# Goro Shimura

### Born: 23 February 1930 in Hamamatsu, Japan

Died: 3 May 2019 in Princeton, New Jersey, USA

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**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]:-

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]:-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.

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.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.

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]:-

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 paperMy 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.

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

*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:

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

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

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 paperIn1957while 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.

*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 who 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:-

In 1971 he publishedThis is a charming little book. It introduces the reader to one of the most beautiful parts of mathematics.

*Introduction to the arithmetic theory of automorphic functions*stating in the Preface that the two major topics in the monograph are:-

In 1977 he was awarded the Cole Prize for Algebra by the American Mathematical Society:-... 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 1996 he received the Steele Prize for Lifetime Achievement from the American Mathematical Society. The citation reads:-... for his two papers "Class fields over real quadratic fields and Hecke operators" and "On modular forms of half integral weight".

He began his response as follows:-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.

Indeed Shimura was right to consider that he was "not so old" for he continued to produce important monographs. He publishedI 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!

*Euler products and Eisenstein series*(1997) which M Ram Murty reviewed - we give the first and last paragraphs of his interesting review:-

Shimura's next bookThis 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.

*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:-

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

**Article by:** *J J O'Connor* and *E F Robertson*

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**List of References** (4 books/articles)

**Mathematicians born in the same country**

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**Honours awarded to Goro Shimura**

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1. | AMS Steele Prize | 1996 |

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