Proceedings of the London Mathematical Society - Advance Access

Theorie ergodique des fractions rationnelles sur un corps ultrametrique

Favre, C., Rivera-Letelier, J. — 2009-06-20

Résumé

On donne les premiers éléments pour l’étude des propriétés ergodi-ques d’une fraction rationnelle à coefficients dans un corps algébriquement clos et complet pour une norme non archimé-dienne. En particulier, pour une telle fraction rationnelle R on montre l’existence d’une mesure naturelle _R représentant la distribution asymptotique des préimages itérées de chaque point non exceptionnel de R. On montre que cette mesure est (exponentiellement) mélangeante, et qu’elle satisfait au théorème limite central. De plus, on donne une estimation de l’entropie métrique de cette mesure, et de l’entropie topologique de R, qui permettent de caractériser les fractions rationnelles d’entropie topologique nulle.

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure _R which represents the asymptotic distribution of preimages of non-exceptional points. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of _R, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction.

The true complexity of a system of linear equations

Gowers, W. T., Wolf, J. — 2009-06-20

In this paper we look for conditions that are sufficient to guarantee that a subset A of a finite Abelian group G contains the ‘expected’ number of linear configurations of a given type. The simplest non-trivial result of this kind is the well-known fact that if G has odd order, A has density and all Fourier coefficients of the characteristic function of A are significantly smaller than (except the one at zero, which equals ), then A contains approximately ³|G|² triples of the form (a, a+d, a+2d). This is ‘expected’ in the sense that a random set A of density has approximately ³|G|² such triples with very high probability. More generally, it was shown by the first author (in the case G = Z_N for N prime, but the proof generalizes) that a set A of density has about ^k|G|² arithmetic progressions of length k if the characteristic function of A is almost as small as it can be, given its density, in a norm that is now called the U^k–1-norm. When investigating linear equations in the primes, Green and Tao found the most general statement that follows from the technique used to prove this result, introducing a notion that they call the complexity of a system of linear forms. They prove that if A has almost minimal U^k+1-norm, then it has the expected number of linear configurations of a given type, provided that the associated complexity is at most k. The main result of this paper is that the converse is not true: in particular there are certain systems of complexity 2 that are controlled by the U²-norm, whereas the result of Green and Tao requires the stronger hypothesis of U³-control. We say that a system of m linear forms L₁, ..., L_m in d variables with integer coeffcients has true complexity k if k is the smallest positive integer such that, for any set A of density and almost minimal U^k+1-norm, the number of d-tuples (x₁, ..., x_d) such that L_i(x₁, ..., x_d) A for every i is approximately ^m|G|^d. We conjecture that the true complexity k is the smallest positive integer s for which the functions L₁^s+1, ... ,L_m^s+1 are linearly independent. Using the ‘quadratic Fourier analysis’ of Green and Tao we prove this conjecture in the case where the complexity of the system (in Green and Tao's sense) is 2, s=1 and G is the group F_pⁿ for some fixed odd prime p. A closely related result in ergodic theory was recently proved independently by Leibman. We end the paper by discussing the connections between his result and ours.

A Maslov cocycle for unitary groups

Kramer, L., Tent, K. — 2009-06-16

We introduce a 2-cocycle for symplectic and skew-hermitian hyperbolic groups over arbitrary fields and skew-fields, with values in the Witt group of hermitian forms. This cocycle has good functorial properties: it is natural under extension of scalars and stable, and so it can be viewed as a universal 2-dimensional characteristic class for these groups. Over R and C, it coincides with the first Chern class.

Spectral properties of matrices associated with some directed graphs

Davies, E. B., Incani, P. A. — 2009-06-09

We study the spectral properties of certain non-self-adjoint matrices associated with large directed graphs. Asymptotically the eigenvalues converge to certain curves, apart from a finite number that have limits not on these curves.

Families of absolutely simple hyperelliptic jacobians

Zarhin, Y. G. — 2009-06-09

We prove that the jacobian of a hyperelliptic curve y² = (x – t)h(x) has no non-trivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and deg(h) is an even number greater than 8. (The case of odd deg(h) > 3 follows easily from previous results of the author.)

The effect of convolving families of L-functions on the underlying group symmetries

Duenez, E., Miller, S. J. — 2009-06-02

Let {F_N} and {G_M} be families of primitive automorphic L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N, M -> , the statistical behavior (1-level density) of the low-lying zeros of L-functions in F_N and G_M agrees with that of the eigenvalues near 1 of matrices in G₁ and G₂, respectively, as the size of the matrices tend to infinity, where each G_i is one of the classical compact groups (unitary U, symplectic Sp, or orthogonal O, SO(even), SO(odd)). Assuming that the convolved families of L-functions F_N x G_M are automorphic, we study their 1-level density. (We also study convolved families of the form f x G_M for a fixed f.) Under natural assumptions on the families (which hold in many cases), we can associate to each family L of L-functions a symmetry constant cL equal to 0, 1, or–1 if the corresponding low-lying zero statistics agree with those of the unitary symplectic, or orthogonal group, respectively. Our main result is that cFxG=cF·cG: the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f x G_M. We provide examples built from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with positive rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N, M -> , as lower-order terms).

A solution to the Douglas-Rudin problem for matrix-valued functions

Barclay, S. — 2009-06-02

We solve the noncommutative Douglas–Rudin problem, showing that any log-integrable essentially bounded square matrix-valued function f can be written in the form h*g, where h and g lie in H . Extensions to other L ^p spaces, with norm bounds on the factors of f, are also provided.

Deformations of finite conformal energy: existence and removability of singularities

Iwaniec, T., Onninen, J. — 2009-05-14

This paper features a class of mappings between bounded domains X, Y Rⁿ, having finite n-harmonic energy, such that we have The fundamental question is whether or not the domains X, Y Rⁿ of the same topological type admit a homeomorphism in a given homotopy class having finite energy. The examples of non-existence, somewhat testing our theory, arise when we remove from bounded smooth domains X and Y thin subsets and , referred to as cracks or fractures. We are looking for homeomorphisms of finite energy for which is the cluster set of h over . In general, infinite energy is required in order to increase the dimension of a crack that is, when . Suppose now that a bounded deformation of finite energy is given. Does h extend continuously to X and, if so, is the extension injective on X? We give affirmative answers to these questions.

Building blocks of etale endomorphisms of complex projective manifolds

Nakayama, N., Zhang, D.-Q. — 2009-05-07

Étale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true. They are the endomorphisms of abelian varieties and the nearly étale rational endomorphisms of weak Calabi–Yau varieties.

Geometric criteria for Landweber exactness

Hollander, S. — 2009-04-29

The purpose of this paper is to give a new presentation of some of the main results concerning Landweber exactness in the context of the homotopy theory of stacks. We present two new criteria for Landweber exactness over a flat Hopf algebroid. The first criterion is used to classify stacks arising from Landweber exact maps of rings. Using as extra input only Lazard's theorem and Cartier's classification of p-typical formal group laws, this result is then applied to deduce many of the main results concerning Landweber exactness in stable homotopy theory and to compute the Bousfield classes of certain BP-algebra spectra. The second criterion can be regarded as a generalization of the Landweber exact functor theorem, and we use it to give a proof of the original theorem.

On the self-similarity problem for ergodic flows

Fraczek, K., Lemanczyk, M. — 2009-04-27

Given an ergodic flow (T_t)_{t R} we study the problem of its self-similarities, that is, we want to describe the set of s R for which the original flow is isomorphic to the flow (T_st)_{t R}. The problem is examined in some classes of special flows over irrational rotations and over interval exchange transformations. In particular, translation flows on translation surfaces are considered: we prove that under the weak mixing condition the set of self-similarities has Lebesgue measure zero. For von Neumann special flows over irrational rotations given by Diophantine numbers, this set is shown to be equal to {1}, while for horocycle flows a weak convergence in case of some singular (with respect to the volume measure) measures is shown to give rise to some new equidistribution result. The problem of self-similarity is also studied from the spectral point of view, especially in the class of Gaussian systems.

Cohomogeneity one disk bundles with normal homogeneous collars

Schwachhofer, L. J., Tapp, K. — 2009-04-24

We consider cohomogeneity one homogeneous disk bundles and address the question when these admit a nonnegatively curved invariant metric with normal collar, that is, such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in {2, 3, 4, 6, 8}. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3.

Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup

Bludov, V. V., Glass, A. M. W. — 2009-04-24

We use permutation groups to give necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable. We obtain several consequences answering previously posed problems and also prove the right-orderable analogues of the Higman Embedding Theorem and the Boone–Higman Theorem.

Solution of the polynomial moment problem

Pakovich, F., Muzychuk, M. — 2009-03-31

In this paper we give a solution of the following ‘polynomial moment problem’ which arose about 10 years ago in connection with Poincaré's center-focus problem: for a given polynomial P(z) to describe polynomials q(z) orthogonal to all powers of P(z) on a segment [a, b].

The Curvature invariant for a class of homogeneous operators

Misra, G., Shyam Roy, S. — 2009-03-25

For an operator T in the class B_n(), introduced by Cowen and Douglas, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle E_T determine the unitary equivalence class of the operator T. In a subsequent paper, the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant derivatives are necessary to determine the unitary equivalence class of the operator T B_n(). Here we show that some of the covariant derivatives are necessary. Our examples consist of homogeneous operators in B_n(D). For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives at any point w in the unit disc D are determined from the simultaneous unitary equivalence class at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples. One of our main results is that the curvature along with its covariant derivative of order (0, 1) at 0 determines the equivalence class of generic homogeneous Hermitian holomorphic vector bundles over the unit disc.

Artin braid groups and homotopy groups

Li, J., Wu, J. — 2009-03-17

We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as summands of the centres of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. The results give new connections between the braid groups and the general higher homotopy groups of spheres.

An orthogonal test of the L-functions Ratios conjecture

Miller, S. J. — 2009-03-16

We test the predictions of (a weakened version of) the L-functions Ratios conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N -> through the primes or N=1 and k -> . We study the main and lower-order terms in the 1-level density. We provide evidence for the Ratios conjecture by computing and confirming its predictions up to a power savings in the family's cardinality, at least for test functions whose Fourier transforms are supported in (–2, 2). We do this both for the weighted and unweighted 1-level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1-level densities differ by a term of size 1/log (k² N). Finally, we show that there is another way of extending the sums arising in the Ratios conjecture, leading to a different answer (although the answer is such a lower-order term that it is hopeless to observe which is correct).

Inversion formulas for elliptic functions

Cooper, S. — 2009-03-16

The aim of this work is to give a unified treatment of the fundamental formulas in Ramanujan's theories of elliptic functions to alternative bases. Our approach relies on well-known results from the theory of theta functions, such as the sum of four squares and sum of eight squares theorems, and their cubic analogues. We prove four inversion theorems, one being classical and the other three belonging to Ramanujan's theories to alternative bases. The connections with iterative means and the corresponding transformation formulas for hypergeometric functions are also established.

Geometry of nilmanifolds with left-invariant complex structure and deformations in the large

Rollenske, S. — 2009-03-04

The relation between nilmanifolds with left-invariant complex structure and iterated principal holomorphic torus bundles is clarified and we give criteria under which deformations in the large are again of such type. As an application we obtain a fairly complete picture in complex dimension 3.

Deformation theory of asymptotically conical coassociative 4-folds

Lotay, J. D. — 2009-03-03

Suppose that a coassociative 4-fold N in R⁷ is asymptotically conical (AC) to a cone C with rate <1. If [–2, 1) is generic, then we show that the moduli space of coassociative deformations of N that are also AC to C with rate is a smooth manifold, and we calculate its dimension. If <–2 and generic, then we show that the moduli space is locally homeomorphic to the kernel of a smooth map between smooth manifolds, and we give a lower bound for its expected dimension. We also derive a test for when N will be planar if <–2 and we discuss examples of AC coassociative 4-folds.

Fundamental groups of symmetric sextics II

Degtyarev, A. — 2009-03-03

We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with the set of inner singularities 2A₈ or A₁₇. We also compute the fundamental groups of a number of other sextics, both of and not of torus type. The groups found are simplest possible, that is, Z₂*Z₃ and Z₆, respectively.

Improving L2 estimates to Harnack inequalities

Filippas, S., Moschini, L., Tertikas, A. — 2009-03-03

We consider operators of the form L = – L – V, where L is an elliptic operator and V is a singular potential, defined on a smooth bounded domain Rⁿ with Dirichlet boundary conditions. We allow the boundary of to be made of various pieces of different codimension. We assume that L has a generalized first eigenfunction of which we know two-sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator L, we show that it generates an intrinsic ultracontractive semigroup and finally we derive a parabolic Harnack inequality up to the boundary as well as sharp heat kernel estimates.

Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials

Kozlovski, O., van Strien, S. — 2009-02-26

We prove that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate. From this we derive that each such polynomial can be approximated by a hyperbolic polynomial. As a by-product we prove that the Julia set of a non-renormalizable polynomial with only hyperbolic periodic points is locally connected, and the Branner–Hubbard conjecture. The main tools are the enhanced nest construction (developed in a previous joint paper with [Rigidity for real polynomials, Ann. of Math. (2) 165 (2007) 749–841.]) and a lemma of Kahn and Lyubich (for which we give an elementary proof in the real case).

Hodge theory for G2-manifolds: intermediate Jacobians and Abel-Jacobi maps

Karigiannis, S., Leung, N. C. — 2009-02-24

We study the moduli space M of torsion-free G₂-structures on a fixed compact manifold M⁷, and define its associated universal intermediate Jacobian J. We define the Yukawa coupling and relate it to a natural pseudo-Kähler structure on J.

We consider natural Chern-Simons-type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang-Mills connections), and also deformed Donaldson-Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in J by means of G₂-analogues of Abel-Jacobi maps.