On nilpotent Moufang loops with central associators

In this paper, we investigate Moufang p-loops of nilpotency class at least three for $p>3$. The smallest examples have order $p^5$ and satisfy the following properties: (1) They are of maximal nilpotency class, (2) their associators lie in the center, and (3) they can be constructed using a general form of the semidirect product of a cyclic group and a group of maximal class. We present some results concerning loops with these properties. As an application, we classify proper Moufang loops of order $p^5$, $p>3$, and collect information on their multiplication groups.     

Elliptic associators

We construct a genus one analogue of the theory of associators and the Grothendieck–Teichmüller (GT) group. The analogue of the Galois action on the profinite braid groups is an action of the arithmetic fundamental group of a moduli space of elliptic curves on the profinite braid groups in genus one. This action factors through an explicit profinite group $\widehat{\mathrm{GT }}_{ell}$ , which admits an interpretation in terms of decorations of braided monoidal categories. This group acts on the tower of profinite braid groups in genus one and has the structure of a semidirect product of the profinite GT group $\widehat{\mathrm{GT }}$ by an explicit radical. We relate $\widehat{\mathrm{GT }}_{ell}$ to its prounipotent group scheme version $\mathrm{GT }_{ell}(-)$ , which also has a semidirect product structure. We construct a torsor over this group, the scheme of elliptic associators. An explicit family of elliptic associators is constructed, based on earlier joint work with Calaque and Etingof on the universal KZB connexion. The existence of elliptic associators enables one to show that the Lie algebra of $\mathrm{GT }_{ell}(-)$ is isomorphic to a graded Lie algebra, on which we obtain several results: it is a semidirect product of the graded GT Lie algebra $\mathfrak grt $ by an explicit radical; we exhibit an explicit Lie subalgebra. Elliptic associators also allow one to compute the Zariski closure of the mapping class group in genus one (isomorphic to the braid group $B_{3}$ ) in the automorphism groups of the prounipotent completions of braid groups in genus one. The analytic study of the family of elliptic associators produces relations between MZVs and iterated integrals of Eisenstein series.     

On associators and the Grothendieck-Teichmuller group, I

We present a formalism within which the relationship (discovered by Drinfel'd in [Dr1], [Dr2]) between associators (for quasi-triangular quasi-Hopf algebras) and (a variant of) the Grothendieck-Teichmuller group becomes simple and natural, leading to a simplification of Drinfel'd's original work. In particular, we reprove that rational associators exist and can be constructed iteratively, though the proof itself still depends on the apriori knowledge that a not-necessarily-rational associator exists.     

On the Drinfeld discriminant function

The discriminant function is a certain rigid analytic modularform defined on Drinfelds upper half-plane . Its absolutevalue may be considered as a function on theassociated Bruhat–Tits tree T. We compare log with the conditionally convergent complex-valued Eisenstein series Edefined on T and thereby obtain results about the growth of and of some related modular forms. We further determine to what extent roots may be extracted of (z)/(nz),regarded as a holomorphic function on . In some cases, this enables us to calculate cuspidal divisor class groups of modular curves.     

On two problems related to associators of Moufang loops

Mathematical Notes - A Moufang loop M of order 319 is constructed, together with a pair a, b of elements of M, such that the set of all elements of M associating with a and b is not a subloop. This...     

On certain families of Drinfeld quasi-modular forms

This paper deals with two problems arising in the study of Drinfeld quasi-modular forms. The first problem is to find the maximal order of vanishing at infinity of a non-zero Drinfeld quasi-modular form and leads to the notion of “extremal” quasi-modular form (highest possible order of vanishing for fixed weight and depth). The second problem is determining differential properties of extremal forms, leading to the notion of “differentially extremal” form. From our investigations, we will obtain an upper bound for the order of vanishing at infinity of non-zero Drinfeld quasi-modular forms of small depths. The paper ends with a collection of tools used in the previous parts. The notion of “extremal” form is similar to one introduced by Kaneko and Koike in [M. Kaneko, M. Koike, On extremal quasimodular forms, Kyushu J. Math. 60 (2006) 457-470].     

On the Support Problem for Drinfeld Modules

Let Φ be a Drinfeld A-module over an A-field K of generic characteristic. We will prove the following two results which are analogous to ones in number fields. Case 1. Φ is of rank one. Suppose that P and Q are two nontorsion points in Φ(K). If for any element a ? A and almost all prime ideals 𝒫 in  one has that Φ _a (P) ≡ 0 (mod 𝒫) ? Φ _a (Q) ≡ 0 (mod 𝒫), then Q = Φ _m (P) for some m ? A. Case 2. Φ is of general rank ≥ 1. Let x, y ? Φ(K) be two K-rational points. Denote  = End _K (Φ) which is commutative and Λ =  · y which is a cyclic -module. Let red _v :Φ(K) → Φ(k _v ) be the reduction map at a place v of K with residue field k _v . If red _v (x) ? red _v (Λ) for almost all places v of K. Then f(x) = g(y), for some nonzero elements f and g in .     

Quasi-reflection algebras and cyclotomic associators

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism , where B _n ¹ is a braid group of type B. The formality isomorphism depends algebraically on a series Ψ_KZ, the “KZ pseudotwist”. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N, k), mapping to Drinfeld’s group GT(k), and whose Lie algebra is isomorphic to its associated graded (N, k). We prove that Ihara’s subgroup GTK of the Grothendieck–Teichmüller group, defined using distribution relations, in fact coincides with it. We show that the subscheme of pseudotwists satisfying distribution relations is a subtorsor. We study the corresponding analogue (N, k) of (N, k); it is a graded Lie algebra with an action of , and we give a lower bound for the character of its space of generators.      

On plane models for Drinfeld modular curves

In this work, we find plane models for certain Drinfeld modular curves X₀(n) which have better properties than the plane models derived from the usual Drinfeld modular equations. As an application, we construct ring class fields over imaginary quadratic fields by using singular values of generators of the function field of X₀(n).     

On the uniform boundedness conjecture for Drinfeld modules

 In analogy with the famous theorems of Mazur and Merel on the torsion subgroups of elliptic curves, one can formulate similar conjectures for the torsion points of Drinfeld modules. We prove some partial results for rank 2 Drinfeld 𝔽 _q [T]-modules, for example the uniform boundedness of the 𝔭-primary torsion. Received: 22 May 2001; in final form 4 September 2002 / Published online: 1 April 2003     

On primitive roots for rank one Drinfeld modules

Let k be a global function field over a finite field and let A be the ring of the elements in k regular outside a fixed place ∞. Let K be a global A-field of finite A-characteristic and let ? be a rank one Drinfeld A-module over K. Given any α∈K, we show that the set of places P of K for which α is a primitive root modulo P under the action of ? possesses a Dirichlet density. We also give conditions for this density to be positive.     

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.     

On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

Let K be a function field over finite field and let be a ring consisting of elements of K regular away from a fixed place of K. Let be a Drinfeld -module defined over an -field L. In the case where L is a finite -field, we study the characteristic polynomial of the geometric Frobenius. A formula for the sign of the constant term of in terms of leading coefficient of is given. General formula to determine signs of other coefficients of is also derived. In the case where L is a global -field of generic characteristic, we apply these formulae to compute the Dirichlet density of places where the Frobenius traces have the maximal possible degree permitted by the Riemann hypothesis.     

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.