Asymptotic relations among Fourier coefficients of automorphic eigenfunctions
Scott
A.
Wolpert
427-456
Abstract: A detailed stationary phase analysis is presented for noncompact parameter ranges of the family of elementary eigenfunctions on the hyperbolic plane $\mathcal{K}(z)=y^{1/2}K_{ir}(2\pi my)e^{2\pi im x}$, $z=x+iy$, $\lambda=\frac14+r^2$ the eigenvalue, $s=2\pi m\lambda^{-1/2}$ and $K_{ir}$ the Macdonald-Bessel function. The phase velocity of $\mathcal{K}$ on $\{\vert s\vert Im z\le1\}$ is a double-valued vector field, the tangent field to the pencil of geodesics $\mathcal{G}$ tangent to the horocycle $\{\vert s\vert Im z =1 \}$. For $A\in SL(2;\mathbb{R} )$ a multi-term stationary phase expansion is presented in $\lambda$ for $\mathcal{K}(Az)e^{2\pi in\,Re z}$ uniform in parameters. An application is made to find an asymptotic relation for the Fourier coefficients of a family of automorphic eigenfunctions. In particular, for $\psi$automorphic with coefficients $\{a_n\}$ and eigenvalue $\lambda$ it is shown for the special range $n\sim \lambda^{1/2}$ that $a_n$ is $O(\lambda^{1/4}\,e^{\pi\lambda^{1/2}/2})$ for $\lambda$ large, improving by an order of magnitude for this special range upon the estimate from the general Hecke bound $O(\vert n\vert^{1/2}\lambda^{1/4}\,e^{\pi\lambda^{1/2}/2})$. An exposition of the WKB asymptotics of the Macdonald-Bessel functions is presented.
Symmetries of flat rank two distributions and sub-Riemannian structures
Yuri
L.
Sachkov
457-494
Abstract: Flat sub-Riemannian structures are local approximations -- nilpotentizations -- of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5.
A version of Gordon's theorem for multi-dimensional Schrödinger operators
David
Damanik
495-507
Abstract: We consider discrete Schrödinger operators $H = \Delta + V$ in $\ell^2(\mathbb{Z} ^d)$with $d \ge 1$, and study the eigenvalue problem for these operators. It is shown that the point spectrum is empty if the potential $V$ is sufficiently well approximated by periodic potentials. This criterion is applied to quasiperiodic $V$ and to so-called Fibonacci-type superlattices.
Simple birational extensions of the polynomial algebra $\mathbb{C}^{[3]}$
Shulim
Kaliman;
Stéphane
Vénéreau;
Mikhail
Zaidenberg
509-555
Abstract: The Abhyankar-Sathaye Problem asks whether any biregular embedding $\varphi:\mathbb{C}^k\hookrightarrow\mathbb{C}^n$ can be rectified, that is, whether there exists an automorphism $\alpha\in{\operatorname{Aut}}\,\mathbb{C}^n$ such that $\alpha\circ\varphi$ is a linear embedding. Here we study this problem for the embeddings $\varphi:\mathbb{C}^3\hookrightarrow \mathbb{C}^4$ whose image $X=\varphi(\mathbb{C}^3)$ is given in $\mathbb{C}^4$ by an equation $p=f(x,y)u+g(x,y,z)=0$, where $f\in\mathbb{C}[x,y]\backslash\{0\}$ and $g\in\mathbb{C}[x,y,z]$. Under certain additional assumptions we show that, indeed, the polynomial $p$ is a variable of the polynomial ring $\mathbb{C}^{[4]}=\mathbb{C}[x,y,z,u]$ (i.e., a coordinate of a polynomial automorphism of $\mathbb{C}^4$). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings $\mathbb{C}^2\hookrightarrow\mathbb{C}^3$. Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial $p$ as above, a criterion for when $X=p^{-1}(0)\simeq\mathbb{C}^3$.
Truncated second main theorem with moving targets
Min
Ru;
Julie
Tzu-Yueh
Wang
557-571
Abstract: We prove a truncated Second Main Theorem for holomorphic curves intersecting a finite set of moving or fixed hyperplanes. The set of hyperplanes is assumed to be non-degenerate. Previously only general position or subgeneral position was considered.
The two-by-two spectral Nevanlinna-Pick problem
Jim
Agler;
N.
J.
Young
573-585
Abstract: We give a criterion for the existence of an analytic $2 \times 2$matrix-valued function on the disc satisfying a finite set of interpolation conditions and having spectral radius bounded by $1$. We also give a realization theorem for analytic functions from the disc to the symmetrised bidisc.
Chern numbers of ample vector bundles on toric surfaces
Sandra
Di Rocco;
Andrew
J.
Sommese
587-598
Abstract: This article shows a number of strong inequalities that hold for the Chern numbers $c_1^2$, $c_2$ of any ample vector bundle $\mathcal{E}$ of rank $r$ on a smooth toric projective surface, $S$, whose topological Euler characteristic is $e(S)$. One general lower bound for $c_1^2$ proven in this article has leading term $(4r+2)e(S)\ln_2\left(\tfrac{e(S)}{12}\right)$. Using Bogomolov instability, strong lower bounds for $c_2$ are also given. Using the new inequalities, the exceptions to the lower bounds $c_1^2> 4e(S)$ and $c_2>e(S)$ are classified.
The distribution of prime ideals of imaginary quadratic fields
G.
Harman;
A.
Kumchev;
P.
A.
Lewis
599-620
Abstract: Let $Q(x, y)$ be a primitive positive definite quadratic form with integer coefficients. Then, for all $(s, t)\in \mathbb R^2$ there exist $(m, n) \in \mathbb Z^2$ such that $Q(m, n)$ is prime and \begin{displaymath}Q(m - s, n - t) \ll Q(s, t)^{0.53} + 1. \end{displaymath} This is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval.
Examples of pleating varieties for twice punctured tori
Raquel
Díaz;
Caroline
Series
621-658
Abstract: We give an explicit description of some pleating varieties (sets with a fixed set of bending lines in the convex hull boundary) in the quasi-Fuchsian space of the twice punctured torus. In accordance with a conjecture of the second author, we show that their closures intersect Fuchsian space in the simplices of minima introduced by Kerckhoff. All computations are done using complex Fenchel-Nielsen coordinates for quasi-Fuchsian space referred to a maximal system of curves.
Variational principles for circle patterns and Koebe's theorem
Alexander
I.
Bobenko;
Boris
A.
Springborn
659-689
Abstract: We prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces. Our method is based on two new functionals--one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdière's, Brägger's and Rivin's functionals can be derived from ours.
On a conjecture of Whittaker concerning uniformization of hyperelliptic curves
Ernesto
Girondo;
Gabino
González-Diez
691-702
Abstract: This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface $y^2=\prod_{i=1}^{2g+2}(x-a_i)$ as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values $a_i$. Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker's conjecture cannot be true in its full generality. Recently, numerical computations have shown that Whittaker's prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms. The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker's conjecture.
Some Picard theorems for minimal surfaces
Francisco
J.
López
703-733
Abstract: This paper deals with the study of those closed subsets $F \subset \mathbb{R} ^3$ for which the following statement holds: If $S$ is a properly immersed minimal surface in $\mathbb{R} ^3$ of finite topology that is eventually disjoint from $F,$ then $S$ has finite total curvature. The same question is also considered when the conclusion is finite type or parabolicity.
Symmetrization, symmetric stable processes, and Riesz capacities
Dimitrios
Betsakos
735-755
Abstract: Let $\texttt{X}_t$ be a symmetric $\alpha$-stable process killed on exiting an open subset $D$ of $\mathbb R^n$. We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set $B$ in the complement of $D$ in the first exit moment from $D$ increases when $D$ and $B$ are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets $K$ in $\mathbb R^n$with given volume, the balls have the least $\alpha$-capacity ( $0<\alpha<2$).
Deriving calculus with cotriples
B.
Johnson;
R.
McCarthy
757-803
Abstract: We construct a Taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors. The tower was inspired by Goodwillie's Taylor tower for functors of spaces, and is related to Dold and Puppe's stable derived functors and Mac Lane's $Q$-construction. We study the layers, $D_{n}F=\text{\rm fiber}(P_{n}F\rightarrow P_{n-1}F)$, and the limit of the tower. For the latter we determine a condition on the cross effects that guarantees convergence. We define differentials for functors, and establish chain and product rules for them. We conclude by studying exponential functors in this setting and describing their Taylor towers.
The geometry of profinite graphs with applications to free groups and finite monoids
K.
Auinger;
B.
Steinberg
805-851
Abstract: We initiate the study of the class of profinite graphs $\Gamma$ defined by the following geometric property: for any two vertices $v$ and $w$ of $\Gamma$, there is a (unique) smallest connected profinite subgraph of $\Gamma$ containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition. We define a pseudovariety of groups $\mathbf{H}$ to be arboreous if all finitely generated free pro- $\mathbf{H}$ groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties $\mathbf{H}$, a pro- $\mathbf{H}$ analog of the Ribes and Zalesski{\u{\i}}\kern.15em product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions $\mathbf{H}$ to the much studied pseudovariety equation $\mathbf{J}\ast\mathbf{H}= \mathbf{J}\mathrel{{\mbox{\textcircled{\petite m}}}}\mathbf{H}$.