Volume 4 (2004), Number 1. Abstracts

J. Bernstein and A. Reznikov. Estimates of Automorphic Functions [PDF]

We present a new method to estimate trilinear period for automorphic representations of SL2(R). The method is based on the uniqueness principle in representation theory. We show how to separate the exponentially decaying factor in the triple period from the essential automorphic factor which behaves polynomially. We also describe a general method which gives an estimate for the average of the automorphic factor and thus prove a convexity bound for the triple period.

Keywords. Automorphic representations, periods, uniqueness.

2000 Mathematics Subject Classification. 11F67, 11F70, 22E45.

H. Bursztyn and A. Weinstein. Picard Groups in Poisson Geometry [PDF]

We study isomorphism classes of symplectic dual pairs PS → \overline{P}, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed P, these Morita self-equivalences of P form a group Pic(P) under a natural "tensor product" operation. Variants of this construction are also studied, for rings (the origin of the notion of Picard group), Lie groupoids, and symplectic groupoids.

Keywords. Picard group, Morita equivalence, Poisson manifold, symplectic groupoid, bimodule.

2000 Mathematics Subject Classification. Primary 53D17, 58H05; Secondary 16D90.

A. Connes and H. Moscovici. Modular Hecke Algebras and Their Hopf Symmetry [PDF]

We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of `polynomial coordinates' for the `transverse space' of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. Its action is shown to span the `holomorphic tangent space' of the noncommutative space, and each of its three basic Hopf cyclic cocycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin—Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon—Vey cocycle gives rise to a 1-cocycle on PSL(2,Q) with values in Eisenstein series of weight 2, which, when coupled with the `period' cocycle, yields a representative of the Euler class.

Keywords. Modular forms, Hecke correspondences, transverse geometry, Hopf cyclic homology, Dedekind eta function, Schwarzian cocycle, Euler class of SL(2,Q), Dedekind sums.

2000 Mathematics Subject Classification. 11F32, 11F75, 58B34.

A. Connes and H. Moscovici. Rankin—Cohen Brackets and the Hopf Algebra of Transverse Geometry [PDF]

We settle in this paper a question left open in our paper "Modular Hecke algebras and their Hopf symmetry", by showing how to extend the Rankin—Cohen brackets from modular forms to modular Hecke algebras. More generally, our procedure yields such brackets on any associative algebra endowed with an action of the Hopf algebra of transverse geometry in codimension one, such that the derivation corresponding to the Schwarzian derivative is inner. Moreover, we show in full generality that these Rankin—Cohen brackets give rise to associative deformations.

Keywords. Rankin—Cohen brackets, modular Hecke algebras, Hopf symmetry, inner Schwarzian cocycle, quadratic differential, transverse fundamental class, Rankin—Cohen deformations of algebras.

2000 Mathematics Subject Classification. 11F32, 11F75, 58B34.

D. Foata and G.-N. Han. Une Nouvelle Transformation pour les Statistiques Euler—Mahoniennes Ensemblistes [PDF]

The construction of a bijection of the symmetric group onto itself is given that has the property of mapping a pair of set-statistics onto another pair. As a consequence, it is shown that a pair of Euler—Mahonian statistics has a symmetric distribution.

Keywords. Permutation statistics, Euler—Mahonian statistics, symmetric groups, set-valued statistic equidistribution.

2000 Mathematics Subject Classification. 05Axx, 05A30, 20B30.

G. Lusztig. Parabolic Character Sheaves, I [PDF]

We study a class of perverse sheaves on the variety of pairs (P,gUP) where P runs through a conjugacy class of parabolics in a connected reductive group and gUP runs through G/UP. This is a generalization of the theory of character sheaves.

Keywords. Reductive group, parabolic group, perverse sheaf, character sheaf.

2000 Mathematics Subject Classification. 20G99.

Yu. Manin. Moduli Stacks \overline{L}g,S [PDF]

This paper is a sequel to the paper by A. Losev and Yu. Manin, in which new moduli stacks \overline{L}g,S of pointed curves were introduced. They classify curves endowed with a family of smooth points divided into two groups, such that the points of the second group are allowed to coincide. The homology of these stacks form components of the extended modular operad whose combinatorial models are further studied in another paper by Losev and Manin. In this paper the basic geometric properties of \overline{L}g,S are established using the notion of weighted stable pointed curves introduced recently by B. Hassett. The main result is a generalization of Keel's and Kontsevich—Manin's theorems on the structure of H*(\overline{M}0,S).

Keywords. Stable pointed curves, moduli spaces, generalized Keel's relations.

2000 Mathematics Subject Classification. Primary 14N35; Secondary 14H10, 53D45.

F. Patras and C. Reutenauer. On Descent Algebras and Twisted Bialgebras [PDF]

Bialgebras in the category of tensor species (twisted bialgebras) deserve a particular attention, in particular in view of applications to algebraic combinatorics. In order to study these bialgebras, a new class of descent algebras is introduced. The fine structure of Barratt's permutation bi-ring (the direct sum of the symmetric group algebras) is investigated in detail from this point of view, leading to the definition of an enveloping algebra structure on it.

Keywords. Descent algebra, tensor species, symmetric group, permutation bi-ring, free Lie algebra.

2000 Mathematics Subject Classification. Primary 16W30, 05E10; Secondary 17A30, 17B01, 17B35.

C. Soulé. Les Variétés sur le Corps à un Élément [PDF]

We propose a definition of varieties over "the field with one element", a notion which had been imagined by Tits, Manin and others. Such a variety has an extension to the integers which is a usual algebraic variety. Examples include smooth toric varieties and euclidean lattices. We also define and compute a zeta function for these objects, and we propose a motivic interpretation of the image of Adams J-homomorphism.

Keywords. Algebraic varieties, toric varieties, euclidean lattices, zeta functions, J-homomorphism.

2000 Mathematics Subject Classification. 14A99, 14M25, 11M99, 55Q50.

M. Waldschmidt. Open Diophantine Problems [PDF]

Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results.

We collect here a number of open questions concerning Diophantine equations (including Pillai's Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel's Conjecture). Some questions related to Mahler's measure and Weil absolute logarithmic height are then considered (e.g., Lehmer's Problem). We also discuss Mazur's question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.

Keywords. Diophantine problems, transcendence, linear and algebraic independence, Schanuel conjecture, four exponential conjecture, multizeta values, abc conjecture, Waring problem, Diophantine approximation, continued fractions, Thue—Siegel—Roth—Schmidt, irrationality measures, Markoff spectrum, height, Lehmer problem, Mazur density conjecture, abelian varieties, special functions, function fields.

2000 Mathematics Subject Classification. Primary 11Jxx; Secondary 11Dxx, 11Gxx, 14Gxx.

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