# Goro Shimura

**Gorō Shimura** (志村 五郎 *Shimura Gorō*, 23 February 1930 – 3 May 2019) was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry.[1] He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem.

Gorō Shimura | |
---|---|

Born | |

Died | 3 May 2019 89) | (aged

Nationality | Japanese |

Alma mater | University of Tokyo |

Known for | Complex multiplication of abelian varieties Modularity theorem Shimura variety Shimura subgroup |

Awards | Guggenheim Fellowship (1970) Cole Prize (1977) Asahi Prize (1991) Steele Prize (1996) |

Scientific career | |

Fields | Mathematics |

Institutions | Princeton University |

Doctoral students | Don Blasius Bill Casselman Melvin Hochster Robert Rumely Alice Silverberg |

## Biography

Gorō Shimura was born in Hamamatsu, Japan on 23 February 1930.[2] Shimura graduated with a B.A. in mathematics and a D.Sc. in mathematics from the University of Tokyo in 1952 and 1958 respectively.[3][2]

After graduating, Shimura became a lecturer at the University of Tokyo, then worked abroad — including ten months in Paris and a seven-month stint at Princeton's Institute for Advanced Study — before returning to Tokyo, where he married Chikako Ishiguro.[4][2] He then moved from Tokyo to join the faculty of Osaka University, but growing unhappy with his funding situation, he decided to seek employment in the United States.[4][2] Through André Weil he obtained a position at Princeton University itself.[4] Shimura joined the Princeton faculty in 1964 and retired in 1999, during which time he advised over 28 doctoral students and received the Guggenheim Fellowship in 1970, the Cole Prize for number theory in 1977, the Asahi Prize in 1991, and the Steele Prize for lifetime achievement in 1996.[1][5]

Shimura described his approach to mathematics as "phenomenological": his interest was in finding new types of interesting behavior in the theory of automorphic forms. He also argued for a "romantic" approach, something he found lacking in the younger generation of mathematicians.[6] Shimura used a two-part process for research, using one desk in his home dedicated to working on new research in the mornings and a second desk for perfecting papers in the afternoon.[2]

Shimura had two children, Tomoko and Haru, with his wife Chikako.[2] Shimura died on 3 May 2019 in Princeton, New Jersey at the age of 89.[1][2]

## Research

Shimura was a colleague and a friend of Yutaka Taniyama, with whom he wrote the first book on the complex multiplication of abelian varieties and formulated the Taniyama–Shimura conjecture.[7] Shimura then wrote a long series of major papers, extending the phenomena found in the theory of complex multiplication of elliptic curves and the theory of modular forms to higher dimensions (e.g. Shimura varieties). This work provided examples for which the equivalence between motivic and automorphic *L*-functions postulated in the Langlands program could be tested: automorphic forms realized in the cohomology of a Shimura variety have a construction that attaches Galois representations to them.[8]

In 1958, Shimura generalized the initial work of Martin Eichler on the Eichler–Shimura congruence relation between the local *L*-function of a modular curve and the eigenvalues of Hecke operators.[9][10] In 1959, Shimura extended the work of Eichler on the Eichler–Shimura isomorphism between Eichler cohomology groups and spaces of cusp forms which would be used in Pierre Deligne's proof of the Weil conjectures.[11][12]

In 1971, Shimura's work on explicit class field theory in the spirit of Kronecker's Jugendtraum resulted in his proof of Shimura's reciprocity law.[13] In 1973, Shimura established the Shimura correspondence between modular forms of half integral weight *k*+1/2, and modular forms of even weight 2*k*.[14]

Shimura's formulation of the Taniyama–Shimura conjecture (later known as the modularity theorem) in the 1950s played a key role in the proof of Fermat's Last Theorem by Andrew Wiles in 1995. In 1990, Kenneth Ribet proved Ribet's theorem which demonstrated that Fermat's Last Theorem followed from the semistable case of this conjecture.[15] Shimura dryly commented that his first reaction on hearing of Andrew Wiles's proof of the semistable case was 'I told you so'.[16]

## Other interests

His hobbies were shogi problems of extreme length and collecting Imari porcelain. *The Story of Imari: The Symbols and Mysteries of Antique Japanese Porcelain* is a non-fiction work about the Imari porcelain that he collected over 30 years that was published by Ten Speed Press in 2008.[2][17]

## Works

### Mathematical books

- Shimura, Goro; Taniyama, Yutaka (1961),
*Complex multiplication of abelian varieties and its applications to number theory*, Publications of the Mathematical Society of Japan,**6**, Tokyo: The Mathematical Society of Japan, MR 0125113 Later expanded and published as Shimura (1997) - Shimura, Goro (1968).
*Automorphic Functions and Number Theory*. Lecture Notes in Mathematics, Vol. 54 (Paperback ed.). Springer. ISBN 978-3-540-04224-2. - Shimura, Goro (1 August 1971).
*Introduction to the Arithmetic Theory of Automorphic Functions*(Paperback ed.). Princeton University Press. ISBN 978-0-691-08092-5. - It is published from Iwanami Shoten in Japan.[18] - Shimura, Goro (1 July 1997).
*Euler Products and Eisenstein Series*. CBMS Regional Conference Series in Mathematics (Paperback ed.). American Mathematical Society. ISBN 978-0-8218-0574-9. - Shimura, Goro (1997).
*Abelian Varieties with Complex Multiplication and Modular Functions*(Hardcover ed.). Princeton University Press. ISBN 978-0-691-01656-6.[19] An expanded version of Shimura & Taniyama (1961). - Shimura, Goro (22 August 2000).
*Arithmeticity in the Theory of Automorphic Forms*. Mathematical Surveys and Monographs (Paperback ed.). American Mathematical Society. ISBN 978-0-8218-2671-3.[20] - Shimura, Goro (1 March 2004).
*Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups*. Mathematical Surveys and Monographs (Hardcover ed.). American Mathematical Society. ISBN 978-0-8218-3573-9. - Shimura, Goro (2007).
*Elementary Dirichlet Series and Modular Forms*. Springer Monographs in Mathematics (Hardcover ed.). Springer. ISBN 978-0-387-72473-7.- Shimura, Goro (28 December 2009).
*Elementary Dirichlet Series and Modular Forms*. Springer Monographs in Mathematics (Paperback ed.). Springer New York. ISBN 978-1-4419-2478-0.

- Shimura, Goro (28 December 2009).
- Shimura, Goro (15 July 2010).
*Arithmetic of Quadratic Forms*. Springer Monographs in Mathematics (Hardcover ed.). Springer. ISBN 978-1-4419-1731-7.

### Non-fiction

- Shimura, Goro (1 June 2008).
*The Story of Imari: The Symbols and Mysteries of Antique Japanese Porcelain*(Hardcover ed.). Ten Speed Press. ISBN 978-1-58008-896-1. - Shimura, Goro (5 September 2008).
*The Map of My Life*(Hardcover ed.). Berlin: Springer-Verlag. ISBN 978-0-387-79714-4. MR 2442779.- Shimura, Goro (28 December 2009).
*The Map of My Life*(Paperback ed.). Springer New York. ISBN 978-1-4419-2724-8.

- Shimura, Goro (28 December 2009).

### Collected papers

- Shimura, Goro (2002).
*Collected Papers*. Ⅰ: 1954–1965 (Hardcover ed.). Springer. ISBN 978-0-387-95406-6. - Shimura, Goro (2002).
*Collected Papers*. Ⅱ: 1967–1977 (Hardcover ed.). Springer. ISBN 978-0-387-95416-5. - Shimura, Goro (2003).
*Collected Papers*. Ⅲ: 1978–1988 (Hardcover ed.). Springer. ISBN 978-0-387-95417-2. - Shimura, Goro (2003).
*Collected Papers*. Ⅳ: 1989–2001 (Hardcover ed.). Springer. ISBN 978-0-387-95418-9.

## References

- "Professor Emeritus Goro Shimura 1930—2019". Princeton University Department of Mathematics. 3 May 2019. Retrieved 3 May 2019.
- Fuller-Wright, Liz (8 May 2019). "Goro Shimura, a 'giant' of number theory, dies at 89". Princeton University Department of Mathematics. Retrieved 9 May 2019.
- Goro Shimura at the Mathematics Genealogy Project
- O'Connor, John J.; Robertson, Edmund F., "Goro Shimura",
*MacTutor History of Mathematics archive*, University of St Andrews. - "The Asahi Prize". The Asahi Shimbun Company. Retrieved 4 May 2019.
- Shimura, Goro (5 September 2008).
*The Map of My Life*(Hardcover ed.). Berlin: Springer-Verlag. ISBN 978-0-387-79714-4. MR 2442779. - Shimura, Goro (1989). "Yutaka Taniyama and his time. Very personal recollections".
*The Bulletin of the London Mathematical Society*.**21**(2): 186–196. doi:10.1112/blms/21.2.186. ISSN 0024-6093. MR 0976064. - Langlands, Robert (1979). "Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen" (PDF). In Borel, Armand; Casselman, William (eds.).
*Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics*. XXXIII Part 1. Chelsea Publishing Company. pp. 205–246. - Shimura, Goro (1958). "Correspondances modulaires et les fonctions ζ de courbes algébriques".
*Journal of the Mathematical Society of Japan*.**10**: 1–28. doi:10.4099/jmath.10.1. ISSN 0025-5645. MR 0095173. - Piatetski-Shapiro, Ilya (1972). "Zeta functions of modular curves".
*Modular functions of one variable II*. Lecture Notes in Mathematics.**349**. Antwerp. pp. 317–360. - Shimura, Goro (1959). "Sur les intégrales attachées aux formes automorphes".
*Journal of the Mathematical Society of Japan*.**11**: 291–311. doi:10.4099/jmath.11.291. ISSN 0025-5645. MR 0120372. - Deligne, Pierre (1971). "Formes modulaires et représentations l-adiques".
*Séminaire Bourbaki vol. 1968/69 Exposés 347-363*. Lecture Notes in Mathematics.**179**. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0058801. ISBN 978-3-540-05356-9. - Shimura, Goro (1971).
*Introduction to the arithmetic theory of automorphic functions*. Publications of the Mathematical Society of Japan.**11**. Tokyo: Iwanami Shoten. Zbl 0221.10029. - Shimura, Goro (1973). "On modular forms of half integral weight".
*Annals of Mathematics*. Second Series.**97**: 440–481. doi:10.2307/1970831. ISSN 0003-486X. JSTOR 1970831. MR 0332663. - Ribet, Kenneth (1990). "From the Taniyama-Shimura conjecture to Fermat's last theorem".
*Annales de la faculté des sciences de Toulouse Sér. 5*.**11**(1): 116–139. doi:10.5802/afst.698. - "Nova Episode: The Proof".
- Shimura, Goro (1 June 2008).
*The Story of Imari: The Symbols and Mysteries of Antique Japanese Porcelain*(Hardcover ed.). Ten Speed Press. ISBN 978-1-58008-896-1. - Goldstein, Larry Joel (1973). "Review of
*Introduction to the Arithmetic Theory of Automorphic Functions*by Goro Shimura".*Bull. Amer. Math. Soc*.**79**: 514–516. doi:10.1090/S0002-9904-1973-13177-5. - Ogg, A. P. (1999). "Review of
*Abelian varieties with complex multiplication and modular functions*by Goro Shimura".*Bull. Amer. Math. Soc. (N.S.)*.**36**: 405–408. doi:10.1090/S0273-0979-99-00784-3. - Yoshida, Hiroyuki (2002). "Review of
*Arithmeticity in the theory of automorphic forms*by Goro Shimura".*Bull. Amer. Math. Soc. (N.S.)*.**39**: 441–448. doi:10.1090/s0273-0979-02-00945-x.

## External links

Quotations related to Goro Shimura at Wikiquote - Goro Shimura at the Mathematics Genealogy Project