志村五郎Gorō Shimura

志村五郎的父亲在一家银行工作，经常从一个分行搬到另一个分行。即使在志村五郎出生后，全家也从一个房子搬到另一个房子，在滨松，一个位于东京以西约240公里的城市。他是父母五个孩子中最小的，有三个姐妹和一个兄弟。1933年3月，全家搬到东京，三年后，1936年4月，志村五郎开始上学。1938年，全家搬到东京一个更大的房子，但志村五郎继续上同一所小学，直到他完成四年级。之后，他上了西大久保区家附近的一所小学，完成了五年级和六年级。志村五郎于1942年开始在第四东京府立中学学习，但数学教学几乎没有让他兴奋[1]：-

数学课不太有趣。我们再次学习了分数和小数的算术运算，这还可以。但我们被要求在不使用代数的情况下解决人为的算术问题。……我从未觉得这样的问题有趣。

由于第二次世界大战，这是困难时期——这意味着生活在一个紧张的氛围中，军事训练是学校课程的一部分。1944年11月，学校关闭，男孩们被送到位于乡村的工厂工作。他写道[1]：-

在战争最后阶段上中学时，我们被迫在一家为战斗机生产零件的工厂里劳动，那时我明白了在这种地方劳动的意义。

他的家在轰炸中被毁，但全家幸存下来。战争结束后，中学重新开学，他继续学业。此时他父母的家在新宿以西的三鹰，他乘火车上学。战争期间食物短缺，但战争结束后，短缺更加严重，志村五郎经常挨饿。

1946年，志村五郎进入第一高等学校。那是一所寄宿学校，他住在宿舍里，但食物短缺意味着几周后每个人都被送回家度假。在高中，他学习数学、英语、德语和法语，但发现数学课程相当令人失望。他觉得他上的解析几何课程由一位不完全理解该科目的老师教授。

志村五郎于1949年开始在东京大学学习。他再次对所学的材料持批评态度，然而[1]：-

我希望能在大学学到大量好的数学，这一愿望很快被现实打破。一方面，在高中时我已经获得了相当多的数学知识，而大学第一年所教的内容并没有多少新东西。但更重要的是，当时的教授和副教授们并没有认真思考应该教什么的问题。这和我中学第一年的经历如出一辙。他们只是在重复旧的东西，而这些本应被更好的材料所取代。

志村五郎最喜欢的课程由岩泽健吉讲授，但志村五郎再次批评说，岩泽健吉“是在为自己讲课，而不是为学生讲课”。他于1952年从东京大学毕业，并被任命为东京大学教养学部的助手。他的第一篇论文On a certain ideal of the center of a Frobeniusean algebra 在他毕业那年发表。正是在这个时候，他开始了作为数学家的职业生涯，他认为这是由两件事引发的。一是谢瓦莱于1953年访问日本。谢瓦莱在东京大学讲授了一门课程，描述他在代数群理论中的最新成果。志村五郎对谢瓦莱的一个引理给出了更好的证明，当讲义于1954年出版时，其中包含了志村五郎的证明，并附有评论：-

这个引理的以下证明是由志村五郎先生告知我的。

志村五郎认为开启他数学职业生涯的第二个事件，是他于1953年3月参加了由Yasuo Akizuki在京都大学组织的一场关于代数几何与数论的会议。Akizuki正在京都建立一个强大的代数几何学派，他邀请志村五郎在会上作报告，谷山丰也参加了这次会议。1954年，志村五郎被任命为东京大学讲师，1957年晋升为副教授。他教授线性代数与微积分，并继续进行研究，发表了A note on the normalization-theorem of an integral domain （1954）和Reduction of algebraic varieties with respect to a discrete valuation of the basic field（1955）等文章。他于1953年与安德烈·韦伊通信，并于1955年在东京-日光举行的代数数论国际研讨会上与他见面，安德烈·韦伊是那次研讨会的主旨演讲者之一。正是在这次国际研讨会上，Shamura-谷山丰猜想有了雏形。该猜想声称：

定义在有理域上的每条椭圆曲线都是某个模函数域的雅可比簇的因子。

这个猜想对数学的发展产生了重大影响，尤其是在皮埃尔·德·费马最后定理的证明中起到了重要作用。（关于Shamura-谷山丰的更多细节，参见有趣的文章[4]和[5]）。志村五郎向国际研讨会提交了他的论文On complex multiplications，并且很可能由于与安德烈·韦伊的会面，志村五郎于1956年收到他的邀请，在1957-58学年去巴黎度过。亨利·嘉当为他在巴黎的国家科学研究中心安排了一个“chargé de recherches”（研究专员）的职位。在他前往巴黎之前，他与谷山丰合著的书Modern number theory（日文）出版了。志村五郎撰写了序言，开头是：

代数几何的进展对数论产生了强大的影响。建立利奥波德·克罗内克经典复乘理论的高维推广，并完成埃里希·赫克留下的工作，一直是一个重要问题。借助代数几何的语言，我们现在可以在那个方向上增添新的知识。我们很难声称该理论以完全令人满意的形式呈现。无论如何，可以说，在进展过程中，我们被允许爬到一定的高度，以便回顾我们的足迹，然后眺望我们的目的地。

巴黎之行对志村五郎来说难以忘怀。他写道[2]：

1957年在巴黎时，我对[儒勒·昂利·庞加莱型的Fuchsian群]产生了兴趣。我刚完成关于椭圆模曲线zeta函数的第一项工作。虽然我知道它需要完善，但我更感兴趣的是寻找其他能确定其ζ函数的曲线。我还试图用多变量自守函数的值——例如卡尔·西格尔模函数——来表述高维复乘理论。结果发现，这两个问题彼此密不可分。而且，没有其他人在研究这类问题。

1958年8月，他在巴黎之行期间，作为日本官方代表出席了在苏格兰爱丁堡举行的国际数学家大会，并宣读了论文Fonctions automorphes et correspondances modulaires。除了访问苏格兰，他还从巴黎出发进行了其他旅行，前往瑞士、德国和意大利。在巴黎期间，他在模函数域和模对应以及儒勒·昂利·庞加莱型的富克斯群方面取得了显著的数学进展。最后这项研究构成了他在爱丁堡国际数学家大会上演讲的主题。此外，他还为他曾在Modern number theory（日文）中提出的理论找到了“一个完全令人满意的形式”。在为期十个月的巴黎访问结束时，志村五郎在普林斯顿的高等爱德华·斯图迪研究院度过了七个月。事实上，安德烈·韦伊已被任命为普林斯顿的教授，因此志村五郎在这次访问期间与安德烈·韦伊保持了联系。他于1959年春天返回东京，并于同年晚些时候与相识六年的石黑千佳子结婚。尽管他在东京的研究进展顺利，但他并不喜欢那里的教学。1961年春天，在松岛与三的劝说下，他转到大阪大学。这次调动意味着他从副教授升为正教授，但薪水保持不变。事实上，由于在东京的额外职责带来了额外资金，他现在不得不靠减少的收入生活。他决定尝试移居美国。

当安德烈·韦伊于1961年访问日本时，机会来了，志村五郎问他是否有可能在美国为他找到一个职位。安德烈·韦伊在普林斯顿安排了一个职位，1962年9月，志村五郎回到普林斯顿，但这次他隶属于大学，而不是他早期职业生涯中曾待过的高等斯图迪研究院。让我们看看他出版的其他一些书。他的书Automorphic functions and number theory （1968年）由S 萨尔瓦达曼·邱拉评论：-

这是一本迷人的小书。它向读者介绍了数学中最美丽的部分之一。

1971年，他出版了Introduction to the arithmetic theory of automorphic functions，在序言中指出该专著的两个主要主题是：-

……椭圆或椭圆模函数的复乘法以及埃里希·赫克算子理论在代数曲线和阿贝尔簇的zeta函数上的应用。

1977年，他被美国数学会授予法兰克·尼尔森·寇尔代数奖：-

……因他的两篇论文《实二次域上的类域与埃里希·赫克算子》和《论半整数权模形式》。

1996年，他获得了美国数学会颁发的凯瑟琳·斯蒂尔终身成就奖。颁奖词写道：-

志村五郎因其在算术几何和自守形式方面重要而广泛的工作；他引入的概念往往具有开创性，并为新的发展提供了肥沃的土壤，数论中许多以他命名的记号以及该领域工作者早已熟悉的记号就是明证。

他的回应开头如下：——

我一直以为这个奖是颁给老人的，肯定比我年长，所以得知自己获奖时，我感到意外，尽管是愉快的意外。虽然我不算年轻，但也不算太老，而且，我一直成功地让我系里每一位新聘的初级成员以为我也是新聘的同事。这一次我失败了，我应该感谢评选委员会发现我至少已经老到其毕生工作可以被人谈论。各类机构颁发许多奖项，但在目前这种情况下，我把它看作来自朋友们的东西，这让我真的很高兴。所以，我只想说，谢谢你们，我的朋友们！

志村五郎认为自己“不算太老”是对的，因为他继续产出重要的专著。他出版了 Euler products and Eisenstein series（1997），M Ram Murty对此作了评论——我们给出他那篇有趣评论的第一段和最后一段：——

本专著聚焦于三个目标：(i) 在经典群上以显式有理形式确定局部莱昂哈德·欧拉因子；(ii) 任意签名的酉群上的莱昂哈德·欧拉乘积和费迪南·艾森斯坦级数；(iii) 全定Hermitian形式的类数公式。它以阐述性风格写成，因此可视为多变量自守形式理论的入门。
...
总之，本专著具有许多教学特点，使其值得研究生和研究人员学习。值得注意的是，附录集，以及关于代数群及其局部化、费迪南·艾森斯坦级数及其解析延拓的材料，这些材料曾分散在研究文献中，有时没有证明，且常被作为“众所周知”而置于背景中，现在都汇集在本卷中。

志村五郎的下一本书Abelian varieties with complex multiplication and modular functions（1998年）是他1961年与谷山丰合著的文本Complex multiplication of abelian varieties and its applications to number theory的扩展版。这个1961年的文本又是他与谷山丰合著的书Modern number theory（日文）（1957年）的重写。由于谷山丰已于1958年去世，即使1961年的文本也主要归功于志村五郎融入了他在巴黎访问期间获得的新理解。当然，该领域在1961年至1998年间有了显著发展（志村五郎做出了相当大的贡献），因此得知他为1998年的专著增加了17个新章节也就不足为奇了。2000年，志村五郎出版了Arithmeticity in the theory of automorphic forms，然后在2004年出版了Arithmetic and analytic theories of quadratic forms and Clifford groups。2007年，他出版了Elementary Dirichlet series and modular forms，2010年出版了Arithmetic of quadratic forms。

然而，并非志村五郎的所有出版物都是关于数学的。The Story of Imari: The Symbols and Mysteries of Antique Japanese Porcelain于2008年8月出版。内容描述如下：-

伊万里瓷器在日本有田的窑炉中烧制，该地以南八英里处有一个海港城镇，伊万里瓷器因此得名。它以其蓝色釉下彩和彩色釉上珐琅所产生的美丽视觉效果而著称。在《威廉·爱德华·史都瑞的伊万里》中，作者志村五郎描述了这些珍贵的瓷碗、盘子、花瓶、茶杯和其他器物的文化和历史意义。通过考察具体作品背后的艺术性和故事，志村五郎分析了它们的釉料、图案、主题和功能，并融入了皇帝、茶道、鹤、冲浪兔等的故事。这就是伊万里瓷器的多彩辉煌，从最宏大的历史到最微小的细节。

最后，让我们提及志村五郎对将棋的热爱，这是一种在9×9棋盘上进行的日本象棋。

Goro Shimura's father worked for a bank and moved frequently from one branch of the bank to another. Even after Goro's birth, the family moved from one house to another in Hamamatsu, a city about 240 km west of Tokyo. He was the youngest of his parents' five children, having three sisters and a brother. In March 1933 the family moved to Tokyo and, three years later, in April 1936, Goro began his schooling. In 1938 the family moved to a larger home in Tokyo but Goro continued to attend the same elementary school until he had completed the forth grade. After that he attended an elementary school near his home in the Nishi-Ohkubu district, completing the fifth and sixth grades. Shimura began his studies in the Fourth Tokyo Prefectural Middle School in 1942 but there was little to excite him in the mathematics teaching [1]:-

Classes in mathematics were not very interesting. We again learned arithmetical operations of fractions and decimals, which was all right. But we were asked to solve artificial arithmetical problems without using algebra. ... I never found such problems interesting.

These were difficult times due to World War II - it meant that life was lived in a strained atmosphere with military training as part of the school curriculum. In November 1944 the school closed and the boys were sent to work in factories located in the countryside. He writes [1]:-

While in middle school during the last period of the war, we were forced to work in a factory that made parts for fighter planes, and at that point I knew the meaning of the labour in such a place.

His home was destroyed in a bombing raid, but the family survived. When the war ended, the middle school opened again and he continued his education. At this time his parents' home was in Mitaka, west of Shinjuku, and he travelled to school by train. There had been food shortages during the war but, after the war ended, the shortages became worse and Shimura was constantly hungry.

In 1946 Shimura entered the First High School. It was a boarding school and he lived in a dormitory but food shortages meant that after a few weeks everyone was sent home for a holiday. At the High School he studied mathematics, English, German and French but found the mathematics courses rather disappointing. He felt that the course he took on analytical geometry was taught by a teacher who did not fully understand the subject.

Shimura began his studies at the University of Tokyo in 1949. Again he is critical of the material he was taught, however [1]:-

My wish that I would be able to learn plenty of good mathematics at the university was soon betrayed by reality. For one thing, while in high school, I had acquired a decent amount of mathematical knowledge, and there was not much new in what was being taught in the first year at the university. But more importantly, the professors and associate professors at that time did not give much serious thought to the question of what should be taught. It was the same story as what I experienced in the first year in middle school. They were simply repeating the old stuff, which should have been replaced by better material.

The course Shimura enjoyed most was taught by Kenkichi Iwasawa, but again Shimura is critical saying that Iwasawa "was lecturing for himself, not for the students". He graduated from the University of Tokyo in 1952 and was appointed as an assistant at the College of General Education of the University of Tokyo. His first paper On a certain ideal of the center of a Frobeniusean algebra was published in the year he graduated. It was at this time that he began his career as a mathematician which, he suggests, was sparked by two events. One was the visit of Claude Chevalley to Japan in 1953. Chevalley gave a lecture course at the University of Tokyo describing his latest results in the theory of algebraic groups. Shimura produced a better proof of one of Chevalley's lemmas and when the lecture notes were published in 1954 they contained Shimura's proof with the comment:-

The following proof of this lemma has been communicated to me by Mr Shimura.

The second event that Shimura considers began his mathematical career was his attendance at a conference on algebraic geometry and number theory in March 1953 organised by Yasuo Akizuki at Kyoto University. Akizuki was building a strong School of Algebraic Geometry in Kyoto and he asked Shimura to talk at the conference, which was also attended by Yutaka Taniyama. In 1954 Shimura was appointed as a lecturer at the University of Tokyo, being promoted to associate professor in 1957. He taught linear algebra and calculus and continued to undertake research publishing articles A note on the normalization-theorem of an integral domain (1954) and Reduction of algebraic varieties with respect to a discrete valuation of the basic field (1955). He had corresponded with André Weil in 1953 and met him in 1955 at the International Symposium on Algebraic Number Theory, Tokyo-Nikko, at which Weil was one of the keynote speakers. It was at this International Symposium that the Shamura-Taniyama conjecture had its genesis. The conjecture claims:-

Every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field.

This conjecture had a major influence on the development of mathematics, and in particular proved important in the proof of Fermat's Last Theorem. (For more details concerning the Shamura-Taniyama see the interesting articles [4] and [5]). Shimura presented his paper On complex multiplications to the International Symposium and, probably as a result of meeting Weil, Shimura received an invitation from him in 1956 to spend the academic year 1957-58 in Paris. Henri Cartan arranged a position of 'chargé de recherches' for him at the Centre National de la Recherche Scientifique (National Centre for Scientific Research) in Paris. Before he left for Paris his book Modern number theory (Japanese), written in collaboration with Yutaka Taniyama, was published. Shimura wrote the Preface which begins:-

The progress of algebraic geometry has had a strong influence on number theory. It has been an important problem to establish a higher-dimensional generalization of the classical theory of complex multiplication by Kronecker and to complete the work left by Hecke. By means of the language of algebraic geometry we can now add new knowledge in that direction. We find it difficult to claim that the theory is presented in a completely satisfactory form. In any case, it may be said, we are allowed in the course of progress to climb to a certain height in order to look back at our tracks and then to take a view of our destination.

The Paris trip was memorable for Shimura. He wrote [2]:-

In 1957 while in Paris I became interested in [the Fuchsian group of Poincaré type]. I had just finished my first work on the zeta functions of elliptic modular curves. Though I knew that it needed elaboration, I was more interested in finding other curves whose zeta functions could be determined. I was also trying to formulate the theory of complex multiplication in higher dimension in terms of the values of automorphic functions of several variables - Siegel modular functions, for example. It turned out that these two problems were inseparably connected to each other. Also, nobody else was working on such questions.

He attended the International Congress of Mathematicians in Edinburgh, Scotland, in August 1958, during the months of the Paris trip, as an official Japanese delegate and presented his paper Fonctions automorphes et correspondances modulaires. In addition to this visit to Scotland, was able to make other trips from Paris, going to Switzerland, Germany and Italy. While he was in Paris he made remarkable mathematical advances on modular function fields and modular correspondences, as well as on the Fuchsian group of Poincaré type. This last investigation formed the topic of his lecture at the International Congress in Edinburgh. Also he was able to find 'a completely satisfactory form' for the theory which he had presented in Modern number theory (Japanese). At the end of his ten-month Paris visit, Shimura spent seven months at the Institute for Advanced Study at Princeton. In fact Weil had been appointed as a professor at Princeton so Shimura remained in contact with Weil during this visit. He returned to Tokyo in the spring of 1959 and, later that year, married Chikako Ishiguro whom he had known for six years. Although his research was going well in Tokyo, he was not enjoying his teaching there. In the spring of 1961 he moved to Osaka University having been persuaded by Yozo Matsushima. His move meant that he moved from associate professor to full professor but his salary remained the same. In fact, due to additional duties in Tokyo which had brought in extra funds, he now had to live on a reduced income. He decided to try to move to the United States.

The opportunity arose when André Weil visited Japan in 1961 and Shimura asked him if it would be possible to find a position for him in the United States. Weil arranged a position at Princeton and, in September 1962, Shimura returned to Princeton but this time he was attached to the University, not the Institute for Advanced Study where he had spent time earlier in his career. Let us look at some of the other books he has published. His book Automorphic functions and number theory (1968) is reviewed by S Chowla:-

This is a charming little book. It introduces the reader to one of the most beautiful parts of mathematics.

In 1971 he published Introduction to the arithmetic theory of automorphic functions stating in the Preface that the two major topics in the monograph are:-

... complex multiplication of elliptic or elliptic modular functions and applications of the theory of Hecke operators to the zeta-functions of algebraic curves and abelian varieties.

In 1977 he was awarded the Cole Prize for Algebra by the American Mathematical Society:-

... for his two papers "Class fields over real quadratic fields and Hecke operators" and "On modular forms of half integral weight".

In 1996 he received the Steele Prize for Lifetime Achievement from the American Mathematical Society. The citation reads:-

To Goro Shimura for his important and extensive work on arithmetical geometry and automorphic forms; concepts introduced by him were often seminal, and fertile ground for new developments, as witnessed by the many notations in number theory that carry his name and that have long been familiar to workers in the field.

He began his response as follows:-

I always thought this prize was for an old person, certainly someone older than I, and so it was a surprise to me, if a pleasant one, to learn that I was chosen as a recipient. Though I am not so young, I am not so old either, and besides, I have been successful in making every newly appointed junior member of my department think that I was also a fellow new appointee. This time I failed, and I should be grateful to the selection committee for discovering that I am a person at least old enough to have his lifetime work spoken of. There are many prizes conferred by various kinds of institutions, but in the present case, I view it as something from my friends, which makes me really happy. So let me just say thank you, my friends!

Indeed Shimura was right to consider that he was "not so old" for he continued to produce important monographs. He published Euler products and Eisenstein series (1997) which M Ram Murty reviewed - we give the first and last paragraphs of his interesting review:-

This monograph focuses on three objectives: (i) the determination of local Euler factors on classical groups, in an explicit rational form; (ii) Euler products and Eisenstein series on a unitary group of an arbitrary signature; (iii) a class number formula for a totally definite Hermitian form. It is written in an expository style, so that it can be viewed as an introduction to the theory of automorphic forms of several variables.
...
In conclusion, this monograph has many didactic features that make it worthy of study by both graduate students and researchers. It is notable that the collection of appendices, as well as the material on algebraic groups and their localizations, Eisenstein series and their analytic continuations, that was scattered in the research literature, sometimes without proof, and often relegated to the background as "well-known", is now gathered together in this volume.

Shimura's next book Abelian varieties with complex multiplication and modular functions (1998) was an expanded edition of his 1961 text Complex multiplication of abelian varieties and its applications to number theory co-authored with Yutaka Taniyama. This 1961 text was, in turn, a rewrite of his joint book with Taniyama Modern number theory (Japanese) (1957). Since Taniyama had died in 1958, even the 1961 text had been largely due to Shimura incorporating the new understanding that he had achieved during his Paris visit. Of course the area had developed markedly between 1961 and 1998 (with considerable contributions by Shimura) so it will come as no surprise to learn that he added 17 new sections for the 1998 monograph. In 2000 Shimura published Arithmeticity in the theory of automorphic forms and then, in 2004, Arithmetic and analytic theories of quadratic forms and Clifford groups. In 2007 he published Elementary Dirichlet series and modular forms and in 2010 Arithmetic of quadratic forms.

Not all of Shimura's publications are on mathematics, however. The Story of Imari: The Symbols and Mysteries of Antique Japanese Porcelain was published in August 2008. The contents are described as follows:-

Fired in the kilns of Arita, Japan, eight miles south of the seaport town after which it was named, Imari porcelain is distinguished by the beautiful visual effects produced by its blue underglaze and colour overglaze enamels. In "The Story of Imari", author Goro Shimura describes the cultural and historical significance of these prized porcelain bowls, plates, vases, teacups, and other wares. Examining the artistry and stories behind specific pieces, Shimura analyses their glazes, patterns, motifs, and functions, weaving in tales of emperors, tea ceremonies, cranes, surfing rabbits, and more. This is Imari in all its colourful glory, from the grandest histories to the smallest details.

Finally let us mention Shimura's love of shogi, a Japanese form of chess played on a 9 × 9 board.