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3  Enriques Diagrams and Adjacency of Planar Curve Singularities AlberichCarramiñana, Maria; Roé, Joaquim
We study adjacency of equisingularity types of planar complex
curve singularities
in terms of their Enriques diagrams. The goal is, given two equisingularity
types, to determine whether one of them is adjacent to the other. For linear
adjacency a complete answer is obtained, whereas for arbitrary (analytic)
adjacency a necessary condition and a sufficient condition are
proved. We also obtain new examples of exceptional deformations,
{\em i.e.,} singular curves of type
$\mathcal{D}'$ that can be deformed to a curve of type $\mathcal{D}$ without
$\mathcal{D}'$ being adjacent to $\mathcal{D}$.


17  On Amenability and CoAmenability of Algebraic Quantum Groups and Their Corepresentations Bédos, Erik; Conti, Roberto; Tuset, Lars
We introduce and study several notions of amenability for unitary
corepresentations and $*$representations of algebraic quantum groups,
which may be used to characterize amenability and coamenability for
such quantum groups. As a background for this study, we investigate
the associated tensor C$^{*}$categories.


61  On Operators with Spectral Square but without Resolvent Points Binding, Paul; Strauss, Vladimir
Decompositions of spectral type are
obtained for closed Hilbert space operators with empty resolvent
set, but whose square has closure which is spectral. Krein space
situations are also discussed.


82  Jordan Structures of Totally Nonnegative Matrices Fallat, Shaun M.; Gekhtman, Michael I.
An $n \times n$ matrix is said to be totally nonnegative if every
minor of $A$ is nonnegative. In this paper we completely
characterize all possible Jordan canonical forms of irreducible
totally nonnegative matrices. Our approach is mostly combinatorial
and is based on the study of weighted planar diagrams associated
with totally nonnegative matrices.


99  Second Order Operators on a Compact Lie Group Fegan, H. D.; Steer, B.
We describe the structure of the space of second order elliptic
differential operators on a homogenous bundle over a compact Lie
group. Subject to a technical condition, these operators are
homotopic to the Laplacian. The technical condition is further
investigated, with examples given where it holds and others where
it does not. Since many spectral invariants are also homotopy
invariants, these results provide information about the invariants
of these operators.


114  Bending Flows for Sums of Rank One Matrices Flaschka, Hermann; Millson, John
We study certain symplectic quotients of $n$fold products of
complex projective $m$space by the unitary group acting
diagonally. After studying nonemptiness and smoothness of these
quotients we construct the actionangle variables, defined on an open
dense subset, of an integrable Hamiltonian system. The semiclassical
quantization of this system reporduces formulas from the
representation theory of the unitary group.


159  Duality and Supports of Induced Representations for Orthogonal Groups Jantzen, Chris
In this paper, we construct a duality for $p$adic orthogonal groups.


180  On the Size of the Wild Set Somodi, Marius
To every pair of algebraic number fields with isomorphic Witt rings
one can associate a number, called the {\it minimum number of wild
primes}. Earlier investigations have established lower bounds for this
number. In this paper an analysis is presented that expresses the
minimum number of wild primes in terms of the number of wild dyadic
primes. This formula not only gives immediate upper bounds, but can be
considered to be an exact formula for the minimum number of wild
primes.


204  On the Duality between Coalescing Brownian Motions Xiong, Jie; Zhou, Xiaowen
A duality formula is found for coalescing Brownian motions on the
real line. It is shown that the joint distribution of a coalescing
Brownian motion can be determined by another coalescing Brownian
motion running backward. This duality is used to study a
measurevalued process arising as the high density limit of the
empirical measures of coalescing Brownian motions.


225  Unbounded Fredholm Operators and Spectral Flow BoossBavnbek, Bernhelm; Lesch, Matthias; Phillips, John
We study the gap (= ``projection norm'' = ``graph distance'') topology
of the space of all (not necessarily bounded) selfadjoint Fredholm
operators in a separable Hilbert space by the Cayley transform and
direct methods. In particular, we show the surprising result that
this space is connected in contrast to the bounded case. Moreover, we
present a rigorous definition of spectral flow of a path of such
operators (actually alternative but mutually equivalent definitions)
and prove the homotopy invariance. As an example, we discuss operator
curves on manifolds with boundary.


251  Some New Results on $L^2$ Cohomology of Negatively Curved Riemannian Manifolds Cocos, M.
The present paper is concerned with the study of the $L^2$ cohomology
spaces of negatively curved manifolds. The first half presents a
finiteness and vanishing result obtained under some curvature
assumptions, while the second half identifies a class of metrics
having nontrivial $L^2$ cohomology for degree equal to the half
dimension of the space. For the second part we rely on the existence
and regularity properties of the solution for the heat equation for
forms.


267  Partial Euler Products on the Critical Line Conrad, Keith
The initial version of the Birch and SwinnertonDyer conjecture
concerned asymptotics for partial Euler products for an elliptic curve
$L$function at $s = 1$. Goldfeld later proved that these asymptotics
imply the Riemann hypothesis for the $L$function and that the
constant in the asymptotics has an unexpected factor of $\sqrt{2}$.
We extend Goldfeld's theorem to an analysis of partial Euler products
for a typical $L$function along its critical line. The general
$\sqrt{2}$ phenomenon is related to second moments, while the
asymptotic behavior (over number fields) is proved to be equivalent to
a condition that in a precise sense seems much deeper than the Riemann
hypothesis. Over function fields, the Euler product asymptotics can
sometimes be proved unconditionally.


298  On the WaringGoldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers Kumchev, Angel V.
We investigate exceptional sets in the WaringGoldbach problem. For
example, in the cubic case, we show that all but
$O(N^{79/84+\epsilon})$ integers subject to the necessary local
conditions can be represented as the sum of five cubes of primes.
Furthermore, we develop a new device that leads easily to similar
estimates for exceptional sets for sums of fourth and higher powers of
primes.


328  On a Conjecture of Birch and SwinnertonDyer Kuo, Wentang; Murty, M. Ram
Let \(E/\mathbb{Q}\) be an elliptic curve defined by the equation
\(y^2=x^3 +ax +b\). For a prime \(p, \linebreak p \nmid\Delta
=16(4a^3+27b^2)\neq 0\), define \[ N_p = p+1 a_p =
E(\mathbb{F}_p). \] As a precursor to their celebrated conjecture,
Birch and SwinnertonDyer originally conjectured that for some
constant $c$, \[ \prod_{p \leq x, p \nmid\Delta } \frac{N_p}{p} \sim c
(\log x)^r, \quad x \to \infty. \] Let \(\alpha _p\) and \(\beta
_p\) be the eigenvalues of the Frobenius at \(p\). Define \[
\tilde{c}_n = \begin{cases} \frac{\alpha_p^k + \beta_p^k}{k}& n =p^k,
p \textrm{ is a prime, $k$ is a natural number, $p\nmid \Delta$} .
\\ 0 & \text{otherwise}. \end{cases}. \] and \(\tilde{C}(x)=
\sum_{n\leq x} \tilde{c}_n\). In this paper, we establish the
equivalence between the conjecture and the condition
\(\tilde{C}(x)=\mathbf{o}(x)\). The asymptotic condition is indeed
much deeper than what we know so far or what we can know under the
analogue of the Riemann hypothesis. In addition, we provide an
oscillation theorem and an \(\Omega\) theorem which relate to the
constant $c$ in the conjecture.


338  Certain Exponential Sums and Random Walks on Elliptic Curves Lange, Tanja; Shparlinski, Igor E.
For a given elliptic curve $\E$, we obtain an upper bound
on the discrepancy of sets of
multiples $z_sG$ where $z_s$ runs through a sequence
$\cZ=\(z_1, \dots, z_T\)$
such that $k z_1,\dots, kz_T $ is a permutation of
$z_1, \dots, z_T$, both sequences taken modulo $t$, for
sufficiently many distinct values of $k$ modulo $t$.


351  Extensions by Simple $C^*$Algebras: Quasidiagonal Extensions Lin, Huaxin
Let $A$ be an amenable separable $C^*$algebra and $B$ be a nonunital
but $\sigma$unital simple $C^*$algebra with continuous scale.
We show that two essential extensions
$\tau_1$ and $\tau_2$ of $A$ by $B$ are approximately
unitarily equivalent if and only if
$$
[\tau_1]=[\tau_2] \text{ in } KL(A, M(B)/B).
$$
If $A$ is assumed to satisfy the Universal Coefficient Theorem,
there is a bijection from approximate unitary equivalence
classes of the above mentioned extensions to
$KL(A, M(B)/B)$.
Using $KL(A, M(B)/B)$, we compute exactly when an essential extension
is quasidiagonal. We show that quasidiagonal extensions
may not be approximately trivial.
We also study the approximately trivial extensions.


400  Generalized $k$Configurations Sabourin, Sindi
In this paper, we find configurations of points in $n$dimensional
projective space ($\proj ^n$) which simultaneously generalize both
$k$configurations and reduced 0dimensional complete intersections.
Recall that $k$configurations in $\proj ^2$ are disjoint unions of
distinct points on lines and in $\proj ^n$ are inductively disjoint
unions of $k$configurations on hyperplanes, subject to certain
conditions. Furthermore, the Hilbert function of a $k$configuration
is determined from those of the smaller $k$configurations. We call
our generalized constructions $k_D$configurations, where $D=\{ d_1,
\ldots ,d_r\}$ (a set of $r$ positive integers with repetition
allowed) is the type of a given complete intersection in $\proj ^n$.
We show that the Hilbert function of any $k_D$configuration can be
obtained from those of smaller $k_D$configurations. We then provide
applications of this result in two different directions, both of which
are motivated by corresponding results about $k$configurations.


416  Approximating Flats by Periodic Flats in \\CAT(0) Square Complexes Wise, Daniel T.
We investigate the problem of whether every immersed flat plane in a
nonpositively curved square complex is the limit of periodic flat
planes. Using a branched cover, we reduce the problem to the case of
$\V$complexes. We solve the problem for malnormal and cyclonormal
$\V$complexes. We also solve the problem for complete square
complexes using a different approach. We give an application towards
deciding whether the elements of fundamental groups of the spaces we
study have commuting powers. We note a connection between the flat
approximation problem and subgroup separability.


449  On the Sizes of Gaps in the Fourier Expansion of Modular Forms Alkan, Emre
Let $f= \sum_{n=1}^{\infty} a_f(n)q^n$ be a cusp form with integer
weight $k \geq 2$ that is not a linear combination of forms with
complex multiplication. For $n \geq 1$, let
$$
i_f(n)=\begin{cases}\max\{ i :
a_f(n+j)=0 \text{ for all } 0 \leq j \leq
i\}&\text{if $a_f(n)=0$,}\\
0&\text{otherwise}.\end{cases}
$$
Concerning bounded values
of $i_f(n)$ we prove that for $\epsilon >0$ there exists $M =
M(\epsilon,f)$ such that $\# \{n \leq x : i_f(n) \leq M\} \geq (1
 \epsilon) x$. Using results of Wu, we show that if $f$ is a weight 2
cusp form for an elliptic curve without complex multiplication, then
$i_f(n) \ll_{f, \epsilon} n^{\frac{51}{134} + \epsilon}$. Using a
result of David and Pappalardi, we improve the exponent to
$\frac{1}{3}$ for almost all newforms associated to elliptic curves
without complex multiplication. Inspired by a classical paper of
Selberg, we also investigate $i_f(n)$ on the average using well known
bounds on the Riemann Zeta function.


471  Small Coverings with Smooth Functions under the Covering Property Axiom Ciesielski, Krzysztof; Pawlikowski, Janusz
In the paper we formulate a Covering Property Axiom, \psmP,
which holds in the iterated perfect set model,
and show that it implies the following facts,
of which (a) and (b) are the generalizations
of results of J. Stepr\={a}ns.
\begin{compactenum}[\rm(a)~~]
\item There exists a family $\F$ of less than continuum many $\C^1$
functions from $\real$ to $\real$ such that $\real^2$ is covered
by functions from $\F$, in the sense that for every $\la
x,y\ra\in\real^2$ there exists an $f\in\F$ such that either
$f(x)=y$ or $f(y)=x$.


494  Summation Formulae for Coefficients of $L$functions Friedlander, John B.; Iwaniec, Henryk
With applications in mind we establish a summation formula for the
coefficients of a general Dirichlet series satisfying a suitable
functional equation. Among a number of consequences we derive a
generalization of an elegant divisor sum bound due to F.~V. Atkinson.


506  Reverse Hypercontractivity for Subharmonic Functions Gross, Leonard; Grothaus, Martin
Contractivity and hypercontractivity properties of semigroups
are now well understood when the generator, $A$, is a Dirichlet form
operator.
It has been shown that in some holomorphic function spaces the
semigroup operators, $e^{tA}$, can be bounded {\it below} from
$L^p$ to $L^q$ when $p,q$ and $t$ are suitably related.
We will show that such lower boundedness occurs also in spaces
of subharmonic functions.


535  On Local $L$Functions and Normalized Intertwining Operators Kim, Henry H.
In this paper we make explicit all $L$functions in the
LanglandsShahidi method which appear as normalizing factors of
global intertwining operators in the constant term of the
Eisenstein series. We prove, in many cases,
the conjecture of Shahidi regarding the
holomorphy of the local $L$functions. We also prove
that the normalized local intertwining operators are holomorphic and
nonvaninishing for $\re(s)\geq 1/2$ in many cases. These local
results are essential in global applications such as Langlands
functoriality, residual spectrum and determining poles of
automorphic $L$functions.


598  Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group Kornelson, Keri A.
Differential operators $D_x$, $D_y$, and $D_z$ are formed using the
action of the $3$dimensional discrete Heisenberg group $G$ on a set
$S$, and the operators will act on functions on $S$. The Laplacian
operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with
variable differences which can be associated to a unitary
representation of $G$ on the Hilbert space $L^2(S)$. Using techniques
from harmonic analysis and representation theory, we show that the
Laplacian operator is locally solvable.


616  Reducibility of Generalized Principal Series Muić, Goran
In this paper we describe reducibility of nonunitary generalized
principal series for classical $p$adic groups in terms of the
classification of discrete series due to M\oe glin and Tadi\'c.


648  Branching Rules for Principal Series Representations of $SL(2)$ over a $p$adic Field Nevins, Monica
We explicitly describe the decomposition into irreducibles of
the restriction of the principal
series representations of $SL(2,k)$, for $k$ a $p$adic field,
to each of its two maximal compact subgroups (up to conjugacy).
We identify these irreducible subrepresentations in the
Kirillovtype classification
of Shalika. We go on to explicitly describe the decomposition
of the reducible principal series of $SL(2,k)$ in terms of the
restrictions of its irreducible constituents to a maximal compact
subgroup.


673  On the Structure of the Spreading Models of a Banach Space Androulakis, G.; Odell, E.; Schlumprecht, Th.; TomczakJaegermann, N.
We study some questions concerning the structure of the
set of spreading models of a separable infinitedimensional Banach
space $X$. In particular we give an example of a reflexive $X$ so that
all spreading models of $X$ contain $\ell_1$ but none of them is
isomorphic to $\ell_1$. We also prove that for any countable set $C$
of spreading models generated by weakly null sequences there is a
spreading model generated by a weakly null sequence which dominates
each element of $C$. In certain cases this ensures that $X$ admits,
for each $\alpha < \omega_1$, a spreading model $(\tilde
x_i^{(\alpha)})_i$ such that if $\alpha < \beta$ then $(\tilde
x_i^{(\alpha)})_i$ is dominated by (and not equivalent to)
$(\tilde x_i^{(\beta)})_i$. Some applications of these ideas are used to
give sufficient conditions on a Banach space for the existence of a
subspace and an operator defined on the subspace, which is not a
compact perturbation of a multiple of the inclusion map.


708  Curvature Estimates in Asymptotically Flat Lorentzian Manifolds Finster, Felix; Kraus, Margarita
We consider an asymptotically flat Lorentzian manifold of
dimension $(1,3)$. An inequality is derived which bounds the
Riemannian curvature tensor in terms of the ADM energy in the
general case with second fundamental form. The inequality
quantifies in which sense the Lorentzian manifold becomes flat in
the limit when the ADM energy tends to zero.


724  Some Results on Surfaces of General Type Purnaprajna, B. P.
In this article we prove some new results on projective normality, normal
presentation and higher syzygies for surfaces of general type, not
necessarily smooth, embedded by adjoint linear series. Some of the
corollaries of more general results include: results on property $N_p$
associated to $K_S \otimes B^{\otimes n}$ where $B$ is basepoint free and
ample divisor with $B\otimes K^*$ {\it nef}, results for pluricanonical
linear systems and results giving effective bounds for adjoint linear series
associated to ample bundles. Examples in the last section show that the results
are optimal.


750  Sur la structure transverse à une orbite nilpotente adjointe Sabourin, Hervé
We are interested in Poisson structures to
transverse nilpotent adjoint orbits in a complex semisimple Lie algebra,
and we study their polynomial nature. Furthermore, in the case
of $sl_n$,
we construct some families of nilpotent orbits with quadratic
transverse structures.


771  The Resolvent of Closed Extensions of Cone Differential Operators Schrohe, E.; Seiler, J.
We study closed extensions $\underline A$ of
an elliptic differential operator $A$ on a manifold with conical
singularities, acting as an unbounded operator on a weighted $L_p$space.
Under suitable conditions we show that the resolvent
$(\lambda\underline A)^{1}$ exists
in a sector of the complex plane and decays like $1/\lambda$ as
$\lambda\to\infty$. Moreover, we determine the structure of the resolvent
with enough precision to guarantee existence and boundedness of imaginary
powers of $\underline A$.


812  On the Vanishing of $\mu$Invariants of Elliptic Curves over $\qq$ Trifković, Mak
Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a
prime $p>2$. It has a welldefined Iwasawa $\mu$invariant $\mu(E)_p$
which encodes part of the information about the growth of the Selmer
group $\sel E{K_n}$ as $K_n$ ranges over the subfields of the
cyclotomic $\zzp$extension $K_\infty/\qq$. Ralph Greenberg has
conjectured that any such $E$ is isogenous to a curve $E'$ with
$\mu(E')_p=0$. In this paper we prove Greenberg's conjecture for
infinitely many curves $E$ with a rational $p$torsion point, $p=3$ or
$5$, no two of our examples having isomorphic $p$torsion. The core
of our strategy is a partial explicit evaluation of the global duality
pairing for finite flat group schemes over rings of integers.


844  Petrie Schemes Williams, Gordon
Petrie polygons, especially as they arise in the study of regular
polytopes and Coxeter groups, have been studied by geometers and group
theorists since the early part of the twentieth century. An open
question is the determination of which polyhedra possess Petrie
polygons that are simple closed curves. The current work explores
combinatorial structures in abstract polytopes, called Petrie schemes,
that generalize the notion of a Petrie polygon. It is established
that all of the regular convex polytopes and honeycombs in Euclidean
spaces, as well as all of the Gr\"unbaumDress polyhedra, possess
Petrie schemes that are not selfintersecting and thus have Petrie
polygons that are simple closed curves. Partial results are obtained
for several other classes of less symmetric polytopes.


871  Hermitian Yang_MillsHiggs Metrics on\\Complete Kähler Manifolds Zhang, Xi
In this paper, first, we will investigate the
Dirichlet problem for one type of vortex equation, which
generalizes the wellknown Hermitian Einstein equation. Secondly,
we will give existence results for solutions of these vortex
equations over various complete noncompact K\"ahler manifolds.


897  Representation of Banach Ideal Spaces and Factorization of Operators Berezhnoĭ, Evgenii I.; Maligranda, Lech
Representation theorems are proved for Banach ideal spaces with the Fatou property
which are built by the Calder{\'o}nLozanovski\u\i\ construction.
Factorization theorems for operators in spaces more general than the Lebesgue
$L^{p}$ spaces are investigated. It is natural to extend the Gagliardo
theorem on the Schur test and the Rubio de~Francia theorem on factorization of the
Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for
the scales far from $L^{p}$spaces this is impossible. For the concrete integral operators
it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces
are not valid. Representation theorems for the Calder{\'o}nLozanovski\u\i\ construction
are involved in the proofs.


941  Some Transformations of Hausdorff Moment Sequences and Harmonic Numbers Berg, Christian; Durán, Antonio J.
We introduce some nonlinear transformations from the set of
Hausdorff moment sequences into itself; among
them is the one defined by
the formula:
$T((a_n)_n)=1/(a_0+\dots +a_n)$. We give some examples of
Hausdorff moment sequences arising from the transformations and
provide the corresponding measures: one of these sequences is the
reciprocal of the harmonic numbers $(1+1/2+\dots +1/(n+1))^{1}$.


961  ConeMonotone Functions: Differentiability and Continuity Borwein, Jonathan M.; Wang, Xianfu
We provide a porositybased approach to the differentiability and
continuity of realvalued functions on separable Banach spaces,
when the function is monotone with respect to an ordering induced
by a convex cone $K$ with nonempty interior. We also show that
the set of nowhere $K$monotone functions has a $\sigma$porous
complement in the space of continuous functions endowed with the
uniform metric.


983  A Symmetric Imprimitivity Theorem for Commuting Proper Actions an Huef, Astrid; Raeburn, Iain; Williams, Dana P.
We prove a symmetric imprimitivity theorem for commuting proper
actions of locally compact groups $H$ and $K$ on a $C^*$algebra.


1012  Deformations of $G_2$ and $\Spin(7)$ Structures Karigiannis, Spiro
We consider some deformations of $G_2$structures on $7$manifolds. We
discover a canonical way to deform a $G_2$structure by a vector field in
which the associated metric gets ``twisted'' in some way by the
vector cross product. We present a system of partial differential
equations for an unknown vector field $w$ whose solution would
yield a manifold with holonomy $G_2$. Similarly we consider analogous
constructions for $\Spin(7)$structures on $8$manifolds. Some of
the results carry over directly, while others do not because of the
increased complexity of the $\Spin(7)$ case.


1056  Hyperbolic Group $C^*$Algebras and FreeProduct $C^*$Algebras as Compact Quantum Metric Spaces Ozawa, Narutaka; Rieffel, Marc A.
Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$
denote the
operator of pointwise multiplication by $\ell$ on $\bell^2(G)$.
Following Connes,
$M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of
$C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $\ell$ is
a wordlength function on $G$, then the topology from this metric
coincides with the
weak$*$ topology (our definition of a ``compact quantum metric
space''). We show that a convenient framework is that of filtered
$C^*$algebras which satisfy a suitable ``Haageruptype'' condition. We
also use this
framework to prove an analogous fact for certain reduced
free products of $C^*$algebras.


1080  The GelfondSchnirelman Method in Prime Number Theory Pritsker, Igor E.
The original GelfondSchnirelman method, proposed in 1936, uses
polynomials with integer coefficients and small norms on $[0,1]$
to give a Chebyshevtype lower bound in prime number theory. We
study a generalization of this method for polynomials in many
variables. Our main result is a lower bound for the integral of
Chebyshev's $\psi$function, expressed in terms of the weighted
capacity. This extends previous work of Nair and Chudnovsky, and
connects the subject to the potential theory with external fields
generated by polynomialtype weights. We also solve the
corresponding potential theoretic problem, by finding the extremal
measure and its support.


1102  Power Residues of Fourier Coefficients of Modular Forms Weston, Tom
Let $\rho \colon G_{\Q} \to \GL_{n}(\Ql)$ be a motivic $\ell$adic Galois
representation. For fixed $m > 1$ we initiate an investigation of the
density of the set of primes $p$ such that the trace of the image of an
arithmetic Frobenius at $p$ under $\rho$ is an $m$th power residue
modulo $p$. Based on numerical investigations with modular forms we
conjecture (with Ramakrishna) that this density equals $1/m$ whenever the
image of $\rho$ is open. We further conjecture that for such $\rho$ the set
of these primes $p$ is independent of any set defined by Cebatorevstyle
Galoistheoretic conditions (in an appropriate sense). We then compute these
densities for certain $m$ in the complementary case of modular forms of
CMtype with rational Fourier coefficients; our proofs are a combination of
the Cebatorev density theorem (which does apply in the CM case) and
reciprocity laws applied to Hecke characters. We also discuss a potential
application (suggested by Ramakrishna) to computing inertial degrees at $p$
in abelian extensions of imaginary quadratic fields unramified away from $p$.


1121  On $\mathcal{CR}$epic Embeddings and Absolute $\mathcal{CR}$epic Spaces Barr, Michael; Raphael, R.; Woods, R. G.
We study Tychonoff spaces $X$ with the property that, for all
topological embeddings $X\to Y $, the induced map $C(Y) \to C(X)$ is an
epimorphism of rings. Such spaces are called \good. The simplest
examples of \good spaces are $\sigma$compact locally compact spaces and
\Lin $P$spaces. We show that \good first countable spaces must be
locally compact.


1139  Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set Burke, Maxim R.; Miller, Arnold W.
We prove that it is relatively consistent with $\ZFC$ that in any
perfect Polish space, for every nonmeager set $A$ there exists a
nowhere dense Cantor set $C$ such that $A\cap C$ is nonmeager in
$C$. We also examine variants of this result and establish a
measure theoretic analog.


1155  The Square Sieve and the LangTrotter Conjecture Cojocaru, Alina Carmen; Fouvry, Etienne; Murty, M. Ram
Let $E$ be an elliptic curve defined over $\Q$ and without
complex multiplication. Let $K$ be a fixed imaginary quadratic field.
We find nontrivial upper bounds for the number of ordinary primes $p \leq x$
for which $\Q(\pi_p) = K$, where $\pi_p$ denotes the Frobenius endomorphism
of $E$ at $p$. More precisely, under a generalized Riemann hypothesis
we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally
we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}}
{(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic
fields $K$, with $\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is
$\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards
a 1976 LangTrotter conjecture.


1178  Asymptotic Behavior of the Length of Local Cohomology Cutkosky, Steven Dale; Hà, Huy Tài; Srinivasan, Hema; Theodorescu, Emanoil
Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring,
and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in
$R$. Let $\lambda(M)$ denote the length of an $R$module $M$. In this paper, we show
that
$$
\lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d}
=\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^n,R(d)\bigr)\bigr)}{n^d}
$$
always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$primary ideals
$I$ in a local CohenMacaulay ring, where $e(I)$ denotes the multiplicity
of $I$. But we find that this limit may not be rational in general. We give an example
for which the limit is an irrational number thereby showing that the lengths of these
extention modules may not have polynomial growth.


1193  Some Conditions for Decay of Convolution Powers and Heat Kernels on Groups Dungey, Nick
Let $K$ be a function on a unimodular locally compact group
$G$, and denote by $K_n = K*K* \cdots * K$ the $n$th convolution
power of $K$.
Assuming that $K$ satisfies certain operator estimates in $L^2(G)$,
we give estimates of
the norms $\K_n\_2$ and $\K_n\_\infty$
for large $n$.
In contrast to previous methods for estimating $\K_n\_\infty$,
we do not need to assume that
the function $K$ is a probability density or nonnegative.
Our results also adapt for continuous time semigroups on $G$.
Various applications are given, for example, to estimates of
the behaviour of heat kernels on Lie groups.


1215  Reciprocity Law for Compatible Systems of Abelian $\bmod p$ Galois Representations Khare, Chandrashekhar
The main result of the paper
is a {\em reciprocity law} which proves that
compatible systems of semisimple, abelian mod $p$ representations
(of arbitrary dimension)
of absolute Galois groups of number fields, arise from Hecke characters.
In the last section analogs for Galois groups of function fields of these
results are explored, and a question is raised whose answer seems to
require developments in transcendence theory in characteristic $p$.


1224  Convex Polynomial Approximation in the Uniform Norm: Conclusion Kopotun, K. A.; Leviatan, D.; Shevchuk, I. A.
Estimating the degree of approximation in the uniform norm, of a
convex function on a finite interval, by convex algebraic
polynomials, has received wide attention over the last twenty
years. However, while much progress has been made especially in
recent years by, among others, the authors of this article,
separately and jointly, there have been left some interesting open
questions. In this paper we give final answers to all those open
problems. We are able to say, for each $r$th differentiable convex
function, whether or not its degree of convex polynomial
approximation in the uniform norm may be estimated by a
Jacksontype estimate involving the weighted DitzianTotik $k$th
modulus of smoothness, and how the constants in this estimate
behave. It turns out that for some pairs $(k,r)$ we have such
estimate with constants depending only on these parameters. For
other pairs the estimate is valid, but only with constants that
depend on the function being approximated, while there are pairs
for which the Jacksontype estimate is, in general, invalid.


1249  Strictly Singular and Cosingular Multiplications Lindström, Mikael; Saksman, Eero; Tylli, HansOlav
Let $L(X)$ be the space of bounded linear operators on the Banach space $X$.
We study the strict singularity andcosingularity of the twosided multiplication
operators $S \mapsto ASB$ on $L(X)$, where $A,B \in L(X)$ are fixed bounded
operators and $X$ is a classical Banach space. Let $1<p<\infty$ and $p \neq 2$.
Our main result establishes thatthe multiplication $S \mapsto ASB$ is strictly
singular on $L\bigl(L^p(0,1)\bigr)$ if and only if the nonzero operators
$A, B \in L\bigl(L^p(0,1)\bigr)$ are strictly singular. We also discuss the
case where $X$ is a $\mathcal{L}^1$ or a $\mathcal{L}^\infty$space, as well as
several other relevant examples.


1279  A Semilinear Problem for the Heisenberg Laplacian on Unbounded Domains Maad, Sara
We study the semilinear equation
\begin{equation*}
\Delta_{\mathbb H} u(\eta) + u(\eta) = f(\eta,
u(\eta)),\quad
u \in \So(\Omega),
\end{equation*}
where $\Omega$ is an unbounded domain of the Heisenberg
group $\mathbb H^N$, $N\ge 1$. The space $\So(\Omega)$ is the
Heisenberg analogue of the Sobolev space $W_0^{1,2}(\Omega)$.
The function $f\colon \overline{\Omega}\times
\mathbb R\to \mathbb R$ is supposed to be odd in $u$,
continuous and satisfy some (superlinear but subcritical) growth
conditions. The operator $\Delta_{\mathbb H}$ is
the subelliptic Laplacian on the Heisenberg group. We
give a condition on $\Omega$ which implies the existence of
infinitely many solutions of the above equation. In the proof we
rewrite the equation as a variational problem, and show that the
corresponding functional satisfies the PalaisSmale
condition. This might be quite surprising since we deal with
domains which are far from bounded. The technique we use rests on
a compactness argument and the maximum principle.


1291  Dupin Hypersurfaces in $\mathbb R^5$ Riveros, Carlos M. C.; Tenenblat, Keti
We study Dupin
hypersurfaces in $\mathbb R^5$ parametrized by lines of curvature, with
four distinct principal curvatures. We characterize locally a generic
family of such hypersurfaces in terms of the principal curvatures and
four vector valued functions of one variable. We show that these vector
valued functions are invariant by inversions and homotheties.


1314  Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra Zhitomirskii, M.
In 1999 V. Arnol'd introduced the local contact algebra: studying the
problem of classification of singular curves in a contact space, he
showed the existence of the ghost of the contact structure (invariants
which are not related to the induced structure on the curve). Our
main result implies that the only reason for existence of the local
contact algebra and the ghost is the difference between the geometric
and (defined in this paper) algebraic restriction of a $1$form to a
singular submanifold. We prove that a germ of any subset $N$ of a
contact manifold is well defined, up to contactomorphisms, by the
algebraic restriction to $N$ of the contact structure. This is a
generalization of the DarbouxGivental' theorem for smooth
submanifolds of a contact manifold. Studying the difference between
the geometric and the algebraic restrictions gives a powerful tool for
classification of stratified submanifolds of a contact manifold. This
is illustrated by complete solution of three classification problems,
including a simple explanation of V.~Arnold's results and further
classification results for singular curves in a contact space. We
also prove several results on the external geometry of a singular
submanifold $N$ in terms of the algebraic restriction of the contact
structure to $N$. In particular, the algebraic restriction is zero if
and only if $N$ is contained in a smooth Legendrian submanifold of
$M$.


1341  Author Index  Index des auteurs 2005, for 2005  pour
No abstract.

