Number TheoryVolume I: Tools and Diophantine Equations

基本信息

Format Hardback | 650 pages

Dimensions 155 x 235 x 43.18mm | 2,470g

Publication date 01 Jun 2007

Publisher Springer-Verlag New York Inc.

Publication City/Country New York, NY, United States

Language English

Edition Statement 2007 ed.

Illustrations note XXIII, 650 p.

ISBN10 0387499229

ISBN13 9780387499222

页面参数仅供参考，具体以实物为准

书籍简介

本书的中心主题是求解狄奥芬丹方程，即必须以整数、有理数或更普遍的代数数来求解的方程或多项式方程系统。特别是这个主题是现代算术代数几何理论的核心动力。在这篇课文中，将通过其最基本的三个方面来考虑这个问题。本书包含350多道习题，文本基本上是自成一体的。更为复杂的技术已经被带到了二方程的主题上，为此，作者包括了五个关于这些技术的附录。

The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.

这本研究生水平的数论教科书的中心主题是求解二方程，即必须在整数、有理数或更普遍的代数中求解的方程或多项式方程系统。特别是这个主题是现代算术代数几何理论的核心动力。在这篇文章中，我们通过三个方面来考虑这个问题。

首先是局部方面：人们可以在p-adic场中进行分析，在这里，作者从研究有限场中的解开始，然后用Hensel提升法将这些解提升为局部解。第二是全球方面：使用数域，特别是使用类群和单位群。这个经典的主题在这里通过广泛的例子进行说明。第三个方面涉及具体的方程类，特别是椭圆曲线的一般研究和索菲特研究，包括2和3的回归以及海格纳点法。这些主题构成前两部分，形成第一卷。

本书第三部分详细论述了伯努利数、伽马函数、Zeta和L函数以及p-adic类似物的研究，包括许多有趣的原创应用。

更为复杂的技术已经被应用到了二方程的主题上，为此，作者在第四部分中加入了关于这些技术的五章，与第三部分一起构成第二卷。这些章节由Yann Bugeaud, Guillaume Hanrot, Maurice Mignotte, Sylvain Duquesne, Samir Siksek和作者撰写，包含了关于伽罗瓦表示的使用、高次元曲线上的点、超费马方程、Mihailescu对卡塔兰猜想的证明以及线性形式在对数中的应用等材料。

本书包含530个难度不同的练习，从正文的直接后果到研究问题，并包含许多重要的附加结果。

The central theme of this graduate-level number theory textbook is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three aspects.

The first is the local aspect: one can do analysis in p-adic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second is the global aspect: the use of number fields, and in particular of class groups and unit groups. This classical subject is here illustrated through a wide range of examples. The third aspect deals with specific classes of equations, and in particular the general and Diophantine study of elliptic curves, including 2 and 3-descent and the Heegner point method. These subjects form the first two parts, forming Volume I.

The study of Bernoulli numbers, the gamma function, and zeta and L-functions, and of p-adic analogues is treated at length in the third part of the book, including many interesting and original applications.

Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five chapters on these techniques forming the fourth part, which together with the third part forms Volume II. These chapters were written by Yann Bugeaud, Guillaume Hanrot, Maurice Mignotte, Sylvain Duquesne, Samir Siksek, and the author, and contain material on the use of Galois representations, points on higher-genus curves, the superfermat equation, Mihailescu's proof of Catalan's Conjecture, and applications of linear forms in logarithms.

The book contains 530 exercises of varying difficulty from immediate consequences of the main text to research problems, and contain many important additional results.

媒体推荐

"正在审查的这本书从数论的角度论述了二项分析。... 每一章最后都有练习题，从简单到相当有挑战性的问题都有。论述的清晰性是我们所期望的，它来自于两本非常成功的计算数论书籍的作者......并使这卷书成为二叉分析研究者的必读之作"。(Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, October, 2013)

"这是一部令人印象深刻的两卷本的《二项分析》教科书的第一卷。... 呈现给读者的是一个几乎压倒性的材料量。这本......教科书必将成为学生和研究人员的重要参考资料"。(C. Baxa, Monatshefte fur Mathematik, Vol. 157 (2), June, 2009)

"科恩（法国波尔多第一大学），一本即时的经典，独特地弥合了老式的、天真的处理方法和许多现代书籍之间的差距，这些书籍开发了刚才提到的工具......。总结一下。推荐。... 从本科生到教师的高年级学生"。(D. V. Feldman，《选择》，第45卷（5），2008年1月)

"《数论》有望填补数论核心教材的空白......。因此，总的来说，亨利-科恩的......。对任何人来说，《数论》都是一项惊人的成就。覆盖面很广，一般来说都是百科全书式的，练习题很好，有些很出色，有些甚至能让准备最充分的学生忙上很长时间，而且这本书的文化水平......非常高"。(Michael Berg, MathDL, 2007年7月)

"审查中的这本书从数论的角度论述了二项分析。... 论述的清晰度是我们所期望的，它来自于两本非常成功的计算数论书籍的作者......并使这卷书成为索菲特分析研究人员的必读之作"。(Franz Lemmermeyer, Zentralblatt MATH, Vol. 1119 (21), 2007)

"The book under review deals with Diophantine analysis from a number-theoretic point of view. ... Each chapter ends with exercises, ranging from simple to quite challenging problems. The clarity of the exposition is the one we expect from the author of two highly successful books on computational number theory ... and makes this volume a must-read for researchers in Diophantine analysis." (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, October, 2013)

"This is the first volume of a highly impressive two-volume textbook on Diophantine analysis. ... Readers are presented with an almost overwhelming amount of material. This ... text book is bound to become an important reference for students and researchers alike." (C. Baxa, Monatshefte fur Mathematik, Vol. 157 (2), June, 2009)

"Cohen (Universite Bordeaux I, France), an instant classic, uniquely bridges the gap between old-fashioned, naive treatments and the many modern books available that develop the tools just mentioned ... . Summing Up: Recommended. ... Upper-division undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (5), January, 2008)

"Number Theory, is poised to fill the gap as a core text in number theory ... . So, all in all, Henri Cohen's ... Number Theory are, to any mind, an amazing achievement. The coverage is thorough and generally all but encyclopedic, the exercises are good, some are excellent, some will keep even the best-prepared student busy for a long time, and the cultural level of the book ... is very high." (Michael Berg, MathDL, July, 2007)

"The book under review deals with Diophantine analysis from a number-theoretic point of view. ... The clarity of the exposition is the one we expect from the author of two highly successful books on computational number theory ... and makes this volume a must-read for researchers in Diophantine analysis." (Franz Lemmermeyer, Zentralblatt MATH, Vol. 1119 (21), 2007)

目录

to Diophantine Equations.

- to Diophantine Equations.

- Tools.

- Abelian Groups, Lattices, and Finite Fields.

- Basic Algebraic Number Theory.

- p-adic Fields.

- Quadratic Forms and Local-Global Principles.

- Diophantine Equations.

- Some Diophantine Equations.

- Elliptic Curves.

- Diophantine Aspects of Elliptic Curves.