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Author: Gerard van der Geer Publisher: 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.
Author: Gerard van der Geer Publisher: 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.
Author: Tsz-On Mario Chan Publisher: 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
Author: Eyal Zvi Goren Publisher: American Mathematical Soc. ISBN: 082181995X Category : Abelian varieties 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.
Author: Jayce Getz Publisher: Springer Science & Business Media ISBN: 3034803516 Category : Mathematics Languages : en Pages : 264
Book Description
In the 1970s Hirzebruch and Zagier produced elliptic modular forms with coefficients in the homology of a Hilbert modular surface. They then computed the Fourier coefficients of these forms in terms of period integrals and L-functions. In this book the authors take an alternate approach to these theorems and generalize them to the setting of Hilbert modular varieties of arbitrary dimension. The approach is conceptual and uses tools that were not available to Hirzebruch and Zagier, including intersection homology theory, properties of modular cycles, and base change. Automorphic vector bundles, Hecke operators and Fourier coefficients of modular forms are presented both in the classical and adèlic settings. The book should provide a foundation for approaching similar questions for other locally symmetric spaces.
Author: Gerard van der Geer
Author: Gerard van der Geer
Author: T. Oda
Author: Friedrich Hirzebruch
Author: Friedrich Hirzebruch
Author: T. Oda
Author: Tsz-On Mario Chan
Author: Eyal Zvi Goren
Author: Eberhard Freitag
Author: Jayce Getz
Author: Gerardus Bartholomeus Maria Van der Geer