Action of Hecke operators on two distinguished Drinfeld modular forms

We study the action of Hecke operators on certain non-standard Fourier expansions for the Drinfeld-Eisenstein series E _q-1 and the Drinfeld discriminant function Δ, and we find an equation which “explains” an old result of D. Goss: these two distinct modular forms have the same eigenvalues.     

Chtoucas de Drinfeld et correspondance de Langlands

One proves Langlands’ correspondence for GL _r over function fields. This is a generalization of Drinfeld’s proof in the case of rank 2 : Langlands’ correspondence is realized in ℓ-adic cohomology spaces of the modular varieties classifying rank r Drinfeld shtukas. Oblatum 13-X-2000 & 7-VI-2001?Published online: 12 October 2001     

Congruence properties of the coefficients of the Drinfeld modular invariant

 Let be the expansion at infinity of the Drinfeld modular invariant. We know that the coefficients c _n 's are in the polynomial ring 𝔽_q[T]. In this text, we prove for these coefficients congruence properties modulo powers of p, where p  𝔽 _q [T] is a polynomial of degree 1. Received: 19 April 2002 Mathematics Subject Classification (2000): 11F52, 11F33, 11G09     

Special values of Drinfeld modular forms and algebraic independence

Let A be a polynomial ring in one variable over a finite field and k be its fraction field. Let f be a Drinfeld modular form of nonzero weight for a congruence subgroup of GL₂(A) so that the coefficients of the q _∞-expansion of f are algebraic over k. We consider n CM points α ₁, . . . , α _n on the Drinfeld upper half plane for which the quadratic fields k(α ₁), . . . , k(α _n ) are pairwise distinct. Suppose that f is non-vanishing at these n points. Then we prove that f(α ₁), . . . , f(α _n ) are algebraically independent over k.     

Tensor products of Drinfeld modules andv-adic representations

We define the tensor product ϕ ⊗ ψ and relatedt-modules Sym²(ϕ), and ∧²(ϕ) for Drinfeld modules ϕ, ψ defined over the rational function fieldK=F _q (T), and describe thev-adic Tate modules of theset-modules by using those of ϕ, ψ.     

Linear relations and congruences for the coefficients of Drinfeld modular forms

We find congruences for the t-expansion coefficients of Drinfeld modular forms for . We give generalized analogies of Siegel’s classical observation on SL ₂(ℤ) by determining all the linear relations among the initial t-expansion coefficients of Drinfeld modular forms for . As a consequence spaces M _k ⁰ are identified, in which there are congruences for the s-expansion coefficients. This work was supported by KOSEF R01-2006-000-10320-0 and by the Korea Research Foundation Grant (KRF-2005-214-M01-2005-000-10100-0)     

Links between cofinite prime ideals in quantum function algebras

We describe the skew primitive elements in a multiparameter enveloping algebraU=U _q,p ⁻¹ (g) and the links between cofinite maximal ideals in the corresponding quantum function algebra ℂ _q [G]. These results are applied to determine the coradical filtration forU, and to obtain a moduli space for multiparameter Drinfeld doubles. Research partially supported by NSA grant MDA 904-93-H3016.     

Quasi-hopf algebras associated withsl 2 and complex curves

We construct quasi-Hopf algebras quantizing double extensions of the Manin pairs of Drinfeld, associated to a curve with a meromorphic differential, and the Lie algebrasl ₂. This construction makes use of an analysis of the vertex relations for the quantum groups obtained in our earlier work, PBW-type results and computation ofR-matrices for them; its key step is a factorization of the twist operator relating “conjugated” versions of these quantum groups.     

Hereditary Torsion Theory of Pseudo-Regular S -Systems

Abstract. The concept of pseudo-regular S -system is introduced. A torsion theory τ _u = ( U _S , \bar U _S ) with its torsion class U _S consisting of pseudo-regular S -systems, is constructed. Its corresponding quasi-filter is completely described. A number of results on the relations between τ _u and two special torsion theories, the stable and Lambek torsion theories, are obtained.     

Central units in salternative loop rings

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. We show that every central unit in the integral loop ring ZL is the product ℓμ₀ of an element ℓ ∈ L and a loop ring element μ₀ whose support is in the torsion subloop of L and use this result to determine when all central units of ZL are trivial. Received: 8 October 2004     

Hauteurs normalisées des sous-variétés de produits de modules de Drinfeld

After establishing some important results on the usual height of projective varieties in positive characteristic, we construct a normalized height for subvarieties of products of Drinfeld modules and investigate its properties. In case the Drinfeld modules are pairwise isogeneous, we obtain in particular that the normalized height vanishes exactly on torsion varieties, that is on translates of sub-T-modules by torsion points.     

Extending modules relative to a torsion theory

An R-module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ-extending module, where τ is a hereditary torsion theory on Mod-R. An R-module M is called type 2 τ-extending if every type 2 τ-closed submodule of M is a direct summand of M. If τ _I is the torsion theory on Mod-R corresponding to an idempotent ideal I of R and M is a type 2 τ _I -extending R-module, then the question of whether or not M/MI is an extending R/I-module is investigated. In particular, for the Goldie torsion theory τ _G we give an example of a module that is type 2 τ _G -extending but not extending.     

Théorie de Lubin-Tate non-abélienne et représentations elliptiques

Non abelian Lubin–Tate theory studies the cohomology of some moduli spaces for p-divisible groups, the broadest definition of which is due to Rapoport–Zink, aiming both at providing explicit realizations of local Langlands functoriality and at studying bad reduction of Shimura varieties. In this paper we consider the most famous examples ; the so-called Drinfeld and Lubin–Tate towers. In the Lubin–Tate case, Harris and Taylor proved that the supercuspidal part of the cohomology realizes both the local Langlands and Jacquet–Langlands correspondences, as conjectured by Carayol. Recently, Boyer computed the remaining part of the cohomology and exhibited two defects : first, the representations of GL _d which appear are of a very particular and restrictive form ; second, the Langlands correspondence is not realized anymore. In this paper, we study the cohomology complex in a suitable equivariant derived category, and show how it encodes Langlands correspondence for elliptic representations. Then we transfer this result to the Drinfeld tower via an enhancement of a theorem of Faltings due to Fargues. We deduce that Deligne’s weight-monodromy conjecture is true for varieties uniformized by Drinfeld’s coverings of his symmetric spaces. This completes the computation of local L-factors of some unitary Shimura varieties.     

On constant <Emphasis Type="Italic">Uq</Emphasis>(<Emphasis Type="Italic">sl</Emphasis><Subscript>2</Subscript>)-invariant <Emphasis Type="Italic">R</Emphasis>-matrices

We consider the spectral resolution of a Uq (sl ₂)-invariant solution R of the constant Yang–Baxter equation in the braid group form. It is shown that if the two highest coefficients in this resolution are not equal, then R is either the Drinfeld R-matrix or its inverse. Bibliography: 13 titles.     

Gruppo delle terne quasi pitagoriche primitive

We study the groupG _m of primitive solution of the diophantine equationx ²+my²=z² (m>1, squarefree). Form∈3 this group is torsion free, form=3 it has a torsion element of order 3; moreover for a finite number of values ofm we prove thatG _m is a direct sum of infinite cyclic groups and we give the generators ofG _m in terms of the primes represented by the quadratic forms of discriminant Δ=−4m.      

A note on reidemeister torsion and period matrix of Riemann surfaces

We consider compact Riemann surfaces Σ _g with genus at least 2. We explain the relation between the Reidemeister torsion of Σ _g and its period matrix.     

A characterization of ℵ1-free abelian groups and its application to the chase radical

A groupA is an ℵ₁-free abelian group iffA is a subgroup of the Boolean power Z^(B) for some complete Boolean algebraB. The Chase radicalvA=Σ{C≦A: Hom(C, Z)=0 &C is countable). The torsion class {A:vA=A} is not closed under uncountable direct products.     

Equidistribution for Torsion Points of a Drinfeld Module

We prove an equidistribution result for torsion points of Drinfeld modules of generic characteristic. We also show that similar equidistribution statements would provide proofs for the Manin–Mumford and the Bogomolov conjectures for Drinfeld modules.     

Uniform boundedness of <Emphasis Type="Italic">p</Emphasis>-primary torsion of abelian schemes

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E _s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves.     

Explicit Classification for Torsion Subgroups of Rational Points of Elliptic Curves

We study the classification of elliptic curves E over the rationals ℚ according to the torsion sugroups E _tors(ℚ). More precisely, we classify those elliptic curves with E _tors(ℚ) being cyclic with even orders. We also give explicit formulas for generators of E _tors(ℚ). These results, together with the recent results of K. Ono for the non-cyclic E _tors(ℚ), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2. Received July 29, 1999, Revised March 9, 2001, Accepted July 20, 2001