1.
Fields of u-invariant 9 - Izhboldin, Oleg T.
Let $F$ be a field of characteristic $\ne2$. The $u$-invariant of the field $F$ is defined as the maximum dimension of anisotropic quadratic forms over $F$. It is well known that the $u$-invariant cannot be equal to 3, 5, or 7. We construct a field $F$ with $u$-invariant 9. It is a first example of a field with odd u-invariant $>1$. The proof uses the computation of the third Chow group of projective quadrics $X_\phi$ corresponding to quadratic forms $\phi$. We compute $\CH^3(X_\phi)$ for all $\phi$ except for the case $\dim\phi=8$. In our computation we use results of B. Kahn,...
2.
Integration on $\scr H\sb p\times\scr H$ and arithmetic applications - Darmon, Henri
This article describes a conjectural $p$-adic analytic construction of global points on (modular) elliptic curves, points which are defined over the ring class fields of real quadratic fields. The resulting conjectures suggest that the classical Heegner point construction, and the theory of complex multiplication on which it is based, should extend to a variety of contexts in which the underlying field is not a CM field.
4.
A question of Rentschler and the Dixmier problem - Bavula, V. V.
The following theorem is the main result of the paper. THEOREM. Let $R$ and $T$ be somewhat commutative algebras with the same holonomic number and let $\v :R\ra T$ be an algebra homomorphism{\rm .} Then every holonomic $T$\/{\rm -}\/module $M$ is {\rm (}\/via $\v$\/{\rm )} a holonomic $R$\/{\rm -}\/module and has finite length as an $R$\/{\rm -}\/module\/{\rm .} When applied to the Weyl algebra this result gives a positive answer to a question of Rentschler. In the important case where $R=\CD (X)$ and $T=\CD (Y)$ are rings of differential operators on smooth irreducible algebraic affine varieties $X$ and $Y$ of...
5.
Invariant differential operators and eigenspace representations on an affine symmetric space - Huang, Jing-Song
Let $G/H$ be an affine symmetric space of split rank $r$. Let $\bold D$ be a preferred polynomial algebra of $G$-invariant differential operators on $G/H$ generated by $r$ elements. We show that the space of $K$-finite joint eigenfunctions of $\bold D$ on $G/H$ form an admissible $(\g,K)$-module which is called an eigenspace representation. The main content of this paper is description of the algebras of invariant differential operators and determination of the eigenspace representations on $G/H$. We also obtain a Poisson transform for $\tau$-spherical eigenfunctions on $G/H$ by Eisenstein integrals.
6.
The Severi bound on sections of rank two semistable bundles on a Riemann surface - Cilleruelo, Javier; Sols, Ignacio
Let $E$ be a semistable, rank two vector bundle of degree $d$ on a Riemann surface $C$ of genus $g\ge 1$, i.e.\ such that the minimal degree $s$ of a tensor product of $E$ with a line bundle having a nonzero section is nonnegative. We give an analogue of Clifford's lemma by showing that $E$ has at most $(d-s)/2 +\delta $ independent sections, where $\delta$ is $2$ or $1$ according to whether the Krawtchouk polynomial $K_r(n,N)$ is zero or not at $r=(d-s)/2+1$, $n=g$, $N=2g-s$ (the analogous bound for nonsemistable rank two bundles being stronger but easier to prove). This gives...
7.
On the differential equations satisfied by weighted orbital integrals - Hoffmann, Werner
Weighted orbital integrals are distributions on reductive groups over local fields appearing both in the local and global trace formulas. There are associated invariant distributions, which play the same role in the invariant trace formulas. In the case of real groups, the Fourier transforms of these distributions satisfy a system of differential equations. As a step towards determining those Fourier transforms, we show that this system is holonomic and has a simple singularity at infinity. We deduce that any solution has a series expansion and is a linear combination of certain canonical solutions. For some groups of small rank, we...
8.
Small amplitude limit cycles and the distribution of zeros of families of analytic functions - Brudnyi, Alexander
We estimate the expected number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin provided that the field in a larger neighbourhood is close enough to a linear center. Our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic functions depending analytically on a parameter. We obtain this inequality by methods of pluripotential theory. This inequality also implies versions of a strong law of large numbers and the central limit theorem for a probabilistic scheme associated with the distribution of zeros.
10.
Global density of reducible quasi-periodic cocycles on T1 x SU(2) - Krikorian, Raphaël
We prove that given $\a$ in a set of total (Haar) measure in ${\T}^1={\R}/{\Z}$, the set of $A\in C^{\infty}({\T}^1,{\rm SU}(2))$ for which the skew-product system $(\a,A):{\T}^1\times {\rm SU}(2)\to {\T}^1\times {\rm SU}(2)$, $(\a,A)(\th,y)=(\th+\a,A(\th)y)$ is reducible --- that is, $A(\cdot)=B(\cdot+\a)A_0B(\cdot)^{-1}$, for some $A_0\in {\rm SU}(2)$, $B\in C^{\infty}({\T}^1,{\rm SU}(2))$,-- is dense for the $C^{\infty}$-topology.
11.
Composition factors of monodromy groups - Frohardt, Daniel; Magaard, Kay
The authors prove the 1990 conjecture of Guralnick and Thompson on composition factors of monodromy groups. Using Riemann's existence theorem, the conjecture translates into a problem on primitive permutation groups. This group theoretic problem had been reduced to a question about actions of classical groups on subspaces of their natural modules. The key ingredients in the present proof are the authors' earlier fixed point ratio estimates for such actions and a result of Scott on the generation of linear groups.
13.
Diameters of finite simple groups: sharp bounds and applications - Liebeck, Martin W.; Shalev, Aner
Let $G$ be a finite simple group and let $S$ be a normal subset of $G$. We determine the diameter of the Cayley graph $\G(G,S)$ associated with $G$ and $S$, up to a multiplicative constant. Many applications follow. For example, we deduce that there is a constant $c$ such that every element of $G$ is a product of $c$ involutions (and we generalize this to elements of arbitrary order). We also show that for any word $w = w(x_1,\ldots , x_d)$, there is a constant $c = c(w)$ such that for any simple group $G$ on which $w$ does not...
14.
Periodic complexes and group actions - Adem, Alejandro; Smith, Jeff H.
In this paper we show that the cohomology of a connected CW-complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on $\Bbb R^n\times \Bbb \bbS^m$; we construct nonstandard free actions of rank-two simple groups on finite complexes $Y\simeq \bbS^n\times\bbS^m$; and we prove that a finite $p$-group $P$ acts freely on such a complex if and only if it does not contain a subgroup isomorphic to $(\bbZ/p)^3$.
15.
Isospectral pairs of metrics on balls, spheres, and other manifolds with different local geometries - Szabó, Z. I.
The first isospectral pairs of metrics are constructed on the most simple simply connected domains, namely, on balls and spheres. This long-standing problem, concerning the existence of such pairs, has been solved by a new method called "anticommutator technique". Among the wide range of such pairs, the most striking examples are provided on the spheres $S^{4k-1}$, where $k\geq 3\,$. One of these metrics is homogeneous (since it is the metric on the geodesic sphere of a 2-point homogeneous space), while the other is locally inhomogeneous. These examples demonstrate the surprising fact that no information about the isometries is encoded in...
16.
Nonvanishing of L-values and the Weyl law - Luo, Wenzhi
This work is concerned with the validity of Weyl law for hyperbolic surfaces on the asymptotic counting of the Laplace eigenvalues. Following Phillips-Sarnak, we show that Weyl law is false for generic hyperbolic surfaces under the standard multiplicity assumption by establishing that a positive proportion of certain critical values of Rankin-Selberg $L$-functions do not vanish.
20.
Stable intersections of regular Cantor sets with large Hausdorff dimensions - de A. Moreira, Carlos Gustavo T.; Yoccoz, Jean-Christophe
In this paper we prove a conjecture by J. Palis according to which the arithmetic difference of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval. More precisely, we prove that if the sum of the Hausdorff dimensions of two regular Cantor sets is bigger than one then, in almost all cases, there are translations of them whose intersection persistently has Hausdorff dimension.
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