformulation of Morse homotopy theory through homological perturbation theory as discussed by Kontsevich and Soibelman [23].

Desingularization of Orbifolds Obtained from Symplectic Reduction at Generic Coadjoint Orbits

Niederkruger, K., Pasquotto, F. — 2009-06-30

Symplectic reduction is a technique that can be used to decrease the dimension of Hamiltonian manifolds. Unfortunately, this only works under strong assumptions on the group action, and in general, even for a compact Lie group, the reduction at a coadjoint orbit that is transverse to the moment map will only yield a symplectic orbifold.

In this article, we show how to construct resolutions of symplectic orbifolds obtained as quotients of presymplectic manifolds with a torus action. As a corollary, this allows us to desingularize generic symplectic quotients for compact Lie group actions. More precisely, if a point in the Lie coalgebra is regular, that is, its stabilizer is a maximal torus, then we may apply our desingularization result. Regular elements of the Lie coalgebra are generic in the sense that the singular strata have codimension at least three.

Additionally, we show that even though the result of a symplectic cut is an orbifold, it can be modified in an arbitrarily small neighborhood of the cut hypersurface to obtain a smooth symplectic manifold.

Moduli of Bundles over Rational Surfaces and Elliptic Curves II: Nonsimply Laced Cases

Leung, N. C., Zhang, J. — 2009-06-27

For any nonsimply laced Lie group G and elliptic curve , we show that the moduli space of flat G bundles over can be identified with the moduli space of rational surfaces with G-configurations which contain as an anticanonical curve. We also construct Lie(G)-bundles over these surfaces. The corresponding results for simply laced groups were obtained by the authors in another paper. Thus, we have established a natural identification for these two kinds of moduli spaces for any Lie group G.

Functoriality for the Classical Groups over Function Fields

Lomeli, L. A. — 2009-06-25

Langlands' functoriality for generic representations from the split classical groups to an appropriate GL_N is established. The functorial lift or transfer to GL_N is obtained with the help of a converse theorem once the analytic properties of L-functions are studied using the Langlands–Shahidi approach. This paper is mostly devoted to understanding L-functions for the classical groups over a global function field, since the Langlands–Shahidi method has only been developed over number fields. To overcome many difficulties, stability of -factors under twists by highly ramified characters is used together with multiplicativity. Finally, by analyzing the image of functoriality, a proof of the Ramanujan conjecture for generic representations is obtained.

Local Structure of the Moduli Space of K3 Surfaces in Positive Characteristic

Yu, J.-D. — 2009-06-23

Let k be a perfect field of characteristic p ≥ 5. Let X be a nonsupersingular K3 surface over k and the enlarged formal Brauer group associated to X. In this paper, we show that the local moduli space of X, with trivial associated deformation of , admits a natural p-divisible formal group structure.

Action Selectors and Maslov Class Rigidity

Kerman, E. — 2009-06-23

In this paper, we detect new restrictions on the Maslov class of displaceable Lagrangian submanifolds of symplectic manifolds which are symplectically aspherical. These restrictions are established using action selectors for Hamiltonian flows. In particular, we construct and utilize a new action selector for the flows of a special class of Hamiltonian functions which arises naturally in the study of Hamiltonian paths which minimize the Hofer length functional.

Nonunitarizable Representations and Random Forests

Epstein, I., Monod, N. — 2009-06-23

We establish a connection between Dixmier's unitarizability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is nonunitarizable if its first L²-Betti number is nonzero or if it is finitely generated with nontrivial cost. Our criterion also applies to torsion groups constructed by Osin, thus providing the first examples of nonunitarizable groups without free subgroups.

Geometric PDEs in the Grushin Plane: Weighted Inequalities and Flatness of Level Sets

Ferrari, F., Valdinoci, E. — 2009-06-23

A geometric Sobolev–Poincaré inequality for stable solutions of semilinear partial differential equations (PDEs) in the Grushin plane will be obtained. Such inequality will bound the weighted L²-norm of a test function by a weighted L²-norm of its gradient, and the weights will be interesting geometric quantities related to the level sets of the solution. From this, we shall see that a geometric PDE holds on the level sets of stable solutions. We shall study in detail the particular case of local minimizers of a Ginzburg–Landau–Allen–Cahn-type phase transition model and provide for them some one-dimensional symmetry results.

L2-Betti Numbers and Non-Unitarizable Groups without Free Subgroups

Osin, D. V. — 2009-06-19

We show that there exist non-unitarizable groups without nonabelian free subgroups. Both torsion and torsion free examples are constructed. As a by-product, we show that there exist finitely generated torsion groups with nonvanishing first L²-Betti numbers. We also relate the well-known problem of whether every hyperbolic group is residually finite to an open question about approximation of L²-Betti numbers.

n-Groupoids and Stacky Groupoids

Zhu, C. — 2009-06-19

We discuss two generalizations of Lie groupoids. One consists of Lie n-groupoids defined as simplicial manifolds with trivial _{k≥ n+1}. The other consists of stacky Lie groupoids with a differentiable stack. We build a 1–1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain Morita equivalence. We prove this in a general setup so that the statement is valid in both differential and topological categories. Hypercovers of higher groupoids in various categories are also described.

Moduli of Crude Limit Linear Series

Liu, F. — 2009-06-19

Eisenbud and Harris introduced the theory of limit linear series and constructed a space parameterizing their limit linear series. Recently, Osserman introduced a new space which compactifies the Eisenbud–Harris construction. In the Eisenbud–Harris space, the set of refined limit linear series is always dense on a general reducible curve. Osserman asks when the same is true for his space. In this paper, we answer his question by characterizing the situations when the crude limit linear series contain a nonempty open subset of his space. We also show that the exact points are always dense.

Depth-Zero Representations of Nonlinear Covers of p-Adic Groups

Howard, T. K., Weissman, M. H. — 2009-06-19

We generalize the methods of Moy–Prasad in order to define and study the genuine depth-zero representations of some nonlinear covers of reductive groups over p-adic local fields. In particular, we construct all depth-zero supercuspidal representations of the metaplectic group Mp_2n over a p-adic field of odd residue characteristic.

Staggered t-Structures on Derived Categories of Equivariant Coherent Sheaves

Achar, P. N. — 2009-06-18

Let X be a scheme, and let G be an affine group scheme acting on X. Under reasonable hypotheses on X and G, we construct a t-structure on the derived category of G-equivariant coherent sheaves that in many ways resembles the perverse coherent t-structure, but which incorporates additional information from the G-action. Under certain circumstances, the heart of this t-structure, called the "staggered t-structure," is a finite-length category, and its simple objects are particularly easy to describe. We also exhibit two small examples in which the staggered t-structure is better-behaved than the perverse coherent t-structure.

Fusion Subcategories of Representation Categories of Twisted Quantum Doubles of Finite Groups

Naidu, D., Nikshych, D., Witherspoon, S. — 2009-06-17

We describe all fusion subcategories of the representation category of a twisted quantum double , where G is a finite group and is a 3-cocycle on G. In view of the fact that every group-theoretical braided fusion category can be embedded into some , this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas for some invariants of the fusion subcategories of . We also give a characterization of group-theoretical braided fusion categories as equivariantizations of pointed categories.

Undecidability in Function Fields of Positive Characteristic

Eisentrager, K., Shlapentokh, A. — 2009-06-17

We prove that the first-order theory of any function field K of characteristic p > 2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. The proof uses a result by Moret-Bailly about ranks of elliptic curves over function fields.

Continuous Extension of Arithmetic Volumes

Moriwaki, A. — 2009-06-17

This article is the sequel of the article [4], in which we established the arithmetic volume function of C-hermitian -invertible sheaves and proved its continuity. The continuity of the volume function has a lot of applications as treated in [4]. In this article, we would like to consider its continuous extension over .

Frame Stabilizers for Framed Vertex Operator Algebras Associated to Lattices Having 4-Frames

Lam, C. H., Shimakura, H. — 2009-06-15

In this paper, we study certain Virasoro frames for lattice vertex operator algebras (VOAs) and their -orbifolds using linear codes over . We also compute the corresponding frame stabilizer from the viewpoint of binary codes and -codes. As an application, we determine the frame stabilizers of several Virasoro frames of the VOA and the moonshine vertex operator algebra V.

Rigidity, Locally Symmetric Varieties, and the Grothendieck-Katz Conjecture

Farb, B., Kisin, M. — 2009-06-13

Using Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck–Katz p-curvature conjecture for many locally symmetric varieties, including Hilbert–Blumenthal modular varieties and the moduli space of abelian varieties when g > 1.

Stability Conditions and Kleinian Singularities

Bridgeland, T. — 2009-06-12

We describe (connected components of) the spaces of stability conditions on certain triangulated categories associated to Dynkin diagrams. These categories can be defined either algebraically via module categories of preprojective algebras, or geometrically via coherent sheaves on resolutions of Kleinian singularities. The resulting spaces of stability conditions are covering spaces of regular subsets of the corresponding complexified Cartan algebras.

On Elements of Prime Order in the Plane Cremona Group over a Perfect Field

Dolgachev, I. V., Iskovskikh, V. A. — 2009-06-12

We show that the plane Cremona group over a perfect field k of characteristic p ≥ 0 contains an element of prime order ≥ 7 not equal to p if and only if there exists a two-dimensional algebraic torus T over k such that T(k) contains an element of order . If p = 0 and k does not contain a primitive th root of unity, we show that there are no elements of prime order > 7 in and all elements of order 7 are conjugate.

A Note on Kahler-Ricci Soliton

Chen, X., Sun, S., Tian, G. — 2009-06-12

On Hochschild Cohomology and Morita Deformations

Keller, B., Lowen, W. — 2009-06-12

In this paper we show that, in general, first-order Morita deformations are too limited to capture the second Hochschild cohomology of a differential graded category. For differential graded categories with bounded above cohomology, the Morita deformations do constitute a part of the Hochschild cohomology.

Counting Integral Points on Universal Torsors

Derenthal, U. — 2009-06-12

Manin's conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of integral points in certain regions on universal torsors. As an application, we prove Manin's conjecture for a singular quartic del Pezzo surface.

Quantum Alpha-Determinants and q-Deformed Hypergeometric Polynomials

Kimoto, K. — 2009-06-11

The quantum -determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic -submodules of the quantum matrix algebra generated by the powers of the quantum -determinant. For such a cyclic module, there exists a collection of polynomials, which describe the irreducible decomposition of it in the following manner: (i) each polynomial corresponds to a certain irreducible -module, (ii) the cyclic module contains an irreducible submodule if the parameter is not a root of the corresponding polynomial. These polynomials are given as a q-deformation of the hypergeometric polynomials. This is a quantum analog of the result obtained in our previous work [Kimoto, K., S. Matsumoto, and M. Wakayama. "Alpha-determinant cyclic modules and Jacobi polynomials." Transactions of the American Mathematical Society (forthcoming)].

A Simple Twisted Relative Trace Formula

Hahn, H. — 2009-06-08

In this article we derive a simple twisted relative trace formula.

Stability of the Periodic Toda Lattice in the Soliton Region

Kruger, H., Teschl, G. — 2009-06-05

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the periodic (and slightly more generally of the quasi-periodic finite-gap) Toda lattice for decaying initial data in the soliton region. In addition, we show how to reduce the problem in the remaining region to the known case without solitons.

Two Applications of Twisted Floer Homology

Ai, Y., Ni, Y. — 2009-06-05

Given an irreducible closed three-manifold Y, we show that its twisted Heegaard Floer homology determines whether Y is a torus bundle over the circle. Another result we will prove is, if K is a genus-1 null-homologous knot in an L-space, and the zero surgery on K is fibered, then K itself is fibered. These two results are the missing cases of earlier results due to the second author.

Coplanar k-Unduloids Are Nondegenerate

2009-06-05

We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic manifold and that a neighborhood of these in the full CMC moduli space is itself a manifold. Nondegeneracy further implies (infinitesimal and local) rigidity in the sense that the asymptotes map is an analytic immersion on these spaces, and also that the coplanar classifying map is an analytic diffeomorphism.

Scalar Curvature Bound for Kahler-Ricci Flows over Minimal Manifolds of General Type

Zhang, Z. — 2009-06-02

In this short paper, we prove that scalar curvature is uniformly bounded for the Kähler–Ricci flow over a minimal manifold of general type. This result can be compared with the result in [6] for the positive first Chern class case. A big part of the computation works for more general situation and we keep track of that for future application.

A Phase Transition for Nonintersecting Brownian Motions, and the Painleve II Equation

Delvaux, S., Kuijlaars, A. B. J. — 2009-05-28

We consider n nonintersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of "large separation" between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between them, as one would intuitively expect. We give a rigorous proof using the Riemann–Hilbert formalism. In the case of "critical separation" between the endpoints, we are led to a model Riemann–Hilbert problem associated to the Hastings–McLeod solution of the Painlevé II equation. We show that the Painlevé II equation also appears in the large n asymptotics of the recurrence coefficients of the multiple Hermite polynomials that are associated with the Riemann–Hilbert problem.

Biextensions of 1-Motives in Voevodsky's Category of Motives

Bertolin, C., Mazza, C. — 2009-05-27

Let k be a perfect field. In this paper, we prove that biextensions of 1-motives define multilinear morphisms between 1-motives in Voevodsky's triangulated category of effective geometrical motives over k with rational coefficients.

On an Average over the Gaussian Unitary Ensemble

Mezzadri, F., Mo, M. Y. — 2009-05-24

We study the asymptotic limit for large matrix dimension N of the partition function of the unitary ensemble (β = 2) with weight . We compute the leading-order term of the partition function and the coefficients of its Taylor expansion. Our results are valid in the region . Such a partition function contains all the information on a new statistics of the eigenvalues of matrices in the Gaussian unitary ensemble that was introduced by Berry and Shukla [2]. It can also be interpreted as the moment-generating function of the singular linear statistics .

Maximal Galois Group of L-Functions of Elliptic Curves

Jouve, F. — 2009-05-21

We give a quantitative version of a result due to Katz about L-functions of elliptic curves over function fields over finite fields. Roughly speaking, Katz's theorem states that, on average over a suitably chosen algebraic family, the L-function of an elliptic curve over a function field becomes "as irreducible as possible" when seen as a polynomial with rational coefficients, as the cardinality of the field of constants grows. A quantitative refinement is obtained as a corollary of our main result, which gives an estimate for the proportion of elliptic curves studied whose L-functions have "big" Galois group. To do so, we make use of Kowalski's idea to apply large sieve methods in algebro-geometric contexts. Besides large sieve techniques, we use results of Hall on finite orthogonal monodromy and previous work of the author on orthogonal groups over finite fields.

Whittaker Limits of Difference Spherical Functions

Cherednik, I. — 2009-05-21

The q-Whittaker function is introduced as a limit at t = 0 of the global -spherical function, which extends the symmetric Macdonald polynomials to arbitrary eigenvalues. The limiting procedure generalizes that due to Etingof. The construction heavily depends on the technique of the q-Gaussians developed by the author (and Stokman in the non-reduced case). In this approach, the q-Whittaker function is given by a series convergent everywhere. One of the applications is a q-version of the Shintani–Casselman–Shalika formula, which is directly connected with the -Mehta–Macdonald identities in terms of the Jackson integral. In type A, this formula generalizes that due to Gerasimov et al. The Harish-Chandra-type asymptotic formula is established for the global -spherical functions, including the Whittaker limit.

Regularity of Dirac-Harmonic Maps

Wang, C., Xu, D. — 2009-05-18

For any n-dimensional compact spin Riemannian manifold M with a given spin structure and a spinor bundle M, and any compact Riemannian manifold N, we show an -regularity theorem for weakly Dirac-harmonic maps (, ):M M -> N ^*TN. As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when n = 2. For n ≥ 3, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in . If, in addition, W^1,p for some , then we obtain an energy monotonicity formula and prove a partial regularity theorem for any such a stationary Dirac-harmonic map.

Diophantine Conditions in Well-Posedness Theory of Coupled KdV-Type Systems: Local Theory

Oh, T. — 2009-05-18

We consider the local well-posedness (LWP) problem of a one-parameter family of coupled Korteweg–de Vries-type systems in both the periodic and nonperiodic settings. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter when != 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish a sharp LWP of the system in , where is determined by the Diophantine characterization of certain constants derived from the coupling parameter . We also present a sharp local (and global) result in . In the Appendix, we briefly discuss the LWP result in for = 1 without the mean zero assumption, by introducing the vector-valued X^s,b spaces.

Finiteness of the Number of Compatibly Split Subvarieties

Kumar, S., Mehta, V. B. — 2009-05-13

We prove that there are only finitely many compatibly split closed subschemes of a Frobenius split scheme.

Counting Open Nodal Lines of Random Waves on Planar Domains

Toth, J. A., Wigman, I. — 2009-05-12

We compute the asymptotic expectation of the number of open nodal lines for random waves on smooth planar domains. We find that for both the long energy window [0, ], and the short one [, +1], the expected number of open nodal lines is proportional to , asymptotically as -> . Our results are consistent with the predictions of Blum, Gnutzmann, and Smilansky [4] in the physics literature.

An Equivariant Index Formula for Almost CR Manifolds

Fitzpatrick, S. — 2009-05-10

We consider the case of a compact manifold M, together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients whose properties we describe.

When E is equipped with a complex structure, we define a class of symbol mappings in terms of the resulting almost CR structure that are H-transversally elliptic whenever the action of H is transverse to E. We determine a formula for the H-equivariant index of such symbols that involves only and standard equivariant characteristic forms. This formula generalizes the formula given in [10] for the case of a contact manifold.

Visible Actions on Irreducible Multiplicity-Free Spaces

Sasaki, A. — 2009-05-08

A holomorphic action of a Lie group G on a connected complex manifold D is called strongly visible with a slice S if D' colone G · S is open in D and there exists an antiholomorphic and orbit-preserving diffeomorphism of D' such that |_S = id _S. In this article, we study linear, strongly visible actions. We prove that irreducible multiplicity-free space V of a connected compact Lie group is strongly visible. Furthermore, we find an explicit description of S and according to Kac's classification. Our result gives an evidence to Kobayashi's conjecture [10, Conjecture 3.2] in the case of irreducible multiplicity-free spaces, asserting that we can take S to have the same dimension as the rank of V.

Prolongement de biextensions et accouplements en cohomologie log plate

Gillibert, J. — 2009-05-08

Nous revisitons, dans le langage des log schémas, le problème de prolongement de biextensions de schémas en groupes commutatifs lisses par le groupe multiplicatif étudié par Grothendieck dans [10]. Nous montrons que ce problème admet en général une solution dans la catégorie des faisceaux pour la topologie log plate, contrairement à ce que l'on peut observer en topologie fppf pour laquelle Grothendieck a défini des obstructions monodromiques. En particulier, dans le cas d'une variété abélienne et de sa duale, il est possible de prolonger la biextension de Weil sur la totalité des modèles de Néron; ceci permet de définir un accouplement sur les points qui combine l'accouplement de classes défini par Mazur et Tate et l'accouplement de monodromie.

We study, using the language of log schemes, the problem of extending biextensions of smooth commutative group schemes by the multiplicative group. This was first considered by Grothendieck in [10]. We show that this problem admits a solution in the category of sheaves for Kato's log flat topology, in contradistinction to what can be observed using the fppf topology, for which monodromic obstructions were defined by Grothendieck. In particular, in the case of an abelian variety and its dual, it is possible to extend the Weil biextension to the whole Néron model. This allows us to define a pairing on the points that combines the class group pairing defined by Mazur and Tate and Grothendieck's monodromy pairing.

Vicious Walkers and Random Contraction Matrices

Novak, J. — 2009-04-29

Random contraction matrices obtained by truncating Haar distributed random unitary matrices were originally introduced and studied in the context of scattering theory. In this paper, we demonstrate a connection between ensembles of random contractions and the random-turns vicious walker model from statistical physics. In particular, we show that the moments of the trace of a random contraction enumerate configurations of vicious walkers.

Spectral Analysis of the Free Orthogonal Matrix

Banica, T., Collins, B., Zinn-Justin, P. — 2009-04-27

We compute the spectral measure of the standard generators u_ij of the Wang algebra A_o(n). We show, in particular, that this measure has support , and that it has no atoms. The computation is done by using various techniques, involving the general Wang algebra A_o(F), a representation of SU^q₂ due to Woronowicz, and several calculations with orthogonal polynomials.

Bounded Harmonic Functions for the Heckman-Opdam Laplacian

Schapira, B. — 2009-04-27

We describe the set of bounded harmonic functions for the Heckman–Opdam Laplacian when the multiplicity function is larger than 1/2. We prove that this set is a vector space of dimension the cardinality of the Weyl group. We give some consequences in terms of the associated hypergeometric functions.

Infinitesimal Derived Torelli Theorem for K3 Surfaces (with an Appendix by Sukhendu Mehrotra)

Macri, E., Stellari, P. — 2009-04-22

We prove that the first-order deformations of two smooth projective K3 surfaces are derived equivalent under a Fourier–Mukai transform if and only if there exists a special isometry of the total cohomology groups of the surfaces which preserves the Mukai pairing, an infinitesimal weight-2 decomposition and the orientation of a positive four-dimensional space. This generalizes the derived version of the Torelli theorem. Along the way we show the compatibility of the actions on Hochschild homology and singular cohomology of any Fourier–Mukai functor.

Representing Sets with Sums of Triangular Numbers

Kane, B. — 2009-04-21

We investigate here sums of triangular numbers where T_n is the nth triangular number. We show that for a set of positive integers S, there is a finite subset S₀ such that f represents S if and only if f represents S₀. However, computationally determining S₀ is ineffective for many choices of S. We give an explicit and efficient algorithm to determine the set S₀ under certain generalized Riemann hypotheses, and implement the algorithm to determine S₀ when S is the set of all odd integers.

Imperfect Mimesis of Weyl Sums

Brudern, J., Daemen, D. — 2009-04-20

New lower bounds are obtained for the error between a Weyl sum and its natural adelic approximation, both pointwise and in mean square. These essentially match with existing upper bounds.

Les classes d'Eisenstein des varietes de Hilbert-Blumenthal

Blottiere, D. — 2009-04-17

Dans [2], on a démontré que les courants de Levin (cf. [11]) permettent de décrire explicitement les classes d'Eisenstein d'un schéma abélien au niveau topologique. On applique ici ce résultat, conjecturé par Levin, au cas où le schéma abélien est une famille de variétés abéliennes de Hilbert-Blumenthal (cf. Proposition 4.3). On étudie ensuite la dégénérescence de ces classes d'Eisenstein en une pointe de la compactification de Baily-Borel de la variété de Hilbert-Blumenthal. Au moyen du Théorème de Burgos-Wildeshaus [3, Theorem 2.9], on démontre un résultat de rigidité (cf. Proposition 3.4) qui permet de restreindre l'étude au niveau topologique. On prouve, en utilisant la description explicite des classes d'Eisenstein obtenue précédemment, que ces classes dégénèrent en des valeurs spéciales d'une fonction L associée au corps de nombres totalement réel sous-jacent (Théorème 5.2). On en déduit une preuve géométrique du Théorème de Klingen-Siegel (Corollaire 5.3) et un résultat de non annulation pour certaines de ces classes d'Eisenstein (Corollaire 5.4).

Petits points et multiplication complexe

Carrizosa, M. — 2009-04-16

We provide a lower bound for the canonical height of a point P in a CM abelian variety A/ K in terms of the degree of the field generated by P over K^ab. This bound is a generalization of results by David, Hindry, Baker, Silverman, Ratazzi and others and is the best known result on the way to proving the relative abelian Lehmer conjecture. Moreover, the given bound allows us to prove some particular cases of Zilber–Pink conjecture.

Special L-Values of t-Motives: A Conjecture

Taelman, L. — 2009-04-16

We propose a conjecture on special values of L-functions in a function field context with positive characteristic coefficients. For M a uniformizable t-motive with everywhere good reduction, we conjecture a relation between the value of the Goss L-function L(M, s) at s = 0 and the uniformization of the abelian t-module associated with M. When M is a power of the Carlitz t-motive, the conjecture specializes to a theorem of Anderson and Thakur on Carlitz zeta values. Beyond this case, we present numerical evidence.

Visibility and the Birch and Swinnerton-Dyer Conjecture for Analytic Rank One

Agashe, A. — 2009-04-15

Let E be an optimal elliptic curve over of conductor N having analytic rank one, i.e. such that the L-function L_E(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the L-function of E over K vanishes to order one at s = 1. Suppose there is another optimal elliptic curve over of the same conductor N whose Mordell–Weil rank is greater than one and whose associated newform is congruent to the newform associated to E modulo an integer r. The theory of visibility then shows that under certain additional hypotheses, r divides the product of the order of the Shafarevich–Tate group of E over K and the orders of the arithmetic component groups of E. We extract an explicit integer factor from the Birch and Swinnerton–Dyer conjectural formula for the product mentioned above, and under some hypotheses similar to the ones made in the situation above, we show that r divides this integer factor. This provides theoretical evidence for the second part of the Birch and Swinnerton–Dyer conjecture in the analytic rank one case.

Polynomial Maps over p -Adics and Residual Properties of Mapping Tori of Group Endomorphisms

Borisov, A., Sapir, M. — 2009-04-14

We continue our study of residual properties of mapping tori of free group endomorphisms. In this paper, we prove that each of these groups are virtually residually (finite p)-groups for all but finitely many primes p. The method involves further studies of polynomial maps over finite fields and p-adic completions of number fields.

On Cluster Algebras Arising from Unpunctured Surfaces

Schiffler, R., Thomas, H. — 2009-04-11

We study cluster algebras that are associated to unpunctured surfaces, with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. In the special case where the cluster algebra is acyclic, we also give a formula for the expansion of cluster variables as a polynomial whose indeterminates are the cluster variables contained in the union of an arbitrary acyclic cluster and all its neighboring clusters in the mutation graph.

A Sharp Inequality for the Strichartz Norm

Carneiro, E. — 2009-04-11

Let be the solution of the linear Schrödinger equation In the first part of this paper, we obtain a sharp inequality for the Strichartz norm , where , and , that admits only Gaussian maximizers. As corollaries, we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi [4] and Hundertmark–Zharnitsky [6]) and also sharp forms of some Sobolev–Strichartz inequalities. In the second part of the paper, we express Foschi's [4] sharp inequalities for the Schrödinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone.

Complex Hessian Equation on Kahler Manifold

Hou, Z. — 2009-04-11

In this paper, complex Hessian equation over Kähler manifold was studied. Under the condition that the underline Kähler manifold has non-negative holomorphic bisectional curvature, the existence and regularity of the solution was proved.

Harmonic Morphisms and Hyperelliptic Graphs

Baker, M., Norine, S. — 2009-04-11

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann–Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a two-edge-connected graph G that is not a cycle, there is at most one involution on G for which the quotient is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a nonconstant harmonic morphism to the graph B₂ consisting of two vertices connected by two edges. Finally, we use the Riemann–Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

Local Zeta Functions Supported on Analytic Submanifolds and Newton Polyhedra

Zuniga-Galindo, W. A. — 2009-04-06

The local zeta functions (also called Igusa's zeta functions) over p-adic fields are connected with the number of solutions of congruences and exponential sums mod p^m. These zeta functions are defined as integrals over open and compact subsets with respect to the Haar measure. In this article, we introduce new integrals defined over submanifolds, or more generally, over nondegenerate complete intersection varieties, and study their connections with some arithmetical problems such as estimation of exponential sums mod p^m. In particular, we extend Igusa's method for estimating exponential sums mod p^m to the case of exponential sums mod p^m along nondegenerate smooth varieties.

Relative Chow-Kunneth Decompositions for Conic Bundles and Prym Varieties

Nagel, J., Saito, M. — 2009-04-01

We construct a relative Chow–Künneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville's theorem on the relation between the intermediate Jacobian of the conic bundle and the Prym variety of the double covering.

Pieri-Type Formulas for Nonsymmetric Macdonald Polynomials

Baratta, W. — 2009-03-24

In symmetric Macdonald polynomial theory, the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function e_r(z) and the Macdonald polynomial . In this paper we give the nonsymmetric analogs for the cases r=1 and r= n–1. We do this by first deducing the decomposition for the product of any nonsymmetric Macdonald polynomial with z_i in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coefficients of we evaluate the generalized binomial coefficients associated with the nonsymmetric Macdonald polynomials for

A Codimension Two CR Singular Submanifold That Is Formally Equivalent to a Symmetric Quadric

Huang, X., Yin, W. — 2009-03-24

Let ( ) be a real analytic submanifold defined by an equation of the form: w=| z|²+ O(| z|³), where we use for the coordinates of . We first derive a pseudonormal form for M near 0. We then use it to prove that (M, 0) is holomorphically equivalent to the quadric if and only if it can be formally transformed to . We also use it to give a necessary and sufficient condition when (M, 0) can be formally flattened. Our main theorem generalizes a classical result of Moser for the case of n=1.

Retract Rationality and Noether's Problem

Kang, M.-c. — 2009-03-23

Let K be any field and G be a finite group. Let G act on the rational function field by K-automorphisms defined by for any . Denote by the fixed field . Noether's problem asks whether is rational (i.e. purely transcendental) over K. We will prove that, if K is any field, p an odd prime number, and G is a nonabelian group of exponent p with or p⁴ satisfying , then is rational over K. Moreover, it will be shown that is retract rational if G belongs to a much larger class of p-groups. In particular, generic G-polynomials of G-Galois extensions exist for these groups.

On Colored Turaev-Viro Invariants for Links in Arbitrary 3-Manifolds

Pervova, E., Petronio, C. — 2009-03-23

We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev–Viro invariants of 3-manifolds, that we call colored Turaev–Viro invariants. Their construction is based on a presentation of a pair (M, L), where M is a closed oriented 3-manifold, and is an oriented link, by a triangulation of M such that each component of L is an edge. We analyze some basic properties of these invariants, including the behavior under connected sums of pairs away and along links. These properties allow us to provide examples of links in having the same HOMFLY polynomial and the same Kauffman polynomial but distinct Turaev–Viro invariants, and similar examples for the Alexander polynomial. We also investigate the relations between the Turaev–Viro invariants of (M, L) and those of , showing that they are sometimes, but not always, determined by each other, and discuss some relations with the Witten–Reshetikhin–Turaev invariants and the Jones polynomial.

Hall-Littlewood Plane Partitions and KP

Foda, O., Wheeler, M. — 2009-03-20

MacMahon's classic generating function of random plane partitions, which is related to Schur polynomials, was recently extended by Vuletic to a generating function of weighted plane partitions that is related to Hall–Littlewood polynomials, , and further to one related to Macdonald polynomials, . Using Jing's one-parameter deformation of charged free fermions, we obtain a Fock space derivation of the Hall–Littlewood extension. Confining the plane partitions to a finite square base, we show that the resulting generating function, , is an evaluation of a -function of KP.

Manin's Conjecture for a Cubic Surface with D5 Singularity

Browning, T. D., Derenthal, U. — 2009-03-18

The Manin conjecture is established for a split singular cubic surface in , with singularity type D₅.

Corrigenda to: Product Sets of Rationals, Multiplicative Translates of Subgroups in Residue Rings and Fixed Points of the Discrete Logarithm

Bourgain, J., Konyagin, S. V., Shparlinski, I. E. — 2009-03-17

On the Korteweg-de Vries Long-Wave Approximation of the Gross-Pitaevskii Equation I

Bethuel, F., Gravejat, P., Saut, J.-C., Smets, D. — 2009-03-13

The fact that the Korteweg–de Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross–Pitaevskii equation was derived several years ago in the physical literature (see e.g. [17]). In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest.

Demazure Embeddings are Smooth

Losev, I. — 2009-02-28

We prove the conjecture of M. Brion stating that the closure of the orbit of a self-normalizing spherical subalgebra in the corresponding Grassmanian is smooth.

The Class as an ME Invariant

Sako, H. — 2009-02-24

We prove that being in Ozawa's class is a measure equivalence invariant.