The Shimura-Taniyama Conjecture and Conformal Field Theory

TL;DR

This paper discusses the proof of the Shimura-Taniyama conjecture and explores the connection between elliptic curves' modular forms and conformal field theory, highlighting a link via affine Kac-Moody algebras.

Contribution

It establishes a natural relation between the Hasse-Weil modular form of Fermat type elliptic curves and conformal field theory characters.

Findings

Proof of the Shimura-Taniyama conjecture by Wiles et al.

Connection between elliptic curve modular forms and conformal field theory.

Identification of modular forms from affine Kac-Moody algebra in this context.

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.