Periods of Hilbert Modular Surfaces
Author : T. Oda
Publisher : Springer Science & Business Media
Page : 141 pages
File Size : 26,41 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 1468492012
Hilbert Modular Surfaces
Author : Gerard van der Geer
Publisher : Springer Science & Business Media
Page : 301 pages
File Size : 46,60 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 3642615538
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 Varieties and Modular Forms
Author : Eyal Zvi Goren
Publisher : American Mathematical Soc.
Page : 282 pages
File Size : 10,3 MB
Release : 2002
Category : Mathematics
ISBN : 082181995X
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 Forms with Coefficients in Intersection Homology and Quadratic Base Change
Author : Jayce Getz
Publisher : Springer Science & Business Media
Page : 264 pages
File Size : 39,51 MB
Release : 2012-03-28
Category : Mathematics
ISBN : 3034803516
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.
Periods of Hilbert Modular Surfaces
Author : T. Oda
Publisher :
Page : 144 pages
File Size : 39,67 MB
Release : 1982-01-01
Category :
ISBN : 9781468492026
Book Description
Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors
Author : Jan H. Bruinier
Publisher : Springer
Page : 159 pages
File Size : 32,15 MB
Release : 2004-10-11
Category : Mathematics
ISBN : 3540458727
Book Description
Around 1994 R. Borcherds discovered a new type of meromorphic modular form on the orthogonal group $O(2,n)$. These "Borcherds products" have infinite product expansions analogous to the Dedekind eta-function. They arise as multiplicative liftings of elliptic modular forms on $(SL)_2(R)$. The fact that the zeros and poles of Borcherds products are explicitly given in terms of Heegner divisors makes them interesting for geometric and arithmetic applications. In the present text the Borcherds' construction is extended to Maass wave forms and is used to study the Chern classes of Heegner divisors. A converse theorem for the lifting is proved.
Rational Points on Modular Elliptic Curves
Author : Henri Darmon
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 45,60 MB
Release : 2004
Category : Mathematics
ISBN : 0821828681
Book Description
The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture.
Hilbert Modular Forms: mod $p$ and $p$-Adic Aspects
Author : Fabrizio Andreatta
Publisher : American Mathematical Soc.
Page : 114 pages
File Size : 21,92 MB
Release : 2005
Category : Mathematics
ISBN : 0821836099
Book Description
We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings. In the characteristic $p$ theory we describe the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; we define operators $U$, $V$ and $\Theta_\chi$ and study the variation of the filtration under these operators. Our methods are geometric - comparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-$p$ structure, whose poles are supported on the non-ordinary locus.In the $p$-adic theory we study congruences between Hilbert modular forms. This applies to the study of congruences between special values of zeta functions of totally real fields. It also allows us to define $p$-adic Hilbert modular forms 'a la Serre' as $p$-adic uniform limit of classical modular forms, and compare them with $p$-adic modular forms 'a la Katz' that are regular functions on a certain formal moduli scheme. We show that the two notions agree for cusp forms and for a suitable class of weights containing all the classical ones. We extend the operators $V$ and $\Theta_\chi$ to the $p$-adic setting.
The 1-2-3 of Modular Forms
Author : Jan Hendrik Bruinier
Publisher : Springer Science & Business Media
Page : 273 pages
File Size : 36,63 MB
Release : 2008-02-10
Category : Mathematics
ISBN : 3540741194
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.
Algorithmic Number Theory
Author : Alf J. van der Poorten
Publisher : Springer
Page : 463 pages
File Size : 35,24 MB
Release : 2008-05-07
Category : Computers
ISBN : 3540794565
Book Description
This book constitutes the refereed proceedings of the 8th International Algorithmic Number Theory Symposium, ANTS 2008, held in Banff, Canada, in May 2008. The 28 revised full papers presented together with 2 invited papers were carefully reviewed and selected for inclusion in the book. The papers are organized in topical sections on elliptic curves cryptology and generalizations, arithmetic of elliptic curves, integer factorization, K3 surfaces, number fields, point counting, arithmetic of function fields, modular forms, cryptography, and number theory.
