Number TheoryVolume II: Analytic and Modern Tools

基本信息

Format Hardback | 596 pages

Dimensions 155 x 235 x 33.27mm | 1,172g

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, 596 p.

ISBN10 0387498931

ISBN13 9780387498935

页面参数仅供参考，具体以实物为准

书籍简介

本书论述了现在被称为 "显性数论 "的几个方面。中心主题是索方程的解，即必须用整数、有理数或更普遍的代数数来解的方程或多项式方程系统。这个主题尤其是现代算术代数几何理论的核心动力。在这篇文章中，我们通过三个最基本的方面来考虑这个问题。局部方面，全局方面，第三个方面是Zeta和L-函数的理论。这最后一个方面可以被认为是整个主题的统一主题。

This book deals with several aspects of what is now called "explicit number theory." The central theme 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 local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.

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

"科恩（法国波尔多第一大学）是一部经典之作，他独特地弥合了老式的、天真的治疗方法和许多开发刚才提到的工具的现代书籍之间的差距......。总结。推荐。... 从本科生到教师的高年级学生"。(D. V. Feldman，《选择》，第45(5)卷，2008年1月)

"这本书涉及到'显性数论'的各个方面。... 中心主题......是二方程的解决。... 它将这一主题的有趣'哲学'与对细节的百科全书式的把握结合起来。通过贡献的章节扩大作者的影响是一个好主意。也许这是一个趋势的开始，因为这个主题会越来越多。... 毫无疑问，它将被导师们在他们的研究生课程中挖掘出来，特别是为了包括一些最近被证明的内容"。(R. C. Baker, Mathematical Reviews, Issue 2008 e)

"这是一本令人印象深刻的两卷本索非亚分析教科书的第二卷。......向读者展示了几乎是压倒性的大量材料。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)

"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)

"The book deals with aspects of 'explicit number theory'. ... The central theme ... is the solution of Diophantine equations. ... It combines an interesting 'philosophy' of the subject with an encyclopedic grasp of detail. The extension of the author's reach via the contributed chapters is a good idea. Perhaps it is the start of a trend, as the subject grows more and more. ... It will undoubtedly be mined by instructors for their graduate courses, particularly for the purpose of including some recently-proved content." (R. C. Baker, Mathematical Reviews, Issue 2008 e)

"This is the second 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)

目录

Analytic Tools.

- Bernoulli Polynomials and the Gamma Function.

- Dirichlet Series and L-Functions.

- p-adic Gamma and L-Functions.

- Modern Tools.

- Applications of Linear Forms in Logarithms.

- Rational Points on Higher-Genus Curves.

- The Super-Fermat Equation.

- The Modular Approach to Diophantine Equations.

- Catalan's Equation.