Shimura then wrote a long series of important papers, extending the phenomena found in the theory of complex multiplication and modular forms to higher dimensions (amongst other results).
Shimura himself has described his approach as 'phenomenological': his interest is in finding new types of interesting behaviour in the theory of automorphic forms.
The central 'Shimura variety' concept has been tamed (by application of Lie group and algebraic group theory, and the extraction of the concept 'parametrises interesting family of Hodge structures' by reference to the algebraic geometry theory of 'motives', which is still largely conjectural).
GOROSHIMURA: I entered the University of Tokyo in 1949, and that was four years after the War, but almost all professors were tired and the lectures were not inspiring.
GOROSHIMURA: That was when I became very close to Taniyama.
GOROSHIMURA: In 1955, there was an international symposium, and Taniyama posed two or three problems.
Shimura conjectured that every elliptic curve over the rational numbers is uniformized by a modular form.
Shimura declared his conjecture that an elliptic curve should always be uniformized by a modular curve, but André Weil did not believe Shimura conjecture.
The conjecture became misquoted as Weil-Taniyama conjecture instead of Shimura- Taniyama conjecture.
I entered the University of Tokyo in 1949 and that was four years after the War, but almost all professors were tired and the lectures were not inspiring.
GoroShimura and his fellow students had to rely on each other for inspiration.
Together, Taniyama and Shimura worked on the complex mathematics of modular functions.
Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions.
In this book, GoroShimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions.
The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals.
That conjecture dates back to 1955, when it was published in Japanese as a research problem by the late Yutaka Taniyama.
GoroShimura of Princeton and Andre Weil of the Institute for Advanced Study provided key insights in formulating the conjecture, which proposes a special kind of equivalence between the mathematics of objects called elliptic curves and the mathematics of certain motions in space.
The equation of Fermat’s last theorem is one example of a type known as a Diophantine equation -- an algebraic expression of several variables whose solutions must to be rational numbers (either whole numbers or fractions, which are ratios of whole numbers).
Taniyama studied mathematics at the University of Tokyo after the end of World War II, and here he developed a friendship with another student named GoroShimura.
Shimura later worked with Taniyama on this idea that modular forms and elliptic curves are linked and this form the basis of the Taniyama-Shimura conjecture:
Every elliptic curve defined over the rational field is a factor of the jacobian of a modular function field.
Moduli of abelian varieties and number theory- Discontinuous groups and abelian varieties- Construction of class fields and zeta functions of algebraic curves- Number fields and zeta functions associated with discontinuous
1996 the AMS awarded GoroShimura the Steele Prize for Lifetime Achievement for his "important and extensive work on arithmetical geometry and automorphic forms." His seminal work has ...
Achievement for his "important and extensive work on arithmetical geometry and automorphic forms." His seminal work has ...
"Yutaka Taniyama and his time: very personal recollections", by GoroShimura; Bulletin of the London Mathematical Society, Volume 21 (1989), pages 186-196.
Taniyama's life and work, as described by the colleague who knew him best.
"Steele Prize for Lifetime Achievement: GoroShimura"; Notices of the American Mathematical Society, Volume 43 (November 1996), pages 1343-1347.
Collected Papers : Volume I. 1954-1966; Author: Shimura, Goro (University of Princeton, USA); English;French; Hardback; ...(Site not responding. Last check: 2007-10-13)
Collected Papers : Volume I. 1954-1966; Author: Shimura, Goro (University of Princeton, USA); English;French; Hardback; Book
In 1996 the AMS awarded GoroShimura the Steele Prize for Lifetime Achievement for his important and extensive work on arithmetical geometry and automorphic forms.
Prices subject to change to be advised on confirmation of order.
