Formal Drinfeld modules

The formal group of an elliptic curve at a finite prime of the field of definition has proven to be a useful tool in studying the elliptic curve. Moreover, these formal groups are interesting in themselves. In this paper we define and study formal Drinfeld modules in a general setting. We also define the formal Drinfeld module associated to a Drinfeld module at a finite prime. The results are applied to the uniform boundedness conjecture for Drinfeld modules.     

Prime representations from a homological perspective

We explore the relation between self extensions of simple representations of quantum affine algebras and the property of a simple representation being prime. We show that every nontrivial simple representation has a nontrivial self extension. Conversely, we prove that if a simple representation has a unique nontrivial self extension up to isomorphism, then its Drinfeld polynomial is a power of the Drinfeld polynomial of a prime representation. It turns out that, in the $\mathfrak{sl }_2$ -case, a simple module is prime if and only if it has a unique nontrivial self extension up to isomorphism. It is tempting to conjecture that this is true in general and we present a large class of prime representations satisfying this homological property.     

Stable anti-Yetter–Drinfeld modules

We define and study a class of entwined modules (stable anti-Yetter–Drinfeld modules) that serve as coefficients for the Hopf-cyclic homology and cohomology. In particular, we explain their relationship with Yetter–Drinfeld modules and Drinfeld doubles. Among sources of examples of stable anti-Yetter–Drinfeld modules, we find Hopf–Galois extensions with a flipped version of the Miyashita–Ulbrich action. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Stacky abelianization of algebraic groups

We prove a conjecture of Drinfeld regarding restriction of central extensions of an algebraic group G to the commutator subgroup. As an application, we construct the “true commutator” of G. The quotient of G by the action of the true commutator is the universal commutative group stack to which G maps.     

Central Extensions of Precrossed Modules

We classify the precrossed module central extensions using the second cohomology group of precrossed modules. We relate these central extensions to the relative central group extensions of Loday, and to other notions of centrality defined in general contexts. Finally we establish a Universal Coefficient Theorem for the (co)homology of precrossed modules, which we use to describe the precrossed module central extensions in terms of the generalized Galois theory developed by Janelidze.     

The local Lehmer inequality for Drinfeld modules

We give a lower bound for the local height of a nontorsion element of a Drinfeld module.     

Explicit formulas for Drinfeld modules and their periods

We provide explicit series expansions for the exponential and logarithm functions attached to a rank r Drinfeld module that generalize well-known formulas for the Carlitz exponential and logarithm. Using these results we obtain a procedure and an analytic expression for computing the periods of rank 2 Drinfeld modules and also a criterion for supersingularity.     

On field extensions given by periods of Drinfeld modules

In this short note, we answer a question raised by M. Papikian on a universal upper bound for the degree of the extension of $$K_\infty $$ given by adjoining the periods of a Drinfeld module of rank 2. We show that contrary to the rank 1 case such a universal upper bound does not exist, and the proof generalies to higher rank. Moreover, we give an upper and lower bound for the extension degree depending on the valuations of the defining coefficients of the Drinfeld module. In particular, the lower bound shows the non-existence of a universal upper bound.     

The Tate-Voloch conjecture for Drinfeld modules

We study the v-adic distance from the torsion of a Drinfeld module to an affine variety.     

A Dirichlet unit theorem for Drinfeld modules

We show that the module of integral points on a Drinfeld module satisfies an analogue of Dirichlet’s unit theorem, despite its failure to be finitely generated. As a consequence, we obtain a construction of a canonical finitely generated sub-module of the module of integral points. We use the results to give a precise formulation of a conjectural analogue of the class number formula.     

Congruences between derivatives of geometric <Emphasis Type="Italic">L</Emphasis>-functions

We prove a natural refinement of a theorem of Lichtenbaum describing the leading terms of Zeta functions of curves over finite fields in terms of Weil-étale cohomology. We then use this result to prove the validity of Chinburg’s Ω(3)-Conjecture for all abelian extensions of global function fields, to prove natural refinements and generalisations of the refined Stark conjectures formulated by, amongst others, Gross, Tate, Rubin and Popescu, to prove a variety of explicit restrictions on the Galois module structure of unit groups and divisor class groups and to describe explicitly the Fitting ideals of certain Weil-étale cohomology groups. In an Appendix coauthored with K.F. Lai and K.-S. Tan we also show that the main conjectures of geometric Iwasawa theory can be proved without using either crystalline cohomology or Drinfeld modules.     

Monoidal Morita invariants for finite group algebras

Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer n, the number of elements of order n is a monoidal Morita invariant for finite group algebras. We also describe relations between our construction and invariants of closed 3-manifolds due to Reshetikhin and Turaev.     

Modular categories from finite crossed modules

It is known that finite crossed modules provide premodular tensor categories. These categories are in fact modularizable. We construct the modularization and show that it is equivalent to the module category of a finite Drinfeld double.     

Hopf-cyclic homology and cohomology with coefficients

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter–Drinfeld compatibility condition leading to the concept of a stable anti-Yetter–Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of a module algebra. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Torsion bounds for elliptic curves and Drinfeld modules

We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.     

Deligne’s congruence and supersingular reduction of Drinfeld modules

Most rank two Drinfeld modules are known to have infinitely many supersingular primes. But how many supersingular primes of a given degree can a fixed Drinfeld module have? In this paper, a congruence between the Hasse invariant and a certain Eisenstein series is used for obtaining a bound on the number of such supersingular primes. Certain exceptional cases correspond to zeros of certain Eisenstein series with rational j-invariants.     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1.     

Essential extensions, excellent extensions and finitely presented dimension

In Section 1 of this paper, we investigate the finitely presented dimension of an essential extension for a module, and obtain results concerning an essential extension of a torsion-free module. We partially answer the question: When is an essential extension of a finitely presented module (an almost finitely presented module) also finitely presented (almost finitely presented)? In Section 2, we study theC-excellent extensions and the finitely presented dimensions. We obtain some results on the homological dimensions of matrix rings and group rings.     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1. Received 21 July 1997; in revised form 5 February 1998     

Extensions galoisiennes non commutatives :normalité cohomologie non abélienne

We study various types of noncommutative Galois extensions and present some examples. We state a criterion which decides whether a given automorphism of a Galois extension belongs to the corresponding Galois group or not. We then compute Borovoi’s nonabelian cohomology sets (in degree 0, 1, 2) of the Galois group with coefficients in a crossed module associated to the Galois extension.