Local Rankin-Selberg convolutions for : explicit conductor formula

Let be a non-Archimedean local field and , positive integers. For , let and let be an irreducible supercuspidal representation of . Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant to the and an additive character of . This object is of central importance in the study of the local Langlands conjecture. It takes the form

where is an integer. The irreducible supercuspidal representations of have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of . This paper gives an explicit formula for in terms of the inducing data for the . It uses, on the one hand, the alternative approach to the local constant due to Shahidi, and, on the other, the general theory of types along with powerful existence theorems for types in , developed by Bushnell and Kutzko.

     

On an near an elliptic complex tangent

In this paper, we study the local biholomorphic property of a real -manifold near an elliptic complex tangent point . In particular, we are interested in the regularity and the unique disk-filling problem of the local hull of holomorphy of near , first considered in a paper of Bishop. When is a -smooth submanifold, using a result established by Kenig-Webster, we show that near , is a smooth Levi-flat -manifold with a neighborhood of in as part of its boundary. Moreover, near , is foliated by a family of disjoint embedded complex analytic disks. We also prove a uniqueness theorem for the analytic disks attached to . This result was proved in the previous work of Kenig-Webster when . When is real analytic, we show that is real analytic with a neighborhood of in as part of its real analytic boundary. Equivalently, we prove the convergence of the formal solutions of a certain functional equation. When or when but the Bishop invariant does not vanish at the point under study, the analyticity was then previously obtained in the work of Moser-Webster, Moser, and in the author's joint work with Krantz.

     

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.

     

Fractional Power Series and Pairings on Drinfeld Modules

Let be an algebraically closed field containing which is complete with respect to an absolute value . We prove that under suitable constraints on the coefficients, the series converges to a surjective, open, continuous -linear homomorphism whose kernel is locally compact. We characterize the locally compact sub--vector spaces of which occur as kernels of such series, and describe the extent to which determines the series. We develop a theory of Newton polygons for these series which lets us compute the Haar measure of the set of zeros of of a given valuation, given the valuations of the coefficients. The ``adjoint' series converges everywhere if and only if does, and in this case there is a natural bilinear pairing

which exhibits as the Pontryagin dual of . Many of these results extend to non-linear fractional power series. We apply these results to construct a Drinfeld module analogue of the Weil pairing, and to describe the topological module structure of the kernel of the adjoint exponential of a Drinfeld module.

     

Asymptotic properties of Banach spaces under renormings

It is shown that a separable Banach space can be given an equivalent norm with the following properties: If is relatively weakly compact and , then converges in norm. This yields a characterization of reflexivity once proposed by V.D. Milman. In addition it is shown that some spreading model of a sequence in is 1-equivalent to the unit vector basis of (respectively, ) implies that contains an isomorph of (respectively, ).

     

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 -automorphism method and noninvariant classes of degrees

A set of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let denote the structure of the computably enumerable sets under inclusion, . Most previously known automorphisms of the structure of sets were effective (computable) in the sense that has an effective presentation. We introduce here a new method for generating noneffective automorphisms whose presentation is , and we apply the method to answer a number of long open questions about the orbits of c.e. sets under automorphisms of . For example, we show that the orbit of every noncomputable ( i.e., nonrecursive) c.e. set contains a set of high degree, and hence that for all the well-known degree classes (the low c.e. degrees) and (the complement of the high c.e. degrees) are noninvariant classes.

     

tensor categories from quantum groups

Let be a semisimple Lie algebra and let be the ratio between the square of the lengths of a long and a short root. Moreover, let be the quotient category of the category of tilting modules of modulo the ideal of tilting modules with zero -dimension for . We show that for a sufficiently large integer, the morphisms of are Hilbert spaces satisfying functorial properties. As an application, we obtain a subfactor of the hyperfinite II factor for each object of .

     

The -method, weak Lefschetz theorems, and the topology of Kähler manifolds

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the -method, avoids moving arguments and gives much stronger results. In particular, it is proved that if and are connected smooth projective varieties of positive dimension and is a holomorphic immersion with ample normal bundle, then the image of in is of finite index. This result is obtained as a consequence of a direct generalization of Nori's theorem. The second part concerns a new approach to the theorem of Burns which states that a quotient of the unit ball in () by a discrete group of automorphisms which has a strongly pseudoconvex boundary component has only finitely many ends. The following generalization is obtained. If a complete Hermitian manifold of dimension has a strongly pseudoconvex end and for some positive constant , then, away from , has finite volume.

     

Polynomial extensions of van der Waerden's and Szemerédi's theorems

An extension of the classical van der Waerden and Szemerédi theorems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following result: Let be a set of positive upper Banach density, let be polynomials with rational coefficients taking integer values on the integers and satisfying , then for any there exist an integer and a vector such that for each .

     

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.

     

Independence of of monodromy groups

Let be a smooth curve over a finite field of characteristic , let be a number field, and let be an -compatible system of lisse sheaves on the curve . For each place of not lying over , the -component of the system is a lisse -sheaf on , whose associated arithmetic monodromy group is an algebraic group over the local field . We use Serre's theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show that when the -compatible system is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is ``independent of '. More precisely, after replacing by a finite extension, there exists a connected split reductive algebraic group over the number field such that for every place of not lying over , the identity component of the arithmetic monodromy group of is isomorphic to the group with coefficients extended to the local field .

     

A new proof of Federer's structure theorem for

We prove that Federer's structure theorem for -dimensional sets in follows from the special case of -dimensional sets in the plane, which was proved earlier by Besicovitch.

     

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.

     

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 .

     

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 Arason invariant and mod 2 algebraic cycles

Let be a field, over a smooth variety with function field and a quadratic vector bundle over . Assuming that the generic fibre of is in , we compute the image of its Arason invariant

in by the differential of the Bloch-Ogus spectral sequence. This gives an obstruction to being a global cohomology class.

     

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 .

     

L-series with nonzero central critical value

Given a cusp form of even integral weight and its associated -function , we expect that a positive proportion of the quadratic twists of will have nonzero central critical value. In this paper we give examples of weight two newforms whose associated -functions have the property that a positive proportion of its quadratic twists have nonzero central critical value.