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 .

     

Pythagoras numbers of fields

A field of characteristic is said to have finite Pythagoras number if there exists an integer such that each nonzero sum of squares in can be written as a sum of squares, in which case the Pythagoras number of is defined to be the least such integer. As a consequence of Pfister's results on the level of fields, of a nonformally real field is always of the form or , and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form , , and can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer there exists a formally real field with . As a refinement, we will show that if and are integers such that , then there exists a uniquely ordered field with and (resp. ), where (resp. ) denotes the supremum of the dimensions of anisotropic forms over which are torsion in the Witt ring of (resp. which are indefinite with respect to each ordering on ).

     

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.

     

Regularity of the free boundary for the porous medium equation

We study the regularity of the free boundary for solutions of the porous medium equation , , on , with initial data nonnegative and compactly supported. We show that, under certain assumptions on the initial data , the pressure will be smooth up to the interface , when , for some . As a consequence, the free-boundary is smooth.

     

The essentially tame local Langlands correspondence, I

Let be a non-Archimedean local field (of characteristic or ) with finite residue field of characteristic . An irreducible smooth representation of the Weil group of is called essentially tame if its restriction to wild inertia is a sum of characters. The set of isomorphism classes of irreducible, essentially tame representations of dimension is denoted . The Langlands correspondence induces a bijection of with a certain set of irreducible supercuspidal representations of . We consider the set of isomorphism classes of certain pairs , called ``admissible', consisting of a tamely ramified field extension of degree and a quasicharacter of . There is an obvious bijection of with . Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of with , generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of with . We show that one obtains the Langlands correspondence by composing the map with a permutation of of the form , where is a tamely ramified character of depending on . This answers a question of Moy (1986). We calculate the character in the case where is totally ramified of odd degree.

     

Plurisubharmonic domination

For a large class of separable Banach spaces we prove the following. Given a pseudoconvex open and that is locally bounded above, there is a plurisubharmonic such that . We also discuss applications of this result.

     

Generalized group characters and complex oriented cohomology theories

Let be the classifying space of a finite group . Given a multiplicative cohomology theory , the assignment

is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories , using the theory of complex representations of finite groups as a model for what one would like to know.

An analogue of Artin's Theorem is proved for all complex oriented : the abelian subgroups of serve as a detecting family for , modulo torsion dividing the order of .

When is a complete local ring, with residue field of characteristic and associated formal group of height , we construct a character ring of class functions that computes . The domain of the characters is , the set of -tuples of elements in each of which has order a power of . A formula for induction is also found. The ideas we use are related to the Lubin-Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, -theory, etc.

The th Morava K-theory Euler characteristic for is computed to be the number of -orbits in . For various groups , including all symmetric groups, we prove that is concentrated in even degrees.

Our results about extend to theorems about , where is a finite -CW complex.

     

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.

     

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.

     

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.

     

Représentations -adiques et normes universelles I. Le cas cristallin

Let be a crystalline -adic representation of the absolute Galois group of an finite unramified extension of , and let be a lattice of stable by . We prove the following result: Let be the maximal sub-representation of with Hodge-Tate weights strictly positive and . Then, the projective limit of is equal up to torsion to the projective limit of . So its rank over the Iwasawa algebra is .

     

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 .

     

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 .

     

The averaging lemma

Averaging lemmas deduce smoothness of velocity averages, such as

from properties of . A canonical example is that is in the Sobolev space whenever and are in . The present paper shows how techniques from Harmonic Analysis such as maximal functions, wavelet decompositions, and interpolation can be used to prove versions of the averaging lemma. For example, it is shown that implies that is in the Besov space , . Examples are constructed using wavelet decompositions to show that these averaging lemmas are sharp. A deeper analysis of the averaging lemma is made near the endpoint .

     

Analytic continuation of overconvergent eigenforms

Let be an overconvergent -adic eigenform of level , , with non-zero -eigenvalue. We show how may be analytically continued to a subset of containing, for example, all the supersingular locus. Using these results we extend the main theorem of our earlier work with R. Taylor to many ramified cases.

     

Potentially semi-stable deformation rings

Let be a finite extension and the absolute Galois group of . For a complete local ring with finite residue and a finite free -module equipped with an action of , we show that has a maximal quotient over which the representation is semi-stable with Hodge-Tate weights in a given range. We show an analogous result for representations which are potentially semi-stable of fixed Galois type and -adic Hodge type.

If is the universal deformation of , then we compute the dimension of and we show that these rings are sometimes smooth.

Finally we apply this theory to show, in some new cases, the compatibility of the -adic Galois representation attached to a Hilbert modular form with the local Langlands correspondence at .

     

Criteria for -ampleness

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a -ample divisor, where is an automorphism of a projective scheme . Many open questions regarding -ample divisors have remained.

We derive a relatively simple necessary and sufficient condition for a divisor on to be -ample. As a consequence, we show right and left -ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms yield a -ample divisor.

     

On the equation and application to control of phases

The main result is the following. Let be a bounded Lipschitz domain in , . Then for every with , there exists a solution of the equation div in , satisfying in addition on and the estimate

where depends only on . However one cannot choose depending linearly on .

Our proof is constructive, but nonlinear--which is quite surprising for such an elementary linear PDE. When there is a simpler proof by duality--hence nonconstructive.

     

Sieving by large integers and covering systems of congruences

An old question of Erdos asks if there exists, for each number , a finite set of integers greater than and residue classes for whose union is . We prove that if is bounded for such a covering of the integers, then the least member of is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number , the complement in of any union of residue classes , for distinct , has density at least for sufficiently large. Here is a positive number depending only on . Either of these new results implies another conjecture of Erdos and Graham, that if is a finite set of moduli greater than , with a choice for residue classes for which covers , then the largest member of cannot be . We further obtain stronger forms of these results and establish other information, including an improvement of a related theorem of Haight.