Hecke actions on certain strongly modular genera of lattices

We calculate the action of some Hecke operators on spaces of modular forms spanned by the Siegel theta-series of certain genera of strongly modular lattices closely related to the Leech lattice. Their eigenforms provide explicit examples of Siegel cusp forms.Received: 10 September 2004     

Origin and infinity manifolds for mechanical systems with homogeneous potentials

We study manifolds describing the behavior of motions close to the origin and at infinity of configuration space, for mechanical systems with homogeneous potentials. We find an inversion between these behaviors when the sign of the degree of homogeneity is changed. In some cases, the blow up equations can be written in canonical form, by first reducing to a contact structure. A motivation for the use of blow-up techniques is given, and some examples are studied in detail.Research partially supported by CONACyT (Mexico), under grants PCCBNAL 790178 and PCCBBNA 022553.Member of CIFMA (Mexico). On sabbatical leave at the University of Barcelona during the year 1987–88.     

Intersections of Magnus subgroups and embedding theorems for cyclically presented groups

It is now known that the intersection of two Magnus subgroups M_i=〈Y_i〉 (1≤i≤2) in a one-relator group is either the free group F on Y₁∩Y₂ or the free product of F together with an infinite cyclic group (so-called exceptional intersection). Using this, we give conditions under which two embedding theorems for cyclically presented groups can be obtained. This provides a new method for proving such groups infinite. We also give a combinatorial method for checking the presence of exceptional intersections.     

Maximal words connected with unimodal maps

Orbits under a unimodal function are commonly characterized by letters of the alphabet (L, C, R). We present an algorithm for recognizing maximality of a word and study the necessary extension of the space of admissible words.     

Vanishing periods of cusp forms over modular symbols

partially supported by NSF grant #DMS-8919696     

Boolean Monomial Dynamical Systems

An important problem in the theory of finite dynamical systems is to link the structure of a system with its dynamics. This paper contains such a link for a family of nonlinear systems over the field with two elements. For systems that can be described by monomials (including Boolean AND systems), one can obtain information about the limit cycle structure from the structure of the monomials. In particular, the paper contains a sufficient condition for a monomial system to have only fixed points as limit cycles. This condition depends on the cycle structure of the dependency graph of the system and can be verified in polynomial time.Received February 2, 2004     

The modular hierarchy of the Toda lattice

The modular vector field plays an important role in the theory of Poisson manifolds and is intimately connected with the Poisson cohomology of the space. In this paper we investigate its significance in the theory of integrable systems. We illustrate in detail the case of the Toda lattice both in Flaschka and natural coordinates.     

Exact formulas for traces of singular moduli of higher level modular functions

Zagier proved that the traces of singular values of the classical j-invariant are the Fourier coefficients of a weight 3/2 modular form and Duke provided a new proof of the result by establishing an exact formula for the traces using Niebur's work on a certain class of non-holomorphic modular forms. In this short note, by utilizing Niebur's work again, we generalize Duke's result to exact formulas for traces of singular moduli of higher level modular functions.     

On simultaneous non-vanishing of modular <Emphasis Type="Italic">L</Emphasis>-functions

For i = 1, , r, let f _i be newforms of weight 2k _i for Γ₀(N _i ) with trivial character. We consider the simultaneous non-vanishing problem for the central values of twisted L-functions of f _i . By using the Shimura correspondence, we give a certain relation between this problem and the kernel fields of 2-adic Galois representations associated to modular forms. Received: 28 January 2006     

Characterizing subgroups of compact abelian groups

We prove that every countable subgroup of a compact metrizable abelian group has a characterizing set. As an application, we answer several questions on maximally almost periodic (MAP) groups and give a characterization of the class of (necessarily MAP) abelian topological groups whose Bohr topology has countable pseudocharacter.     

On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.     

Interaction cohomology of forward or backward self-similar systems

We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings U of the invariant set, where each U consists of (backward) images of the invariant set under elements of finite word length. We give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of semigroups of polynomials. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given.     

Maass–Jacobi Poincaré series and Mathieu Moonshine

Mathieu moonshine attaches a weak Jacobi form of weight zero and index one to each conjugacy class of the largest sporadic simple group of Mathieu. We introduce a modification of this assignment, whereby weak Jacobi forms are replaced by semi-holomorphic Maass–Jacobi forms of weight one and index two. We prove the convergence of some Maass–Jacobi Poincaré series of weight one, and then use these to characterize the semi-holomorphic Maass–Jacobi forms arising from the largest Mathieu group.     

The modular automorphisms of the Drinfeld modular curve $$ X_{1}(\frak n) $$

In this work we determine the group of modular automorphisms of the Drinfeld modular curve for any nonconstant polynomial .Received: 19 September 2002     

Variants of recognition problems for modular forms

We consider the problem of distinguishing two modular forms, or two elliptic curves, by looking at the coefficients of their L-functions for small primes (compared to their conductor). Using analytic methods based on large-sieve type inequalities we give various upper bounds on the number of forms having the first few coefficients equal to those of a fixed one. In addition, we consider similar questions of recognizing symmetric squares and CM forms from the behavior of small primes.Received: 30 August 2004     

About the separability of completely integrable quasi-bi-Hamiltonian systems with compact levels

We study completely integrable quasi-bi-Hamiltonian systems whose common level surfaces are compact and prove in particular their separability in the sense of Falqui and Pedroni.