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Rational Points on Elliptic Curves by Joseph H. Silverman,John Tate Pdf
The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.
This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere. The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points. —Yuri Manin, Max-Planck-Institute, Bonn It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student. —Hendrik Lenstra, University Leiden Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics. —Jean-Louis Colliot-Thélène, Université Paris-Sud A beautiful subject, handled by a master. —Joseph Silverman, Brown University
Rational Points on Modular Elliptic Curves by Henri Darmon Pdf
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.
Rational Points and Arithmetic of Fundamental Groups by Jakob Stix Pdf
The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.
Rational Points on Algebraic Varieties by Emmanuel Peyre,Yuri Tschinkel Pdf
This book is devoted to the study of rational and integral points on higher- dimensional algebraic varieties. It contains research papers addressing the arithmetic geometry of varieties which are not of general type, with an em- phasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies. The book gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric con- structions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups. In recent years there has been substantial progress in our understanding of the arithmetic of algebraic surfaces. Five papers are devoted to cubic surfaces: Basile and Fisher study the existence of rational points on certain diagonal cubics, Swinnerton-Dyer considers weak approximation and Broberg proves upper bounds on the number of rational points on the complement to lines on cubic surfaces. Peyre and Tschinkel compare numerical data with conjectures concerning asymptotics of rational points of bounded height on diagonal cubics of rank 2. Kanevsky and Manin investigate the composition of points on cubic surfaces. Satge constructs rational curves on certain Kummer surfaces. Colliot-Thelene studies the Hasse principle for pencils of curves of genus 1. In an appendix to this paper Skorobogatov produces explicit examples of Enriques surfaces with a Zariski dense set of rational points.
Rational Points on Algebraic Varieties by Emmanuel Peyre,Yuri Tschinkel Pdf
This book is devoted to the study of rational and integral points on higher-dimensional algebraic varieties. It contains carefully selected research papers addressing the arithmetic geometry of varieties which are not of general type, with an emphasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies. The present volume gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric constructions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups.
Since the publication of the first edition of this work, considerable progress has been made in many of the questions examined. This edition has been updated and enlarged, and the bibliography has been revised. The variety of topics covered here includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian integers.
When discussing normative reasons, oughts, requirements of rationality, motivating reasons, and so on, we often have to use verbs like "believe" and "want" to capture a relevant subject's perspective. According to the received view about sentences involving these verbs, what they do is describe the subject's mental states. Many puzzles concerning normative discourse have to do with the role that mental states consequently appear to play in normative discourse. Tim Henning uses tools from semantics and the philosophy of language to develop an alternative account of sentences involving these verbs. According to this view, which is called parentheticalism, we very commonly use these verbs in a parenthetical sense. These verbs themselves express backgrounded side-remarks on the contents they embed, and these latter, embedded contents constitute the at-issue contents. This means that instead of speaking about the subject's mental states, we often use sentences involving "believe" and "want" to speak about the world from her point of view. Henning makes this notion precise, and uses it to solve various puzzles concerning normative discourse. The final result is a new, unified understanding of normative discourse, which gets by without postulating conceptual breaks between objective and subjective normative reasons, or normative reasons and rationality, or indeed between the reasons we ascribe to an agent and the reasons she herself can be expected to cite. Instead of being connected to either subjective mental states or objective facts, all of these normative statuses are can be adequately articulated by citing worldly considerations from a subject's point of view.
Complex Analysis and Algebraic Geometry by Kunihiko Kodaira,W. L. Jr Baily,T. Shioda Pdf
The articles in this volume cover some developments in complex analysis and algebraic geometry. The book is divided into three parts. Part I includes topics in the theory of algebraic surfaces and analytic surface. Part II covers topics in moduli and classification problems, as well as structure theory of certain complex manifolds. Part III is devoted to various topics in algebraic geometry analysis and arithmetic. A survey article by Ueno serves as an introduction to the general background of the subject matter of the volume. The volume was written for Kunihiko Kodaira on the occasion of his sixtieth birthday, by his friends and students. Professor Kodaira was one of the world's leading mathematicians in algebraic geometry and complex manifold theory: and the contributions reflect those concerns.
Information Security and Cryptology – ICISC 2017 by Howon Kim,Dong-Chan Kim Pdf
This book constitutes revised selected papers from the 20th International Conference on Information Security and Cryptology, ICISC 2017, held in Seoul, South Korea, in November/December 2017. The total of 20 papers presented in this volume were carefully reviewed and selected from 70 submissions. The papers were organized in topical sections named: symmetric key encryption; homomorphic encryption, side channel analysis and implementation; broadcast encryption; elliptic curve; signature and protocol; and network and system security.