On a correspondence between cuspidal representations of

Let be an irreducible, automorphic, self-dual, cuspidal representation of , where is the adele ring of a number field . Assume that has a pole at and that . Given a nontrivial character of , we construct a nontrivial space of genuine and globally -generic cusp forms on -the metaplectic cover of . is invariant under right translations, and it contains all irreducible, automorphic, cuspidal (genuine) and -generic representations of , which lift (``functorially, with respect to ") to . We also present a local counterpart. Let be an irreducible, self-dual, supercuspidal representation of , where is a -adic field. Assume that has a pole at . Given a nontrivial character of , we construct an irreducible, supercuspidal (genuine) -generic representation of , such that has a pole at , and we prove that is the unique representation of satisfying these properties.

     

Homoclinic points of algebraic -actions

Let be an action of by continuous automorphisms of a compact abelian group . A point in is called homoclinic for if as . We study the set of homoclinic points for , which is a subgroup of . If is expansive, then is at most countable. Our main results are that if is expansive, then (1) is nontrivial if and only if has positive entropy and (2) is nontrivial and dense in if and only if has completely positive entropy. In many important cases is generated by a fundamental homoclinic point which can be computed explicitly using Fourier analysis. Homoclinic points for expansive actions must decay to zero exponentially fast, and we use this to establish strong specification properties for such actions. This provides an extensive class of examples of -actions to which Ruelle's thermodynamic formalism applies. The paper concludes with a series of examples which highlight the crucial role of expansiveness in our main results.

     

The Dolbeault complex in infinite dimensions II

We study the equation on a ball , and prove that it is solvable if is a Lipschitz continuous, closed -form.

     

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.

     

On the image of the -adic Abel-Jacobi map for a variety over the algebraic closure of a finite field

Let be a smooth projective variety of dimension at most 4 defined over the algebraic closure of a finite field of characteristic . It is shown that the Tate conjecture implies the surjectivity of the -adic Abel-Jacobi map, , for all and almost all . For a special class of threefolds the surjectivity of is proved without assuming any conjectures.

     

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

     

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.

     

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.

     

The homotopy theory of fusion systems

We define and characterize a class of -complete spaces which have many of the same properties as the -completions of classifying spaces of finite groups. For example, each such has a Sylow subgroup , maps for a -group are described via homomorphisms , and is isomorphic to a certain ring of ``stable elements' in . These spaces arise as the ``classifying spaces' of certain algebraic objects which we call ``-local finite groups'. Such an object consists of a system of fusion data in , as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.

     

Periods of automorphic forms

Let be a quadratic extension of number fields and , where is a reductive group over . We define the integral (in general, non-convergent) of an automorphic form on over via regularization. This regularized integral is used to derive a formula for the integral over of a truncated Eisenstein series on . More explicit results are obtained in the case . These results will find applications in the expansion of the spectral side of the relative trace formula.

     

An A Bailey lemma and Rogers-Ramanujan-type identities

Using new -functions recently introduced by Hatayama et al. and by (two of) the authors, we obtain an A version of the classical Bailey lemma. We apply our result, which is distinct from the A Bailey lemma of Milne and Lilly, to derive Rogers-Ramanujan-type identities for characters of the W algebra.

     

A new proof of D. Popescu's theorem on smoothing of ring homomorphisms

We give a new proof of D. Popescu's theorem which says that if is a regular homomorphism of noetherian rings, then is a filtered inductive limit of smooth finite type -algebras. We strengthen Popescu's theorem in two ways. First, we show that a finite type -algebra , mapping to , has a desingularization which is smooth wherever possible (roughly speaking, above the smooth locus of ). Secondly, we give sufficient conditions for to be a filtered inductive limit of its smooth finite type -subalgebras. We also give counterexamples to the latter statement in cases when our sufficient conditions do not hold.

     

Fractional isoperimetric inequalities and subgroup distortion

It is shown that there exist infinitely many non-integers such that the Dehn function of some finitely presented group is . Explicit examples of such groups are constructed. For each rational number pairs of finitely presented groups are constructed so that the distortion of in is .

     

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.     

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.

     

Two-primary algebraic -theory of rings of integers in number fields

We relate the algebraic -theory of the ring of integers in a number field to its étale cohomology. We also relate it to the zeta-function of when is totally real and Abelian. This establishes the -primary part of the ``Lichtenbaum conjectures.' To do this we compute the -primary -groups of and of its ring of integers, using recent results of Voevodsky and the Bloch-Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory.

     

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.

     

Cyclotomic integers and finite geometry

We obtain an upper bound for the absolute value of cyclotomic integers which has strong implications on several combinatorial structures including (relative) difference sets, quasiregular projective planes, planar functions, and group invariant weighing matrices. Our results are of broader applicability than all previously known nonexistence theorems for these combinatorial objects. We will show that the exponent of an abelian group containing a -difference set cannot exceed where is the number of odd prime divisors of and is a number-theoretic parameter whose order of magnitude usually is the squarefree part of . One of the consequences is that for any finite set of primes there is a constant such that for any abelian group containing a Hadamard difference set whose order is a product of powers of primes in . Furthermore, we are able to verify Ryser's conjecture for most parameter series of known difference sets. This includes a striking progress towards the circulant Hadamard matrix conjecture. A computer search shows that there is no Barker sequence of length with . Finally, we obtain new necessary conditions for the existence of quasiregular projective planes and group invariant weighing matrices including asymptotic exponent bounds for cases which previously had been completely intractable.

     

systems of hyperbolic conservation laws

In this paper, we study the evolution of the distance of solutions for systems of hyperbolic conservation laws. For the approximate solutions constructed by Glimm's scheme with the aid of the wave tracing method, we introduce a nonlinear functional which is equivalent to the distance between solutions, nonincreasing in time, and expressed explicitly in terms of the wave patterns of the solutions. This functional reveals the nonlinear mechanism of wave interactions and coupling which affect the topology.

     

Principe local-global pour les zéro-cycles sur les surfaces réglées

Let be a number field, a smooth projective curve, and a smooth projective surface which is a conic bundle over . Let be the relative Chow group, which is the kernel of the projection map on Chow groups of zero-cycles. For each place of , one may consider the relative Chow group . Under minor assumptions, we identify the diagonal image of in the product of all as the kernel of the natural pairing with the Brauer group of . When is an elliptic curve with finite Tate-Shafarevich group, under minor assumptions, we show that the Brauer-Manin obstruction to the existence of a zero-cycle of degree one on is the only obstruction.