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 .

     

Homological methods for hypergeometric families

We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems  arising from a integer matrix  and a parameter . To do so we introduce an Euler-Koszul functor for hypergeometric families over  , whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter is rank-jumping for if and only if lies in the Zariski closure of the set of -graded degrees  where the local cohomology of the semigroup ring supported at its maximal graded ideal  is nonzero. Consequently, has no rank-jumps over  if and only if is Cohen-Macaulay of dimension .

     

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.

     

Orbit equivalence for Cantor minimal -systems

We show that every minimal, free action of the group on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, -actions and -actions.

     

boundedness of discrete singular Radon transforms

We prove that if is a Calderón-Zygmund kernel and is a polynomial of degree with real coefficients, then the discrete singular Radon transform operator

extends to a bounded operator on , . This gives a positive answer to an earlier conjecture of E. M. Stein and S. Wainger.

     

On a lattice problem of H. Steinhaus

It is shown that there is a subset of such that each isometric copy of (the lattice points in the plane) meets in exactly one point. This provides a positive answer to a problem of H. Steinhaus.

     

Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi

In 1978 De Giorgi formulated the following conjecture. Let be a solution of in all of such that and 0$"> in . Is it true that all level sets of are hyperplanes, at least if ? Equivalently, does depend only on one variable? When , this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for . The question, however, remains open for . The results for and 3 apply also to the equation for a large class of nonlinearities .

     

A solution to the L space problem

In this paper I will construct a non-separable hereditarily Lindelöf space (L space) without any additional axiomatic assumptions. The constructed space is a subspace of where is the unit circle. It is shown to have a number of properties which may be of additional interest. For instance it is shown that the closure in of any uncountable subset of contains a canonical copy of .

I will also show that there is a function such that if are uncountable and , then there are in and respectively with . Previously it was unknown whether such a function existed even if was replaced by . Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality .

The results all stem from the analysis of oscillations of coherent sequences of finite-to-one functions. I expect that the methods presented will have other applications as well.

     

Integral motives and special values of zeta functions

For each field , we define a category of rationally decomposed mixed motives with -coefficients. When is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain groups.

     

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.

     

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.

     

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.

     

Fuglede-Kadison determinants and entropy for actions of discrete amenable groups

In 1990, Lind, Schmidt, and Ward gave a formula for the entropy of certain -dynamical systems attached to Laurent polynomials , in terms of the (logarithmic) Mahler measure of . We extend the expansive case of their result to the noncommutative setting where gets replaced by suitable discrete amenable groups. Generalizing the Mahler measure, Fuglede-Kadison determinants from the theory of group von Neumann algebras appear in the entropy formula.

     

Presentations of finite simple groups: A quantitative approach

There is a constant such that all nonabelian finite simple groups of rank over , with the possible exception of the Ree groups , have presentations with at most generators and relations and total length at most . As a corollary, we deduce a conjecture of Holt: there is a constant such that for every finite simple group , every prime and every irreducible -module .

     

Random polynomials with prescribed Newton polytope

The Newton polytope of a polynomial is well known to have a strong impact on its behavior. The Bernstein-Kouchnirenko Theorem asserts that even the number of simultaneous zeros in of a system of polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes also have a strong impact on the distribution of zeros and pointwise norms of polynomials, the basic theme being that Newton polytopes determine allowed and forbidden regions in for these distributions.

Our results are statistical and asymptotic in the degree of the polynomials. We equip the space of polynomials of degree in complex variables with its usual SU-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope . We then determine the asymptotics of the conditional expectation of simultaneous zeros of polynomials with Newton polytope as . When , the unit simplex, it is clear that the expected zero distributions are uniform relative to the Fubini-Study form. For a convex polytope , we show that there is an allowed region on which is asymptotically uniform as the scaling factor . However, the zeros have an exotic distribution in the complementary forbidden region and when (the case of the Bernstein-Kouchnirenko Theorem), the expected percentage of simultaneous zeros in the forbidden region approaches 0 as .

     

Isoperimetric inequalities in crystallography

Given a cubic space group (viewed as a finite group of isometries of the torus ), we obtain sharp isoperimetric inequalities for -invariant regions. These inequalities depend on the minimum number of points in an orbit of and on the largest Euler characteristic among nonspherical -symmetric surfaces minimizing the area under volume constraint (we also give explicit estimates of this second invariant for the various crystallographic cubic groups ). As an example, we prove that any surface dividing into two equal volumes with the same (orientation-preserving) symmetries as the A. Schoen minimal Gyroid has area at least (the conjectured minimizing surface in this case is the Gyroid itself whose area is ).

     

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.

     

Minkowski's conjecture, well-rounded lattices and topological dimension

Let be the diagonal subgroup, and identify with the space of unimodular lattices in . In this paper we show that the closure of any bounded orbit

meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for and yields bounds for the density of algebraic integers in totally real sextic fields.

The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of and .

     

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.