The Shimura–Taniyama conjecture and conformal field theory

Abstract

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.