Takurō Shintani

Takurō Shintani ( Japanese 新 谷 卓 郎 , Shintani Takurō ; born February 4, 1943 in Kitakyūshū ; † November 14, 1980 in Tokyo ) was a Japanese mathematician who dealt with algebraic number theory .

Shintani studied at the University of Tokyo with a diploma in 1967 with Nagayoshi Iwahori . In 1971 he received his doctorate ( On Dirichlet series whose coefficients are class numbers of integral binary cubic forms ) and assistant professor at the University of Tokyo. As a post-doctoral student , he was at the Institute for Advanced Study from 1971 to 1973 . In 1978 he became an associate professor in Tokyo. In 1979 he attended the Tata Institute in Bombay and in 1980 the IHES .

In the 1970s he worked with Mikio Satō on zeta functions of pre-homogeneous vector spaces (which Sato introduced in 1970), on modular forms (he found a reversal of the Shimura correspondence between modular forms of half-number and whole-number weights) and in the Langlands program (definition of lifting automorphic representations over local bodies in case of base change).

In 1976 he introduced a zeta function named after him and proved a theorem on the geometry of the action of the unit group (structure of its fundamental domain) of algebraic number fields in the Minkowski space of the geometry of numbers (Shintani unit theorem). He developed all of this as part of his work on the values ​​of the zeta function of totally real algebraic number fields on non-positive whole numbers, which he also presented at the 1978 International Congress of Mathematicians. He found a simple expression for the values ​​of the zeta function at these points through the fundamental quantities of the number field. The further development of the theory led him to a conjecture about the construction of class fields of real quadratic number fields by special values ​​of double gamma functions. Shintani recognized the possibility of expanding to include imaginary square number fields and connections to works by Harold Stark . Conjectures about the possibility of constructing class fields of real quadratic number fields along this theory are named after Shintani and Stark. Shintani classified these investigations as an approach to Hilbert's twelfth problem , the search for analytic functions whose special values ​​are suitable for the construction of Abelian extensions of algebraic number fields.

In 1978 he received the Iyanaga Prize . In the same year he was invited speaker at the International Congress of Mathematicians in Helsinki ( On special values ​​of zeta functions of totally real algebraic number fields ).

Fonts

• with M. Sato: On Zeta functions associated with prehomogeneous vector spaces, Annals of Mathematics, Volume 100, 1974, 131-170
• On Zeta functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo, 22, 1975, pp. 26-65
• On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, Journal of the Faculty of Science. University of Tokyo, Section IA. Mathematics, Vol. 23, 1976, pp. 393-417
• A remark on zeta functions of algebraic number fields, in: Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math. 10, Bombay: Tata Inst. Fundamental Res., 1981, pp. 255-260

literature

• Yasutaka Ihara , obituary in J. Fac. Sci. Tokyo, Vol. 28, 1981, iii-vi Number Theory Web

Individual evidence

1. ↑ Takuro Shintani: On Dirichlet series whose coefficients are class numbers of integral binary cubic forms . In: Proceedings of the Japan Academy . tape 46 , no. 9 , 1970, pp. 909–911 , doi : 10.3792 / pja / 1195520156 (preliminary version).
2. ↑ Takuro Shintani: On Dirichlet series whose coefficients are class numbers of integral binary cubic forms . In: Journal of the Mathematical Society of Japan . tape 24 , no. 1 , 1972, p. 132-188 , doi : 10.2969 / jmsj / 02410132 (full text).
3. ↑ Paul Gunnels on the Shintani set of units