The Shimura-taniyama Conjecture and Conformal Field Theory
نویسندگان
چکیده
The Shimura-Taniyama conjecture states that the Mellin transform of the Hasse-Weil Lfunction of any elliptic curve defined over the rational numbers is a modular form. Recent work of Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor has provided a proof of this longstanding conjecture. Elliptic curves provide the simplest framework for a class of CalabiYau manifolds which have been conjectured to be exactly solvable. It is shown that the Hasse-Weil modular form determined by the arithmetic structure of the Fermat type elliptic curve is related in a natural way to a modular form arising from the character of a conformal field theory derived from an affine Kac-Moody algebra. PACS Numbers and
منابع مشابه
A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced
On June 23, 1993, Andrew Wiles unveiled his strategy for proving the Shimura-Taniyama-Weil conjecture for semistable elliptic curves defined over the field Q of rational numbers. Thanks to the work of Gerhard Frey, JeanPierre Serre and Kenneth Ribet, this was known to imply Fermat’s Last Theorem. Six years later, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor have finally ann...
متن کاملA Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced, Volume 46, Number 11
On June 23, 1993, Andrew Wiles unveiled his strategy for proving the Shimura-Taniyama-Weil conjecture for semistable elliptic curves defined over the field Q of rational numbers. Thanks to the work of Gerhard Frey, Jean-Pierre Serre, and Kenneth Ribet, this was known to imply Fermat’s Last Theorem. Six years later Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor have finally an...
متن کاملThe Japanese approach to the Shimura - Taniyama conjecture
In this note we point out links between the Shimura Taniyama conjecture and certain ideas in physics. Since all the seminal references are by strange coincidence Japanese we wish to call this the Japanese approach. The note elaborates on some inspired comments made by Barry Mazur in his popular article “Number theory as gadfly.”
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The Conjecture of Shimura and Taniyama is a special case of a general philosophy according to which a motive of a certain type should correspond to a special type of automorphic forms on a reductive group. Now familiar extensions of the conjecture include (i) essentially the same statement for elliptic curves over totally real fields and (ii) the statement that for each irreducible abelian surf...
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When Weil arrived in Tokyo in 1955, planning to speak about his ideas on the extension to abelian varieties of the classical theory of complex multiplication, he was surprised to learn that two young Japanese mathematicians had also made decisive progress on this topic. They were Shimura and Taniyama. While Weil wrote nothing on complex multiplication except for the report on his talk, Shimura ...
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