Author: Gerardus Bartholomeus Maria van der GeerPublisher:
ISBN:
Category :
Languages : en
Pages : 80
Book Description
On Hilbert Modular Surfaces of Principal Congruence Subgroups
Author: Gerardus Bartholomeus Maria van der GeerPublisher:
ISBN:
Category :
Languages : en
Pages : 80
Book Description
Hilbert Modular Surfaces
Author: Gerard van der GeerPublisher: Springer Science & Business Media
ISBN: 3642615538
Category : Mathematics
Languages : en
Pages : 301
Book Description
Over the last 15 years important results have been achieved in the field of Hilbert Modular Varieties. Though the main emphasis of this book is on the geometry of Hilbert modular surfaces, both geometric and arithmetic aspects are treated. An abundance of examples - in fact a whole chapter - completes this competent presentation of the subject. This Ergebnisbericht will soon become an indispensible tool for graduate students and researchers in this field.
Lectures on Hilbert Modular Surfaces
Author: Friedrich HirzebruchPublisher:
ISBN:
Category : Discontinuous groups
Languages : en
Pages : 200
Book Description
Periods of Hilbert Modular Surfaces
Author: T. OdaPublisher: Springer Science & Business Media
ISBN: 1468492012
Category : Mathematics
Languages : en
Pages : 141
Book Description
Hilbert Modular Surfaces
Author: Friedrich HirzebruchPublisher:
ISBN:
Category : Discontinuous groups
Languages : en
Pages : 108
Book Description
On Hilbert Modular Surfaces Which Are of the General Type
Author: Tsz-On Mario ChanPublisher: Open Dissertation Press
ISBN: 9781361479797
Category :
Languages : en
Pages :
Book Description
This dissertation, "On Hilbert Modular Surfaces Which Are of the General Type" by Tsz-on, Mario, Chan, 陳子安, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled ON HILBERT MODULAR SURFACES WHICH ARE OF THE GENERAL TYPE submitted by Chan Tsz On Mario for the degree of Master of Philosophy at The University of Hong Kong in November 2007 Compact Riemann surfaces are classified according to their genera. For a surface of genus>= 2, the uniformization theorem says that it is a quotient Γ\∆ of the unit disc ∆ by a discrete subgroup Γ of Aut(∆), acting freely on ∆. In general, the quotient Γ\∆ for an arbitrary dis- crete subgroup Γ∈ Aut(∆) is considered. It is equivalent to consider X = Γ\H, where H is the upper half plane and Γ a discrete subgroup ofAut(H) =PSL (R). Thisspacecanbegivenastructureofmanifold, but may not be compact in general. When Γ is a subgroup commensu- rable withPSL (Z), X is called a modular curve. There is a procedure to compactify X by adding finite number of points, and the resultinge spaceX canbegiventhestructureofacompactRiemannsurface. The properties of X can be studied according to the genus of X. In the theory of compact complex surfaces, there is a rough clas- sification according to the Kodaira dimensions. A surface of Kodaira dimension 2 is called a surface of general type and is analogous to the Riemann surfaces of genus>= 2. Parallel to modular curves, one would study the quotient of HH by a discrete group commensurable with a Hilbert modular groupPSL (o ), where o is the ring of integers of 2 K K a real quadratic field K overQ. These spaces are called Hilbert modu- lar surfaces. PSL (o ) is irreducible, i.e. whenPSL (K) is embedded 2 K 2 into PSL (R)PSL (R), the image of PSL (o ) under each projec- 2 2 2 K tion is dense in PSL (R). Therefore the Hilbert modular surfaces are not simply products of modular curves. There is also a procedure to compactify such quotients by adding finite number of points. Contrary to the case of modular curves, the compact spaces thus obtained are highly singular. Hirzebruch gave a procedure to desingularize them. As a result, Hilbert modular surfaces can be studied using theory of compact complex surfaces. Hilbert modular surfaces have a deep rootin number theory. Because of this nature, one can calculate explic- itly the geometric invariants of them in terms of algebraic parameters. Their types according to the rough classification can then be found. This thesis aims at demonstrating how a Hilbert modular surface can be identified to be of general type. To provide necessary back- ground of the one-dimensional theory, it presents the basic theories of compactRiemannsurfacesandmodularcurvesindetail, andillustrates how the theory of compact Riemann surfaces can be applied to study modular curves. Hilbert modular surfaces were then introduced as an analogue of modular curves. Hirzebruch's procedure of desingulariza- tion was described. Analogous to the one-dimensional cases, the application of the the- ory of compact complex surfaces to Hilbert modular surfaces was illus- trated by demonstrating how the geometric invariants of the surfaces canbecalculatedfromthealgebraicparameters. Attheendofthethe- sis, a sufficient condition for a Hilbert modular surface to be of general type was given. DOI: 10.5353/th_b3955766 Subjects: Hilbert modular surfaces
Arithmetic Groups and Their Generalizations
Author: Lizhen JiPublisher: American Mathematical Soc.
ISBN: 0821848666
Category : Mathematics
Languages : en
Pages : 282
Book Description
In one guise or another, many mathematicians are familiar with certain arithmetic groups, such as $\mathbf{Z}$ or $\textrm{SL}(n, \mathbf{Z})$. Yet, many applications of arithmetic groups and many connections to other subjects within mathematics are less well known. Indeed, arithmetic groups admit many natural and important generalizations. The purpose of this expository book is to explain, through some brief and informal comments and extensive references, what arithmetic groups and their generalizations are, why they are important to study, and how they can be understood and applied to many fields, such as analysis, geometry, topology, number theory, representation theory, and algebraic geometry. It is hoped that such an overview will shed a light on the important role played by arithmetic groups in modern mathematics. Titles in this series are co-published with International Press, Cambridge, MA.Table of Contents: Introduction; General comments on references; Examples of basic arithmetic groups; General arithmetic subgroups and locally symmetric spaces; Discrete subgroups of Lie groups and arithmeticity of lattices in Lie groups; Different completions of $\mathbb{Q}$ and $S$-arithmetic groups over number fields; Global fields and $S$-arithmetic groups over function fields; Finiteness properties of arithmetic and $S$-arithmetic groups; Symmetric spaces, Bruhat-Tits buildings and their arithmetic quotients; Compactifications of locally symmetric spaces; Rigidity of locally symmetric spaces; Automorphic forms and automorphic representations for general arithmetic groups; Cohomology of arithmetic groups; $K$-groups of rings of integers and $K$-groups of group rings; Locally homogeneous manifolds and period domains; Non-cofinite discrete groups, geometrically finite groups; Large scale geometry of discrete groups; Tree lattices; Hyperbolic groups; Mapping class groups and outer automorphism groups of free groups; Outer automorphism group of free groups and the outer spaces; References; Index. Review from Mathematical Reviews: ...the author deserves credit for having done the tremendous job of encompassing every aspect of arithmetic groups visible in today's mathematics in a systematic manner; the book should be an important guide for some time to come.(AMSIP/43.
Classical Algebraic Geometry
Author: Igor V. DolgachevPublisher: Cambridge University Press
ISBN: 1107017653
Category : Mathematics
Languages : en
Pages : 653
Book Description
This detailed exposition makes classical algebraic geometry accessible to the modern mathematician.
Abelian Varieties
Author: Wolf P. BarthPublisher: Walter de Gruyter
ISBN: 3110889439
Category : Mathematics
Languages : en
Pages : 353
Book Description
The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.
Number Theory
Author: Michel WaldschmidtPublisher: American Mathematical Soc.
ISBN: 0821806068
Category : Mathematics
Languages : en
Pages : 410
Book Description
To observe the tenth anniversary of the founding of the Ramanujan Mathematical Society, an international conference on Discrete Mathematics and Number Theory was held in January 1996 in Tiruchirapalli, India. This volume contains proceedings from the number theory component of that conference. Papers are divided into four groups: arithmetic algebraic geometry, automorphic forms, elementary and analytic number theory, and transcendental number theory. This work deals with recent progress in current aspects of number theory and covers a wide variety of topics.