Author: Friedrich HirzebruchPublisher:
ISBN:
Category : Discontinuous groups
Languages : en
Pages : 200
Book Description
Lectures on Hilbert Modular Surfaces
Author: Friedrich HirzebruchPublisher:
ISBN:
Category : Discontinuous groups
Languages : en
Pages : 200
Book Description
Lectures on Hilbert Modular Varieties and Modular Forms
Author: Eyal Zvi GorenPublisher: American Mathematical Soc.
ISBN: 082181995X
Category : Mathematics
Languages : en
Pages : 282
Book Description
This book is devoted to certain aspects of the theory of $p$-adic Hilbert modular forms and moduli spaces of abelian varieties with real multiplication. The theory of $p$-adic modular forms is presented first in the elliptic case, introducing the reader to key ideas of N. M. Katz and J.-P. Serre. It is re-interpreted from a geometric point of view, which is developed to present the rudiments of a similar theory for Hilbert modular forms. The theory of moduli spaces of abelianvarieties with real multiplication is presented first very explicitly over the complex numbers. Aspects of the general theory are then exposed, in particular, local deformation theory of abelian varieties in positive characteristic. The arithmetic of $p$-adic Hilbert modular forms and the geometry ofmoduli spaces of abelian varieties are related. This relation is used to study $q$-expansions of Hilbert modular forms, on the one hand, and stratifications of moduli spaces on the other hand. The book is addressed to graduate students and non-experts. It attempts to provide the necessary background to all concepts exposed in it. It may serve as a textbook for an advanced graduate course.
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.
Hilbert Modular Surfaces
Author: Friedrich HirzebruchPublisher:
ISBN:
Category :
Languages : en
Pages : 100
Book Description
Lectures on Hilbert Schemes of Points on Surfaces
Author: Hiraku NakajimaPublisher: American Mathematical Soc.
ISBN: 0821819569
Category : Mathematics
Languages : en
Pages : 146
Book Description
It has been realized that Hilbert schemes originally studied in algebraic geometry are closely related to several branches of mathematics, such as singularities, symplectic geometry, representation theory - even theoretical physics. This book reflects this feature of Hilbert schemes.
Hilbert Modular Surfaces
Author: Friedrich HirzebruchPublisher:
ISBN:
Category : Discontinuous groups
Languages : en
Pages : 108
Book Description
The 1-2-3 of Modular Forms
Author: Jan Hendrik BruinierPublisher: Springer Science & Business Media
ISBN: 3540741194
Category : Mathematics
Languages : en
Pages : 273
Book Description
This book grew out of three series of lectures given at the summer school on "Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004. The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture. Each part treats a number of beautiful applications.
Hilbert Modular Surfaces
Author: Friedrich HirzebruchPublisher:
ISBN:
Category : Discontinuous groups
Languages : en
Pages : 103
Book Description
Periods of Hilbert Modular Surfaces
Author: T. OdaPublisher: Springer Science & Business Media
ISBN: 1468492012
Category : Mathematics
Languages : en
Pages : 141
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