The Shimura–Taniyama conjecture and conformal field theory

✍ Scribed by Rolf Schimmrigk; Sean Underwood

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 170 KB
• Volume: 48
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(03)00030-5

✦ Synopsis

The Shimura-Taniyama conjecture states that the Mellin transform of the Hasse-Weil L-function 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 Calabi-Yau 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.