Diophantine approximation on hyperbolic Riemann surfaces

For finite volume, noncompact Riemann surfaces with their canonical hyperbolic metrics, there is a notion of pseudo-Laplace operators which include all embedded eigenvalues $(> \frac {1}{4})$ of the Laplacian as a part of their eigenvalues. Similarly, we define pseudo-Laplace operators for compact hyperbolic Riemann surfaces with short geodesics. Then, for any degenerating family of hyperbolic Riemann surfaces ${S_l} (l \geq 0)$, we show that normalized pseudoeigenfunctions and pseudoeigenvalues of ${S_l}$ converge to normalized pseudoeigenfunctions and pseudoeigenvalues of ${S_0}$ as $l \to 0$. In particular, normalized embedded eigenfunctions and their embedded eigenvalues of ${S_0}$ can be approximated by normalized pseudoeigenfunctions and pseudoeigenvalues of ${S_l}$ and $l \to 0$.

Let S be a compact hyperbolic Riemann surface of genus \({g \geq 2}\). We call a systole a shortest simple closed geodesic in S and denote by \({{\rm sys}(S)}\) its length. Let \({{\rm msys}(g)}\) be the maximal value that \({{\rm sys}(\cdot)}\) can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface Smax a compact Riemann surface of genus g whose systole has length \({{\rm msys}(g)}\). In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating \({{\rm msys}(\cdot)}\) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.

For finite volume, noncompact Riemann surfaces with their canonical hyperbolic metrics, there is a notion of pseudo-Laplace operators which include all embedded eigenvalues ( > 1 4 ) (> \frac {1}{4}) of the Laplacian as a part of their eigenvalues. Similarly, we define pseudo-Laplace operators for compact hyperbolic Riemann surfaces with short geodesics. Then, for any degenerating family of hyperbolic Riemann surfaces S l ( l ≥ 0 ) {S_l} (l \geq 0) , we show that normalized pseudoeigenfunctions and pseudoeigenvalues of S l {S_l} converge to normalized pseudoeigenfunctions and pseudoeigenvalues of S 0 {S_0} as l → 0 l \to 0 . In particular, normalized embedded eigenfunctions and their embedded eigenvalues of S 0 {S_0} can be approximated by normalized pseudoeigenfunctions and pseudoeigenvalues of S l {S_l} and l → 0 l \to 0 .

In this paper a hyperbolic Riemann surface is described, and the uniformizing function for this surface is constructed.The uniformizing function for this hyperbolic surface is an infinite product similar to the ones constructed for some parabolic surfaces in earlier studiesDescription of the surface.Let I denote the positive integers.For each iEI, let C, be the vertical line segment in the Z-plane of length 2 centered at the point (1/t, 0).Let Si be the Z-sphere cut along G and for each iEI, i>l, let Si be the Z-sphere cut along C,_i and along d.For each iEI, join S, to Si+i along C; in such a way as to form first-order branch points over {l/i, 1) and il/i, -1).The resulting Riemann surface F is simply connected and is easily seen to be hyperbolic because the surface has a free edge along the vertical line segment from -i to i.Approximating surfaces.For each iEI, let F{ be the Riemann surface formed from the first i sheets of F with the points of the curve d on Si deleted.Let Ff be the closed surface formed from the first i sheets of F with C, on 5, healed.There exists a unique function g,mapping Ff onto the Z-sphere such that fi=gl1, ftiO) =0ESi, fi (0) = 1, and fii oo ) = oo ESi.Let g be the unique function mapping

This textbook covers with the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense, but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. Nine main areas of the subject are treated: (i) divisibility and multiplicative functions; (ii) congruences and quadratic residues; (iii) the basics of algebraic numbers and sums of squares; (iv) continued fractions, diophantine approximations and transcendence; (v) quadratic forms; (vi) partitions; (vii) the prime numbers; (viii) diophantine equations; and (ix) elliptic curves (this part has been considerably extended for this edition). The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on fermat's last theorem is also breifly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables.

Szekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

Let be an arbitrary non‐compact hyperbolic Riemann surface, that is, not the complex plane or punctured plane. Let be , or . We show that, on a Higgs bundle in the ‐Hitchin section over , there always exists a harmonic metric satisfying certain nice properties. Moreover, when the Higgs field has bounded spectral curve with respect to the complete hyperbolic metric on , such a harmonic metric is unique. As a result, we obtain a new proof of the existence and uniqueness of a harmonic metric for Higgs bundles in the ‐Hitchin section over a compact hyperbolic Riemann surface. Moreover, we show an existence result of harmonic metrics for a more general family of Higgs bundles that admit a full holomorphic filtration.

By the Collar Theorem, every puncture on a hyperbolic Riemann surface with punctures has a horocyclic neighborhood of area 2 2 . Furthermore two such neighborhoods associated to different punctures are disjoint. This result can be improved if we omit the condition that horocyclic neighborhoods of different punctures must be disjoint. Using arguments of the second author we show, in this paper, that each puncture of a hyperbolic Riemann surface has a horocyclic neighborhood of area 4 4 .

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

This chapter is our second digression into number theory. We saw in §1.8 that Jensen’s Formula (or the First Main Theorem) can be viewed as an analogue of the Artin-Whaples product formula in number theory. In the present chapter, we discuss a celebrated number theory theorem known as Roth’s Theorem, and we explain how this is analogous to a weak form of the Second Main Theorem. In fact, it was the formal similarity between Nevanlinna’s Second Main Theorem and Roth’s Theorem that led C. F. Osgood (see [Osg 1981] and [Osg 1985]) to the discovery of an analogy between Nevanlinna theory and Diophantine approximations. As we said previously, such an analogy was later, but independently, explored in greater depth by P. Vojta. In this chapter we will discuss, without proof, some of the key points in Vojta’s monograph [Vojt 1987], and we include Vojta’s so called “dictionary” relating Nevanlinna theory and number theory. This section is intended for the analytically inclined and is only intended to provide the most basic insight into the beautiful analogy between Nevanlinna theory and Diophantine approximation theory. By omitting proofs, we have tried to make this section less demanding on the reader than [Vojt 19871. However, a true appreciation for Vojta’s analogy cannot be obtained without also studying the proofs of the number theoretic analogues in their full generality. Any reader that is seriously interested in the connection between Nevanlinna theory and Diophantine approximation is highly encouraged to carefully read Vojta’s monograph [Vojt 1987].

Under a certain geometric assumption on a hyperbolic Riemann surface, we prove an asymptotic version of the fixed point theorem for the Teichmüller modular group, which asserts that every finite subgroup of the asymptotic Teichmüller modular group has a common fixed point in the asymptotic Teichmüller space. For its proof, we use a topological characterization of the asymptotically trivial mapping class group, which has been obtained in the authors’ previous paper, but a simpler argument is given here. As a consequence, every finite subgroup of the asymptotic Teichmüller modular group is realized as a group of quasiconformal automorphisms modulo coincidence near infinity. Furthermore, every finite subgroup of a certain geometric automorphism group of the asymptotic Teichmüller space is realized as an automorphism group of the Royden boundary of the Riemann surface. These results can be regarded as asymptotic versions of the Nielsen realization theorem.

the Martin boundary of R and the minimal Martin boundary of R, respectively. We refer to

We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathfrak{F})$ of a compact minimal lamination $(M,\mathfrak{F})$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal and examples where this action is not minimal. In the first case, we prove that if $\mathfrak{F}$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.

We study diophantine properties of a typical point with respect to measures on \(\mathbb{R}^n .\) Namely, we identify geometric conditions on a measure μ on \(\mathbb{R}^n \) guaranteeing that μ-almost every \({\bf y}\,\in\,\mathbb{R}^n \) is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called ‘friendly’. Examples include smooth measures on nondegenerate manifolds; thus this paper generalizes the main result of [KM]. Another class of examples is given by measures supported on self-similar sets satisfying the open set condition, as well as their products and pushforwards by certain smooth maps.

p-adic dynamical systems of (2,2)-rational functions with unique fixed point