Heegner Modules and Elliptic Curves
Author:
Publisher: Springer Science & Business Media
Published: 2004
Total Pages: 532
ISBN-13: 9783540222903
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Heegner Modules and Elliptic Curves
Author: Martin L. Brown
Publisher: Springer
Published: 2004-08-30
Total Pages: 523
ISBN-13: 3540444750
DOWNLOAD EBOOKHeegner points on both modular curves and elliptic curves over global fields of any characteristic form the topic of this research monograph. The Heegner module of an elliptic curve is an original concept introduced in this text. The computation of the cohomology of the Heegner module is the main technical result and is applied to prove the Tate conjecture for a class of elliptic surfaces over finite fields, this conjecture is equivalent to the Birch and Swinnerton-Dyer conjecture for the corresponding elliptic curves over global fields.
Rational Points on Modular Elliptic Curves
Author: Henri Darmon
Publisher: American Mathematical Soc.
Published: 2004
Total Pages: 146
ISBN-13: 0821828681
DOWNLOAD EBOOKThe 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.
Arithmetic of L-functions
Author: Cristian Popescu
Publisher: American Mathematical Soc.
Published:
Total Pages: 517
ISBN-13: 0821886983
DOWNLOAD EBOOKDrinfeld Modules, Modular Schemes And Applications
Author: M Van Der Put
Publisher: World Scientific
Published: 1997-08-27
Total Pages: 378
ISBN-13: 9814546402
DOWNLOAD EBOOKIn his 1974 seminal paper 'Elliptic modules', V G Drinfeld introduced objects into the arithmetic geometry of global function fields which are nowadays known as 'Drinfeld Modules'. They have many beautiful analogies with elliptic curves and abelian varieties. They study of their moduli spaces leads amongst others to explicit class field theory, Jacquet-Langlands theory, and a proof of the Shimura-Taniyama-Weil conjecture for global function fields.This book constitutes a carefully written instructional course of 12 lectures on these subjects, including many recent novel insights and examples. The instructional part is complemented by research papers centering around class field theory, modular forms and Heegner points in the theory of global function fields.The book will be indispensable for everyone who wants a clear view of Drinfeld's original work, and wants to be informed about the present state of research in the theory of arithmetic geometry over function fields.
Heegner Points and Rankin L-Series
Author: Henri Darmon
Publisher: Cambridge University Press
Published: 2004-06-21
Total Pages: 386
ISBN-13: 9780521836593
DOWNLOAD EBOOKThirteen articles by leading contributors on the history of the Gross-Zagier formula and its developments.
Number Theory I
Author: Yu. I. Manin
Publisher: Springer Science & Business Media
Published: 2013-04-17
Total Pages: 311
ISBN-13: 3662080052
DOWNLOAD EBOOKA unified survey of both the status quo and the continuing trends of various branches of number theory. Motivated by elementary problems, the authors present todays most significant results and methods. Topics covered include non-Abelian generalisations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. The book is rounded off with an overview of the major conjectures, most of which are based on analogies between functions and numbers, and on connections with other branches of mathematics such as analysis, representation theory, geometry and algebraic topology.
Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians
Author: Francis Nier
Publisher: Springer Science & Business Media
Published: 2005-02-11
Total Pages: 228
ISBN-13: 9783540242000
DOWNLOAD EBOOKThere has recently been a renewal of interest in Fokker-Planck operators, motivated by problems in statistical physics, in kinetic equations, and differential geometry. Compared to more standard problems in the spectral theory of partial differential operators, those operators are not self-adjoint and only hypoelliptic. The aim of the analysis is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. While exploring and improving recent results in this direction, this volume proposes a review of known techniques on: the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudo-differential calculus, the spectral theory of non-self-adjoint operators, the semi-classical analysis of Schrödinger-type operators, the Witten complexes, and the Morse inequalities.
Abstract Harmonic Analysis of Continuous Wavelet Transforms
Author:
Publisher: Springer Science & Business Media
Published: 2005
Total Pages: 212
ISBN-13: 9783540242598
DOWNLOAD EBOOKIntroduction to Modern Number Theory
Author: Yu. I. Manin
Publisher: Springer Science & Business Media
Published: 2006-03-30
Total Pages: 519
ISBN-13: 3540276920
DOWNLOAD EBOOKThis edition has been called ‘startlingly up-to-date’, and in this corrected second printing you can be sure that it’s even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.
