Mating Siegel quadratic polynomials

Let be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers and . Using a new degree Blaschke product model for the dynamics of and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers and .

     

Locally analytic distributions and

In this paper we study continuous representations of locally -analytic groups in locally convex -vector spaces, where is a finite extension of and is a spherically complete nonarchimedean extension field of . The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of , along with interesting new objects such as the action of on global sections of equivariant vector bundles on -adic symmetric spaces. We introduce a restricted category of such representations that we call ``strongly admissible' and we show that, when is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of . As an application we prove the topological irreducibility of generic members of the -adic principal series for . Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous -valued representations of locally -analytic groups.

     

The McKay correspondence as an equivalence of derived categories

Let be a finite group of automorphisms of a nonsingular three-dimensional complex variety , whose canonical bundle is locally trivial as a -sheaf. We prove that the Hilbert scheme parametrising -clusters in is a crepant resolution of and that there is a derived equivalence (Fourier-Mukai transform) between coherent sheaves on and coherent -sheaves on . This identifies the K theory of with the equivariant K theory of , and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.

     

The strange duality conjecture for generic curves

Let be a smooth connected projective algebraic curve of genus . The strange duality conjecture connects non-abelian theta functions of rank and level and those of rank and level on (for and , respectively). In this paper we prove this conjecture for generic in the moduli space of curves of genus .

     

Vaught's conjecture on analytic sets

Let be a Polish group. We characterize when there is a Polish space with a continuous -action and an analytic set (that is, the Borel image of some Borel set in some Polish space) having uncountably many orbits but no perfect set of orbit inequivalent points.

Such a Polish -space and analytic exist exactly when there is a continuous, surjective homomorphism from a closed subgroup of onto the infinite symmetric group, , consisting of all permutations of equipped with the topology of pointwise convergence.

     

Double affine Hecke algebras and 2-dimensional local fields

We give an interpretation of the double affine Hecke algebra of Cherednik as a (suitably regularized) algebra of double cosets of a group by a subgroup , extending the well-known interpretations of the finite and affine Hecke algebras. In this interpretation, consists of -points of a simple algebraic group, where is a 2-dimensional local field such as or , and is a certain analog of the Iwahori subgroup.

     

On the number of zero-patterns of a sequence of polynomials

Let be a sequence of polynomials of degree in variables over a field . The zero-pattern of at is the set of those ( ) for which . Let denote the number of zero-patterns of as ranges over . We prove that for and

for . For , these bounds are optimal within a factor of . The bound () improves the bound proved by J. Heintz (1983) using the dimension theory of affine varieties. Over the field of real numbers, bounds stronger than Heintz's but slightly weaker than () follow from results of J. Milnor (1964), H.E.  Warren (1968), and others; their proofs use techniques from real algebraic geometry. In contrast, our half-page proof is a simple application of the elementary ``linear algebra bound'.

Heintz applied his bound to estimate the complexity of his quantifier elimination algorithm for algebraically closed fields. We give several additional applications. The first two establish the existence of certain combinatorial objects. Our first application, motivated by the ``branching program' model in the theory of computing, asserts that over any field , most graphs with vertices have projective dimension (the implied constant is absolute). This result was previously known over the reals (Pudlák-Rödl). The second application concerns a lower bound in the span program model for computing Boolean functions. The third application, motivated by a paper by N. Alon, gives nearly tight Ramsey bounds for matrices whose entries are defined by zero-patterns of a sequence of polynomials. We conclude the paper with a number of open problems.

     

The enumerative geometry of surfaces and modular forms

Let be a surface, and let be a holomorphic curve in representing a primitive homology class. We count the number of curves of geometric genus with nodes passing through generic points in in the linear system for any and satisfying .

When , this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Göttsche has generalized their conjecture to arbitrary in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus constrained to points are also given in terms of quasi-modular forms.     

Computing roadmaps of semi-algebraic sets on a variety

We consider a semi-algebraic set defined by polynomials in variables which is contained in an algebraic variety . The variety is assumed to have real dimension the polynomial and the polynomials defining have degree at most . We present an algorithm which constructs a roadmap on . The complexity of this algorithm is . We also present an algorithm which, given a point of defined by polynomials of degree at most , constructs a path joining this point to the roadmap. The complexity of this algorithm is These algorithms easily yield an algorithm which, given two points of defined by polynomials of degree at most , decides whether or not these two points of lie in the same semi-algebraically connected component of and if they do computes a semi-algebraic path in connecting the two points.

     

Compatibility of local and global Langlands correspondences

We prove the compatibility of local and global Langlands correspondences for , which was proved up to semisimplification in M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Studies 151, Princeton Univ. Press, Princeton-Oxford, 2001. More precisely, for the -dimensional -adic representation of the Galois group of an imaginary CM-field attached to a conjugate self-dual regular algebraic cuspidal automorphic representation of , which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of to the decomposition group of a place of not dividing corresponds to by the local Langlands correspondence. If is square integrable for some finite place we deduce that is irreducible. We also obtain conditional results in the case .

     

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point the local real-analytic CR automorphisms of fixing can be parametrized real-analytically by their jets at . As a direct application, we derive a Lie group structure for the topological group . Furthermore, we also show that the order of the jet space in which the group embeds can be chosen to depend upper-semicontinuously on . As a first consequence, it follows that given any compact real-analytic hypersurface in , there exists an integer depending only on such that for every point germs at of CR diffeomorphisms mapping into another real-analytic hypersurface in are uniquely determined by their -jet at that point. Another consequence is the following boundary version of H. Cartan's uniqueness theorem: given any bounded domain with smooth real-analytic boundary, there exists an integer depending only on such that if is a proper holomorphic mapping extending smoothly up to near some point with the same -jet at with that of the identity mapping, then necessarily .

Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.

     

Order -adic field

Let be an algebraically closed field of characteristic the ring of Witt vectors and a complete discrete valuation ring dominating and containing a primitive -th root of unity. Let denote a uniformizing parameter for We study order automorphisms of the formal power series ring which are defined by a series

The set of fixed points of is denoted by and we suppose that they are -rational and that for Let be the minimal semi-stable model of the -adic open disc over in which specializes to distinct smooth points. We study the differential data that can be associated to each irreducible component of the special fibre of Using this data we show that if , then the fixed points are equidistant, and that there are only a finite number of conjugacy classes of order automorphisms in which are not the identity

     

Functoriality for the exterior square of

In this paper we prove the functoriality of the exterior square of cusp forms on as automorphic forms on and the symmetric fourth of cusp forms on as automorphic forms on . We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain -functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of and the absolute convergence of the exterior square -functions of . Second, we prove that the fourth symmetric power -functions of cuspidal representations of are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound for Hecke eigenvalues of Maass forms over any number field.

     

Braid groups are linear

The braid group can be defined as the mapping class group of the -punctured disk. A group is said to be linear if it admits a faithful representation into a group of matrices over . Recently Daan Krammer has shown that a certain representation of the braid groups is faithful for the case . In this paper, we show that it is faithful for all .

     

Purity of the stratification by Newton polygons

Let be a variety in characteristic . Suppose we are given a nondegenerate -crystal over , for example the th relative crystalline cohomology sheaf of a family of smooth projective varieties over . At each point of we have the Newton polygon associated to the action of on the fibre of the crystal at . According to a theorem of Grothendieck the Newton polygon jumps up under specialization. The main theorem of this paper is that the jumps occur in codimension on (the Purity Theorem). As an application we prove some results on deformations of iso-simple -divisible groups.

     

Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

Let be a connected semisimple Lie group with finite center. Let be the maximal compact subgroup of corresponding to a fixed Cartan involution . We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary -module contains a -type with highest weight , then has infinitesimal character . Here is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary -modules with non-zero Dirac cohomology, provided has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant's cubic Dirac operator.

     

Curvature and injectivity radius estimates for Einstein 4-manifolds

Let denote an Einstein -manifold with Einstein constant, , normalized to satisfy . For , a metric ball, we prove a uniform estimate for the pointwise norm of the curvature tensor on , under the assumption that the -norm of the curvature on is less than a small positive constant, which is independent of , and which in particular, does not depend on a lower bound on the volume of . In case , we prove a lower injectivity radius bound analogous to that which occurs in the theorem of Margulis, for compact manifolds with negative sectional curvature, . These estimates provide key tools in the study of singularity formation for -dimensional Einstein metrics. As one application among others, we give a natural compactification of the moduli space of Einstein metrics with negative Einstein constant on a given .

     

On the size of -fold sum and product sets of integers

In this paper, we show that for all 1$"> there is a positive integer such that if is an arbitrary finite set of integers, 2$">, then either N^{b}$"> or N^{b}$">. Here (resp. ) denotes the -fold sum (resp. product) of . This fact is deduced from the following harmonic analysis result obtained in the paper. For all 2$"> and 0$">, there is a 0$"> such that if satisfies , then the -constant of (in the sense of W. Rudin) is at most .

     

Random polynomials having few or no real zeros

Consider a polynomial of large degree whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly real zeros with probability as through integers of the same parity as the fixed integer . In particular, the probability that a random polynomial of large even degree has no real zeros is . The finite, positive constant is characterized via the centered, stationary Gaussian process of correlation function . The value of depends neither on nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability one may specify also the approximate locations of the zeros on the real line. The constant is replaced by in case the i.i.d. coefficients have a nonzero mean.