Spectral exponential sums on hyperbolic surfaces I

We study an exponential sum over Laplace eigenvalues $\lambda_{j} = 1/4+t_{j}^{2}$ with $t_{j} \leqslant T$ for Maass cusp forms on $\Gamma \backslash \mathbb{H}$ as $T$ grows, where $\Gamma \subset PSL_{2}(\mathbb{R})$ is a cofinite Fuchsian group acting on the upper half-plane $\mathbb{H}$. Specifically, for the congruence subgroups $\Gamma_{0}(q), \, \Gamma_{1}(q)$ and $\Gamma(q)$, we explicitly describe each sum in terms of a certain oscillatory component, von Mangoldt-like functions and the Selberg zeta function. We also establish a new expression of the spectral exponential sum for a general cofinite group $\Gamma \subset PSL_{2}(\mathbb{R})$, and in particular we find that the behavior of the sum is decisively determined by whether $\Gamma$ is essentially cuspidal or not. We also work with certain moonshine groups for which our plotting of the spectral exponential sum alludes to the fact that the conjectural bound $O(X^{1/2+\epsilon})$ in the Prime Geodesic Theorem may be allowable. In view of our numerical evidence, the conjecture of Petridis and Risager is generalized.Comment: 20 pages, 3 figures, comments welcom

Bifurcations in the Space of Exponential Maps

This article investigates the parameter space of the exponential family $z\mapsto \exp(z)+\kappa$. We prove that the boundary (in \C) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon. In fact, we prove the stronger statement that the exponential bifurcation locus is connected in \C, which is an analog of Douady and Hubbard's celebrated theorem that the Mandelbrot set is connected. We show furthermore that $\infty$ is not accessible through any nonhyperbolic ("queer") stable component. The main part of the argument consists of demonstrating a general "Squeezing Lemma", which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees of hyperbolic components.Comment: 29 pages, 3 figures. The main change in the new version is the introduction of Theorem 1.1 on the connectivity of the bifurcation locus, which follows from the results of the original version but was not explicitly stated. Also, some small revisions have been made and references update

A pp-adic analogue of the Borel regulator and the Bloch-Kato exponential map

In this paper we define a $p$-adic analogue of the Borel regulator for the $K$-theory of $p$-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this $p$-adic regulator to the Bloch-Kato exponential and the Soul\'e regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups.Comment: 38 page

L1-determined ideals in group algebras of exponential Lie groups

A locally compact group $G$ is said to be $\ast$-regular if the natural map \Psi:\Prim C^\ast(G)\to\Prim_{\ast} L^1(G) is a homeomorphism with respect to the Jacobson topologies on the primitive ideal spaces \Prim C^\ast(G) and \Prim_{\ast} L^1(G). In 1980 J. Boidol characterized the $\ast$-regular ones among all exponential Lie groups by a purely algebraic condition. In this article we introduce the notion of $L^1$-determined ideals in order to discuss the weaker property of primitive $\ast$-regularity. We give two sufficient criteria for closed ideals $I$ of $C^\ast(G)$ to be $L^1$-determined. Herefrom we deduce a strategy to prove that a given exponential Lie group is primitive $\ast$-regular. The author proved in his thesis that all exponential Lie groups of dimension $\le 7$ have this property. So far no counter-example is known. Here we discuss the example $G=B_5$, the only critical one in dimension $\le 5$

Periodic Structure of the Exponential Pseudorandom Number Generator

We investigate the periodic structure of the exponential pseudorandom number generator obtained from the map $x\mapsto g^x\pmod p$ that acts on the set $\{1, \ldots, p-1\}$