Modular Curves And Abelian Varieties

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Modular Curves and Abelian Varieties

John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet
Modular Curves and Abelian Varieties Book PDF

Author : John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet
Publisher : Birkhäuser
Page : 291 pages
File Size : 54,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879194

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Modular Curves and Abelian Varieties by John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet Pdf

This book presents lectures from a conference on "Modular Curves and Abelian Varieties'' at the Centre de Recerca Matemtica (Bellaterra, Barcelona). The articles in this volume present the latest achievements in this extremely active field and will be of interest both to specialists and to students and researchers. Many contributions focus on generalizations of the Shimura-Taniyama conjecture to varieties such as elliptic Q-curves and Abelian varieties of GL_2-type. The book also includes several key articles in the subject that do not correspond to conference lectures.

Abelian l-Adic Representations and Elliptic Curves

Jean-Pierre Serre
Abelian l Adic Representations and Elliptic Curves Book PDF

Author : Jean-Pierre Serre
Publisher : CRC Press
Page : 203 pages
File Size : 51,9 Mb
Release : 1997-11-15
Category : Mathematics
ISBN : 9781439863862

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Abelian l-Adic Representations and Elliptic Curves by Jean-Pierre Serre Pdf

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one

Arithmetic on Modular Curves

G. Stevens
Arithmetic on Modular Curves Book PDF

Author : G. Stevens
Publisher : Springer Science & Business Media
Page : 233 pages
File Size : 52,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468491654

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Arithmetic on Modular Curves by G. Stevens Pdf

One of the most intriguing problems of modern number theory is to relate the arithmetic of abelian varieties to the special values of associated L-functions. A very precise conjecture has been formulated for elliptic curves by Birc~ and Swinnerton-Dyer and generalized to abelian varieties by Tate. The numerical evidence is quite encouraging. A weakened form of the conjectures has been verified for CM elliptic curves by Coates and Wiles, and recently strengthened by K. Rubin. But a general proof of the conjectures seems still to be a long way off. A few years ago, B. Mazur [26] proved a weak analog of these c- jectures. Let N be prime, and be a weight two newform for r 0 (N) . For a primitive Dirichlet character X of conductor prime to N, let i\ f (X) denote the algebraic part of L (f , X, 1) (see below). Mazur showed in [ 26] that the residue class of Af (X) modulo the "Eisenstein" ideal gives information about the arithmetic of Xo (N). There are two aspects to his work: congruence formulae for the values Af(X) , and a descent argument. Mazur's congruence formulae were extended to r 1 (N), N prime, by S. Kamienny and the author [17], and in a paper which will appear shortly, Kamienny has generalized the descent argument to this case.

Moduli of Curves and Abelian Varieties

Carel Faber,Eduard Looijenga
Moduli of Curves and Abelian Varieties Book PDF

Author : Carel Faber,Eduard Looijenga
Publisher : Springer Science & Business Media
Page : 205 pages
File Size : 44,5 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9783322901729

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Moduli of Curves and Abelian Varieties by Carel Faber,Eduard Looijenga Pdf

The Dutch Intercity Seminar on Moduli, which dates back to the early eighties, was an initiative of G. van der Geer, F. Oort and C. Peters. Through the years it became a focal point of Dutch mathematics and it gained some fame, also outside Holland, as an active biweekly research seminar. The tradition continues up to today. The present volume, with contributions of R. Dijkgraaf, C. Faber, G. van der Geer, R. Hain, E. Looijenga, and F. Oort, originates from the seminar held in 1995--96. Some of the articles here were discussed, in preliminary form, in the seminar; others are completely new. Two introductory papers, on moduli of abelian varieties and on moduli of curves, accompany the articles.

A First Course in Modular Forms

Fred Diamond,Jerry Shurman
A First Course in Modular Forms Book PDF

Author : Fred Diamond,Jerry Shurman
Publisher : Springer Science & Business Media
Page : 450 pages
File Size : 40,5 Mb
Release : 2006-03-30
Category : Mathematics
ISBN : 9780387272269

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A First Course in Modular Forms by Fred Diamond,Jerry Shurman Pdf

This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. The authors assume no background in algebraic number theory and algebraic geometry. Exercises are included.

Introduction to Abelian Varieties

V. Kumar Murty
Introduction to Abelian Varieties Book PDF

Author : V. Kumar Murty
Publisher : American Mathematical Soc.
Page : 136 pages
File Size : 44,9 Mb
Release : 1986
Category : Abelian groups
ISBN : 082187005X

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Introduction to Abelian Varieties by V. Kumar Murty Pdf

This book presents an elementary and self-contained approach to Abelian varieties, a subject that plays a central role in algebraic and analytic geometry, number theory, and complex analysis. The book is based on notes from a course given at Concordia University and would be useful for independent study or as a textbook for graduate courses in complex analysis, Riemann surfaces, number theory, or analytic geometry. Murty works mostly over the complex numbers, discussing the theorem of Abel-Jacobi and Lefschetz's theorem on projective embeddings. After presenting some examples, Murty touches on Abelian varieties over number fields, as well as the conjecture of Tate (Faltings's theorem) and its relation to Mordell's conjecture. References are provided to guide the reader in further study.

Introduction to Abelian Varieties

Vijaya Kumar Murty
Introduction to Abelian Varieties Book PDF

Author : Vijaya Kumar Murty
Publisher : Unknown
Page : 112 pages
File Size : 40,9 Mb
Release : 1993
Category : Abelian varieties
ISBN : 1470438496

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Introduction to Abelian Varieties by Vijaya Kumar Murty Pdf

The book represents an introduction to the theory of abelian varieties with a view to arithmetic. The aim is to introduce some of the basics of the theory as well as some recent arithmetic applications to graduate students and researchers in other fields. The first part contains proofs of the Abel-Jacobi theorem, Riemann's relations and the Lefschetz theorem on projective embeddings over the complex numbers in the spirit of S. Lang's book Introduction to algebraic and abelian functions. Then the Jacobians of Fermat curves as well as some modular curves are discussed. Finally, as an application.

Abelian Varieties and Number Theory

Moshe Jarden,Tony Shaska
Abelian Varieties and Number Theory Book PDF

Author : Moshe Jarden,Tony Shaska
Publisher : American Mathematical Soc.
Page : 200 pages
File Size : 49,8 Mb
Release : 2021-05-03
Category : Education
ISBN : 9781470452070

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Abelian Varieties and Number Theory by Moshe Jarden,Tony Shaska Pdf

This book is a collection of articles on Abelian varieties and number theory dedicated to Gerhard Frey's 75th birthday. It contains original articles by experts in the area of arithmetic and algebraic geometry. The articles cover topics on Abelian varieties and finitely generated Galois groups, ranks of Abelian varieties and Mordell-Lang conjecture, Tate-Shafarevich group and isogeny volcanoes, endomorphisms of superelliptic Jacobians, obstructions to local-global principles over semi-global fields, Drinfeld modular varieties, representations of etale fundamental groups and specialization of algebraic cycles, Deuring's theory of constant reductions, etc. The book will be a valuable resource to graduate students and experts working on Abelian varieties and related areas.

A First Course in Modular Forms

Fred Diamond,Jerry Shurman
A First Course in Modular Forms Book PDF

Author : Fred Diamond,Jerry Shurman
Publisher : Springer Science & Business Media
Page : 448 pages
File Size : 51,5 Mb
Release : 2006-03-30
Category : Mathematics
ISBN : 9780387272269

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A First Course in Modular Forms by Fred Diamond,Jerry Shurman Pdf

This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. The authors assume no background in algebraic number theory and algebraic geometry. Exercises are included.

Curves, Jacobians, and Abelian Varieties

Ron Donagi
Curves Jacobians and Abelian Varieties Book PDF

Author : Ron Donagi
Publisher : American Mathematical Soc.
Page : 342 pages
File Size : 46,9 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821851432

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Curves, Jacobians, and Abelian Varieties by Ron Donagi Pdf

This volume contains the proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on the Schottky Problem, held in June 1990 at the University of Massachusetts at Amherst. The conference explored various aspects of the Schottky problem of characterizing Jacobians of curves among all abelian varieties. Some of the articles study related themes, including the moduli of stable vector bundles on a curve, Prym varieties and intermediate Jacobians, and special Jacobians with exotic polarizations or product structures.

The Arithmetic of Elliptic Curves

Joseph H. Silverman
The Arithmetic of Elliptic Curves Book PDF

Author : Joseph H. Silverman
Publisher : Springer Science & Business Media
Page : 414 pages
File Size : 46,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475719208

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The Arithmetic of Elliptic Curves by Joseph H. Silverman Pdf

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.

Moduli of Abelian Varieties

Gerard van der Geer,C. Faber,Frans Oort
Moduli of Abelian Varieties Book PDF

Author : Gerard van der Geer,C. Faber,Frans Oort
Publisher : Birkhäuser
Page : 526 pages
File Size : 42,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883030

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Moduli of Abelian Varieties by Gerard van der Geer,C. Faber,Frans Oort Pdf

Abelian varieties and their moduli are a topic of increasing importance in today`s mathematics, applications ranging from algebraic geometry and number theory to mathematical physics. This collection of 17 refereed articles originates from the third "Texel Conference" held in 1999. Leading experts discuss and study the structure of the moduli spaces of abelian varieties and related spaces, giving an excellent view of the state of the art in this field.

Geometric Modular Forms and Elliptic Curves

Haruzo Hida
Geometric Modular Forms and Elliptic Curves Book PDF

Author : Haruzo Hida
Publisher : World Scientific
Page : 468 pages
File Size : 54,8 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814368643

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Geometric Modular Forms and Elliptic Curves by Haruzo Hida Pdf

This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the Shimura?Taniyama conjecture, is given. In addition, the book presents an outline of the proof of diverse modularity results of two-dimensional Galois representations (including that of Wiles), as well as some of the author's new results in that direction.In this new second edition, a detailed description of Barsotti?Tate groups (including formal Lie groups) is added to Chapter 1. As an application, a down-to-earth description of formal deformation theory of elliptic curves is incorporated at the end of Chapter 2 (in order to make the proof of regularity of the moduli of elliptic curve more conceptual), and in Chapter 4, though limited to ordinary cases, newly incorporated are Ribet's theorem of full image of modular p-adic Galois representation and its generalization to ?big? ?-adic Galois representations under mild assumptions (a new result of the author). Though some of the striking developments described above is out of the scope of this introductory book, the author gives a taste of present day research in the area of Number Theory at the very end of the book (giving a good account of modularity theory of abelian ?-varieties and ?-curves).

Classification of Irregular Varieties

Edoardo Ballico,Fabrizio Catanese,Ciro Ciliberto
Classification of Irregular Varieties Book PDF

Author : Edoardo Ballico,Fabrizio Catanese,Ciro Ciliberto
Publisher : Springer
Page : 155 pages
File Size : 51,6 Mb
Release : 2006-12-08
Category : Mathematics
ISBN : 9783540470168

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Classification of Irregular Varieties by Edoardo Ballico,Fabrizio Catanese,Ciro Ciliberto Pdf

M. Andreatta, E. Ballico, J. Wisniewski: Projective manifolds containing large linear subspaces; - F. Bardelli: Algebraic cohomology classes on some specialthreefolds; - Ch. Birkenhake, H. Lange: Norm-endomorphisms of abelian subvarieties; - C. Ciliberto, G.van der Geer: On the jacobian of ahyperplane section of a surface; - C. Ciliberto, H. Harris, M. Teixidor i Bigas: On the endomorphisms of Jac (W1d(C)) when p=1 and C has general moduli; - B. van Geemen: Projective models of Picard modular varieties; - J. Kollar, Y. Miyaoka, S. Mori: Rational curves on Fano varieties; - R. Salvati Manni: Modular forms of the fourth degree; A. Vistoli: Equivariant Grothendieck groups and equivariant Chow groups; - Trento examples; Open problems.

Lectures on Hilbert Modular Varieties and Modular Forms

Eyal Zvi Goren,Zvi Goren
Lectures on Hilbert Modular Varieties and Modular Forms Book PDF

Author : Eyal Zvi Goren,Zvi Goren
Publisher : American Mathematical Soc.
Page : 282 pages
File Size : 42,7 Mb
Release : 2002
Category : Abelian varieties
ISBN : 9780821819951

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Lectures on Hilbert Modular Varieties and Modular Forms by Eyal Zvi Goren,Zvi Goren Pdf

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.