约翰·泰特John Tate

首先让我们澄清，弗瑞兹·约翰 Torrence 泰特的名字后面应该有Jr，因为他的父亲也叫约翰 Torrence 泰特。约翰·泰特 Sr（1889年7月28日出生于爱荷华州Lennox）的父亲是苏格兰后裔，母亲是爱尔兰后裔。他在内布拉斯加大学获得物理学学位，然后被柏林大学授予博士学位。他于1917年12月28日与Lois Beatrice Fossler结婚，她是一名高中英语教师，当时他已在明尼苏达大学任职。泰特 Sr.是明尼苏达大学的物理学正教授，当时他的儿子约翰 Torrence 泰特 Jr出生。泰特 Jr，本传记的主题，在明尼阿波利斯长大。

1939年，约翰的母亲去世。第二次世界大战期间，约翰的父亲在国家国防研究委员会任职，负责研究水下作战的部门。约翰在成长过程中对数学谜题着迷，尤其喜欢读他父亲拥有的亨利·杜德耐的书。高中时，他读了埃里克·坦普尔·贝尔的Men of mathematics，从中了解了二次互反律和约翰·彼得·古斯塔夫·勒热纳·狄利克雷关于等差数列中素数的定理。然而，尽管他喜欢读到的这些思想，他认为数学是比他更聪明的人才能学的科目，因此他决定在大学学习物理。他于1946年毕业于哈佛大学，并前往普林斯顿大学，仍然打算从事物理研究。然而，在普林斯顿研究生第一年期间，他清楚地认识到数学不仅是他最喜欢的科目，也是他最有天赋的科目。他获准转入数学研究生学习，并被指派埃米尔·阿廷作为他的学位论文导师。纯属巧合的是，他的学位论文导师对泰特还是学童时最着迷的课题做出了重大贡献。

1950年，泰特因其学位论文Fourier Analysis in Number Fields and Hecke's Zeta Functions[5]被授予博士学位：-

在他的博士论文中，泰特将调和分析引入数论，为自守形式的adelic方法和罗伯特·朗兰兹纲领铺平了道路。

在[4]中，作者们写道：-

在他的学位论文中，他通过一种涉及idele群上傅里叶分析的新方法，证明了埃里希·赫克的L-级数的函数方程，这篇论文后来成为经典。

该学位论文于1967年发表。回到1950年，即泰特获得博士学位的年份，我们注意到他的父亲于当年5月去世。

1950年，泰特被任命为普林斯顿的研究助理和讲师。1951-1952年在普林斯顿大学举办的埃米尔·阿廷-泰特类域论讨论班涵盖了群的上同调理论、代数数论的基础、类构造的初步讨论、局部类域论、整体类域论以及类构造和安德烈·韦伊群的抽象理论。其中部分内容由埃米尔·阿廷和泰特写成Class field theory一书，于1968年出版。在普林斯顿担任研究助理的三年（1950-53）期间，泰特发表了如下论文：On the relation between extremal points of convex sets and homomorphisms of algebras (1951)；（与埃米尔·阿廷）A note on finite ring extensions (1951)；Genus change in inseparable extensions of function fields (1952)；（与塞尔日·兰）On Chevalley's proof of Luroth's theorem (1952)；以及The higher dimensional cohomology groups of class field theory (1952)。对于最后提到的这篇论文，泰特于1956年获得了美国数学会颁发的法兰克·尼尔森·寇尔数论奖。他在1953-54年作为访问教授在哥伦比亚大学度过，然后于1954年被任命到哈佛大学。他一直担任此职位直到1990年，当时他接受了德克萨斯大学奥斯汀分校的Sid Richardson Regents讲席。

London Mathematical Society于1999年选举泰特为荣誉会员。我们引用[5]中的引文，其中概述了泰特对数学的卓越贡献：-

他在类域论以及局部和整体域上的埃瓦里斯特·伽罗瓦上同调方面的工作，尤其是他的对偶理论，为现代数论的许多内容奠定了基础；而他为类域论而发明的有限群的泰特上同调群，是代数学家的标准工具。泰特的深刻洞见对六十年代以来算术代数几何的发展产生了关键影响。也许最著名的是他关于有限域和整体域上簇的代数闭链的猜想，这些猜想提出于35年前，但至今在很大程度上仍未证明。同样引人注目的是他1966年的开创性论文“p-可除群”，该文首次认识到p进域绝对埃瓦里斯特·伽罗瓦群的p进表示的丰富性，并指出了威廉·瓦兰斯·道格拉斯·霍奇理论的p进类似物的存在。这现在是理解代数簇算术的关键工具。泰特关于有限域上阿贝尔簇分类的工作是标准理论的核心部分，为几乎所有关于志村五郎簇的L-函数的工作奠定了基础，也是研究有限域上动机的起点。通过发现刚性解析空间，他为p进整体分析建立了新的基础，这在数论、代数几何和表示论中有广泛的应用。椭圆曲线理论极大地归功于他的贡献，无论是理论上的还是计算上的；高度函数理论（Neron-泰特和Mazur-泰特）和下降理论（包括他构造的著名的伊戈尔·沙法列维奇-泰特群）对于理解椭圆曲线的算术至关重要，而泰特确定椭圆曲线坏约化的算法在计算中同样重要。其他具有深远意义的贡献包括他与让-皮埃尔·塞尔关于阿贝尔簇形变理论的工作，他对代数K-理论及其与埃瓦里斯特·伽罗瓦上同调关系的贡献，他在Stark猜想方面的工作，以及最近他在非交换环论方面的工作。

泰特在1959-61年间获得Sloan Fellowship，在1965-66年间获得Guggenheim Fellowship。他是1970年在尼斯举行的国际数学家大会的全会演讲者，当时他作了Symbols in Arithmetic的演讲。1972年，他是美国数学会的Colloquium Lecturer，并作了关于The arithmetic of elliptic curves的演讲。他是1974年决定菲尔兹奖奖项的委员会成员。该委员会只颁发了两个奖项（给恩里科·邦别里和戴维·芒福德），令数学界感到惊讶。在温哥华国际数学家大会的颁奖仪式上，是泰特报告了戴维·芒福德的工作。

泰特于1969年荣幸地当选为美国国家科学院成员，并于1992年在巴黎当选为Académie de Sciences成员。1995年，他获得了来自美国数学会的Leroy P 凯瑟琳·斯蒂尔终身成就奖[6]：-

……以表彰其跨越四十五年的科学成就。在此期间，他在代数、代数几何和数论的许多重要发展中产生了深远影响。

2003年，他获得了沃尔夫奖：-

……因其在代数数论中创建了基本概念。

尽管这段引文与我们上面引用的伦敦数学会的引文相似，但我们给出以下摘录：-

四分之一个多世纪以来，泰特教授的思想主导了算术代数几何的发展。泰特引入了开创性的技术和概念，开创了许多至今仍充满活力的理论。这些包括局部域和adele环上的约瑟夫·傅里叶分析、埃瓦里斯特·伽罗瓦上同调、刚性解析簇理论、p-可除群和p进威廉·瓦兰斯·道格拉斯·霍奇分解，仅举几例。泰特激励了所有从事数论工作的人。众多概念以他的名字命名：有限群的泰特上同调、阿贝尔簇的泰特模、泰特-伊戈尔·沙法列维奇群、Lubin-泰特群、Neron-泰特高度、泰特动机、佐藤干夫-泰特猜想、泰特扭转、泰特椭圆曲线等。泰特是代数数论中一个受人尊敬的名字。

在1980-81学年的第一学期，泰特在巴黎南大学（奥赛）讲授了一门关于Stark猜想的课程。这门课程于1984年以Les conjectures de Stark sur les fonctions L d'Artin en s = 0为书名出版。这并不是泰特唯一一本基于他先前讲授的课程而写成的书。1992年，他与Joseph H Silverman合著出版了Rational points on elliptic curves。这本书基于泰特30多年前于1961年在哈弗福德学院讲授的一门课程。Andrew Bremner的书评开头如下：-

作者们的目标是在一个技术上困难的领域写一本普通数学专业本科生也能读懂的教科书，看来他们出色地做到了这一点。这本书相当令人愉快。……对于这样一个领域的本科生教科书来说，最明显的缺点是它不可能完全严格，因此，正如作者们所声明的，“第一章中关于椭圆曲线的大部分基础材料旨在解释和说服，而不是严格证明。”一个附录确实展开了必要的代数几何，但整本书中对底层几何的处理是非正式的，从而允许更快速、更直观地进入数论。

2000年5月24日，迈克尔·阿蒂亚和泰特在巴黎介绍了克莱数学研究所的千禧年大奖难题。泰特的演讲涵盖了波恩哈德·黎曼假设、Birch-Swinnerton-Dyer猜想和$P = NP$问题。他解释了这些问题，并将它们置于其历史背景中。

2010年3月24日，挪威科学与文学院院长宣布，泰特将于5月25日在奥斯陆被授予尼尔斯·阿贝尔奖：-

……因其对数论广泛而持久的影响。

新闻稿如下：-

数论从素数之谜延伸到我们在现代计算机中存储、传输和保护信息的方式。在过去一个世纪里，它已发展成为数学中最精细、最复杂的分支之一，与其他关键领域有着深刻的互动。泰特是这一发展的主要缔造者。泰特的科学成就跨越六十年。大量基本的数学思想和构造由泰特开创，后来以他命名，例如泰特模、泰特曲线、泰特循环、威廉·瓦兰斯·道格拉斯·霍奇-泰特分解、泰特上同调、让-皮埃尔·塞尔-泰特参数、Lubin-泰特群、泰特迹、伊戈尔·沙法列维奇-泰特群、Néron-泰特高度，仅举几例。据尼尔斯·阿贝尔委员会称：“代数数论和算术几何中的许多主要研究方向之所以可能，仅仅是因为泰特的深刻贡献和启发性洞见。他确实在现代数学上留下了显著的印记。”

我们应以最后一点说明作结。泰特是尼古拉·布尔巴基团队中较年轻的成员之一，并且在该团队中几乎独一无二，因为他不是法国人。

First let us clarify that John Torrence Tate should have Jr after his name since his father was also named John Torrence Tate. John Tate Sr (born Lennox, Iowa, 28 July 1889) had a father of Scottish descent and a mother of Irish descent. He obtained a degree in physics from the University of Nebraska and was then awarded a doctorate by the University of Berlin. He married Lois Beatrice Fossler, a high school teacher of English, on 28 December 1917 by which time he was on the staff at the University of Minnesota. John Tate Sr. was a full professor of physics at the University of Minnesota when his son John Torrence Tate Jr was born. John Tate Jr, the subject of this biography, was brought up in Minneapolis.

In 1939 John's mother died. During World War II, John's father served on the National Defense Research Committee, in charge of the Division which researched undersea warfare. John grew up with a fascination for mathematical puzzles, in particular reading books by Henry Dudeney that his father owned. When at high school, he read E T Bell's Men of mathematics from which he learnt about quadratic reciprocity and Dirichlet's theorem on primes in an arithmetic progression. However, despite loving the ideas he had read about, he thought that mathematics was a subject for people who were cleverer than he was, so he decided to study physics at university. He graduated from Harvard University in 1946 and went to Princeton University, still with the intention of undertaking research in physics. However, during his first year of graduate study at Princeton it became clear to him that mathematics was not only the subject he liked best but it was also the subject for which he had the most talent. He was allowed to transfer to graduate study in mathematics and was assigned Emil Artin as his thesis advisor. It was pure coincidence that his thesis advisor had made major contributions to the topics that had most fascinated Tate when he was a schoolboy.

In 1950 Tate was awarded his doctorate for his thesis Fourier Analysis in Number Fields and Hecke's Zeta Functions [5]:-

In his doctoral thesis, Tate introduced harmonic analysis into number theory, paving the way for the adelic approach to automorphic forms and the Langlands programme.

In [4] the authors write:-

In his thesis, which has become a classic, he proved the functional equation for Hecke's L-series by a novel method involving Fourier analysis on idele groups.

The thesis was published in 1967. Returning to 1950, the year Tate was awarded his doctorate, we note that his father died in May of that year.

Tate was appointed as a research assistant and instructor at Princeton in 1950. The Artin-Tate seminar on class field theory given at Princeton University in 1951-1952 covered cohomology theory of groups, the fundamentals of algebraic number theory, a preliminary discussion of class formations, local class field theory, global class field theory, and the abstract theory of class formations and Weil group. Parts of this was written up as the book Class field theory by Artin and Tate and published in 1968. During his three years (1950-53) as a research assistant at Princeton, Tate published papers such as: On the relation between extremal points of convex sets and homomorphisms of algebras (1951); (with Emil Artin) A note on finite ring extensions (1951); Genus change in inseparable extensions of function fields (1952); (with Serge Lang) On Chevalley's proof of Luroth's theorem (1952); and The higher dimensional cohomology groups of class field theory (1952). For this last mentioned paper, Tate received the Frank Nelson Cole Prize in Number Theory from the American Mathematical Society in 1956. He spent the year 1953-54 as a visiting professor at Columbia University then, in 1954, he was appointed to Harvard University. He remained in this position until 1990 when he accepted the Sid Richardson Regents Chair at the University of Texas at Austin.

The London Mathematical Society elected Tate to Honorary Membership in 1999. We quote from the citation in [5] which gives an overview of Tate's remarkable contributions to mathematics:-

His work on class field theory and Galois cohomology over local and global fields, especially his duality theory, underpins much of modern number theory; and the Tate cohomology groups for finite groups, which he invented for use in class field theory, are a standard tool of algebraists. Tate's deep insights have had a crucial impact on the development of arithmetic algebraic geometry from the sixties onwards. Perhaps most celebrated are his conjectures about algebraic cycles on varieties over finite and global fields, formulated 35 years ago but still largely unproved. Equally striking is his seminal 1966 paper 'p-divisible groups', which for the first time recognised the richness of p-adic representations of the absolute Galois group of a p-adic field, as well as indicating the existence of a p-adic analogue of Hodge theory. This is now a key tool in understanding the arithmetic of algebraic varieties. Tate's work on classification of abelian varieties over finite fields is a core part of standard theory, underpinning almost all work on the L-functions of Shimura varieties as well as being the starting point for the study of motives over finite fields. Through his discovery of rigid analytic spaces, he established new foundations for p-adic global analysis which have wide applicability in number theory, algebraic geometry and representation theory. The theory of elliptic curves owes an enormous amount to his contributions, both theoretical and computational; the theory of height functions (Neron-Tate and Mazur-Tate) and descent theory (including his construction of the notorious Shafarevich-Tate group) are of key importance in understanding the arithmetic of elliptic curves, and Tate's algorithm for determining the bad reduction of an elliptic curve plays an equally important role in computation. Other contributions of deep significance include his work with Serre on the deformation theory of abelian varieties, his contributions to algebraic K-theory and its relation with Galois cohomology, his work on the Stark conjectures, and most recently his work in non-commutative ring theory.

Tate received a Sloan Fellowship during 1959-61, and a Guggenheim Fellowship during 1965-66. He was a plenary speaker at the International Congress of Mathematicians held in Nice in 1970 when he gave the lecture Symbols in Arithmetic. In 1972 he was the American Mathematical Society's Colloquium Lecturer and spoke on The arithmetic of elliptic curves. He was a member of the committee that decided on the awards of the Fields Medals in 1974. The committee surprised the mathematical world by only making two awards (to Enrico Bombieri and David Mumford). It was Tate who reported on Mumford's work at the awarding ceremony at the International Congress of Mathematicians in Vancouver.

Tate was honoured with election to the U.S. National Academy of Sciences in 1969 and to the Académie de Sciences in Paris in 1992. In 1995 he received the Leroy P Steele Prize For Lifetime Achievement from the American Mathematical Society [6]:-

... for scientific accomplishments spanning four and a half decades. He has been deeply influential in many of the important developments in algebra, algebraic geometry, and number theory during this time.

In 2003 he received the Wolf prize:-

... for his creation of fundamental concepts in algebraic number theory.

Although the citation is similar to that of the London Mathematical Society which we quoted above, we give the following extract:-

For over a quarter of a century, Professor John Tate's ideas have dominated the development of arithmetic algebraic geometry. Tate has introduced path breaking techniques and concepts, that initiated many theories which are very much alive today. These include Fourier analysis on local fields and adele rings, Galois cohomology, the theory of rigid analytic varieties, and p-divisible groups and p-adic Hodge decompositions, to name but a few. Tate has been an inspiration to all those working on number theory. Numerous notions bear his name: Tate cohomology of a finite group, Tate module of an abelian variety, Tate-Shafarevich group, Lubin-Tate groups, Neron-Tate heights, Tate motives, the Sato-Tate conjecture, Tate twist, Tate elliptic curve, and others. John Tate is a revered name in algebraic number theory.

In the first semester of the academic year 1980-81 Tate gave a course of lectures on Stark's conjectures at Université de Paris-Sud (Orsay). This was published in 1984 as Les conjectures de Stark sur les fonctions L d'Artin en s = 0. This is not the only book by Tate based on a lecture course he had given previously. In 1992 he published Rational points on elliptic curves coauthored with Joseph H Silverman. This book was based on a course Tate had given over 30 years earlier in 1961 at Haverford College. Andrew Bremner begins a review as follows:-

The authors' goal has been to write a textbook in a technically difficult field which is accessible to the average undergraduate mathematics major, and it seems that they have succeeded admirably. The book is quite delightful. ... The most obvious drawback to a text for undergraduates in a field such as this is that it is not possible to be entirely rigorous, and so, as the authors declare, "much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than rigorously prove." An appendix does develop the necessary algebraic geometry, but throughout the book the approach to the underlying geometry is informal, allowing a more rapid and intuitive access to the number theory.

On 24 May 2000, Atiyah and Tate presented the Clay Mathematics Institute Millennium Prize Problems in Paris. Tate's lecture covered the Riemann hypothesis, the Birch-Swinnerton-Dyer conjecture and the $P = NP$ problem. He explained the problems and put them into their historical context.

On 24 March 2010 the President of the Norwegian Academy of Science and Letters announced that Tate would be presented with the Abel Prize in Oslo on 25 May:-

... for his vast and lasting impact on the theory of numbers.

The press release reads:-

The theory of numbers stretches from the mysteries of prime numbers to the ways in which we store, transmit, and secure information in modern computers. Over the past century it has developed into one of the most elaborate and sophisticated branches of mathematics, interacting profoundly with other key areas. John Tate is a prime architect of this development. John Tate's scientific accomplishments span six decades. A wealth of essential mathematical ideas and constructions were initiated by Tate and later named after him, such as the Tate module, Tate curve, Tate cycle, Hodge-Tate decompositions, Tate cohomology, Serre-Tate parameter, Lubin-Tate group, Tate trace, Shafarevich-Tate group, Néron-Tate height, to mention just a few. According to the Abel committee, "Many of the major lines of research in algebraic number theory and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate. He has truly left a conspicuous imprint on modern mathematics."

We should end with one final note. Tate was one of the younger members of the Bourbaki team and almost unique in that team in that he was not French.