Endomorphisms of abelian varieties, cyclotomic extensions and Lie algebras

We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.Comment: 9 page

Automorphisms and forms of simple infinite-dimensional linearly compact Lie superalgebras

We describe the group of continuous automorphisms of all simple infinite-dimensional linearly compact Lie superalgebras and use it in order to classify F-forms of these superalgebras over any field F of characteristic zero.Comment: 24 page

Finiteness properties of cubulated groups

We give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let H_1,...,H_s be relatively quasiconvex codimension-1 subgroups of a group G that is hyperbolic relative to P_1,...,P_r. We prove that G acts relatively cocompactly on the associated dual CAT(0) cube complex C. This generalizes Sageev's result that C is cocompact when G is hyperbolic. When P_1,...,P_r are abelian, we show that the dual CAT(0) cube complex C has a G-cocompact CAT(0) truncation.Comment: 58 pages, 12 figures. Version 3: Revisions and slightly improved results in Sections 7 and 8. Several theorem numbers have changed from the previous versio

Homotopy nilpotent groups

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $\pi_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $\Omega F$ have values in homotopy n-nilpotent groups.Comment: 16 pages, uses xy-pic, improved exposition, submitte

Experience in the exploitation of a large control system

Experience of a four-year exploitation of the large control system of the CERN PS accelerator complex is presented with special emphasis on the parameters which are very sensitive to the exploitation team productivity. The software tools suite used in this daily maintenance is described and a particular analysis of the power and benefits of advance software technology used for the architecture of this suite is explained. The integration of this suite in the Control System is presented, as well as its use in the Control System development phase. Some considerations of the potential benefit of an Object Oriented equipment access are outlined

On p-adic lattices and Grassmannians

It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive group G over a field k, carry the geometric structure of an inductive limit of projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for G. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of p-typical Witt vectors, k is a perfect field of characteristic p and G is a reductive group scheme over W(k). The present paper is an attempt to describe which constructions carry over from the function field case to the p-adic case, more precisely to the situation of the p-adic affine Grassmannian for the special linear group G=SL_n. We start with a description of the R-valued points of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R), where R is a perfect k-algebra. In order to obtain a link with geometry we further construct projective k-subvarieties of the multigraded Hilbert scheme which map equivariantly to the p-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the p-adic setting. Further, for any reduced k-algebra R these morphisms induce bijective maps between the sets of R-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the p-adic affine Grassmannian for SL_n.Comment: 36 pages. This is a thorough revision, in the form accepted by Math. Zeitschrift, of the previously published preprint "On p-adic loop groups and Grassmannians

Integral closure of rings of integer-valued polynomials on algebras

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic polynomial of least degree such that $\mu_a(a)=0$. The ring ${\rm Int}_K(A)$ consists of polynomials in $K[X]$ that send elements of $A$ back to $A$ under evaluation. If $D$ has finite residue rings, we show that the integral closure of ${\rm Int}_K(A)$ is the ring of polynomials in $K[X]$ which map the roots in an algebraic closure of $K$ of all the $\mu_a(X)$, $a\in A$, into elements that are integral over $D$. The result is obtained by identifying $A$ with a $D$-subalgebra of the matrix algebra $M_n(K)$ for some $n$ and then considering polynomials which map a matrix to a matrix integral over $D$. We also obtain information about polynomially dense subsets of these rings of polynomials.Comment: Keywords: Integer-valued polynomial, matrix, triangular matrix, integral closure, pullback, polynomially dense set. accepted for publication in the volume "Commutative rings, integer-valued polynomials and polynomial functions", M. Fontana, S. Frisch and S. Glaz (editors), Springer 201

Two limit cases of Born-Infeld equations

International audienceWe study two limit cases \l \rightarrow \infty and \l \rightarrow 0 in Born-Infeld equations. Here the parameter \l >0 is interpreted as the maximal electric field in the electromagnetic theory and the case \l = 0 corresponds to the string theory. Formal limits are governed by the classical Maxwell equations and pressureless magnetohydrodynamics system, respectively. For studying the limit \l \rightarrow \infty, a new scaling is introduced. We give the relations between these limits and Brenier high and low field limits. Finally, using compensated compactness arguments, the limits are rigorously justified for global entropy solutions in $L^\infty$ in one space dimension, based on derived uniform estimates and techniques for linear Lagrangian systems

Strongly bounded groups and infinite powers of finite groups

We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the so-called uncountable strong cofinality, recently introduced by G. Bergman. Our main result is that G^I is strongly bounded when G is a finite, perfect group and I is any set. This strengthens a result of Koppelberg and Tits. We also prove that omega_1-existentially closed groups are strongly bounded.Comment: 10 pages, no figure. Versions 1-3 were entitled "Uncountable groups with Property (FH)". To appear in Comm. Algebr