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.

References in corpus (1)

• Arithmetic of Calabi-Yau Varieties and Rational Conformal Field Theory

Cited by in corpus (11)

• The Langlands Program and String Modular K3 Surfaces
• Emergent spacetime from modular motives
• String Modular Phases in Calabi-Yau Families
• Arithmetic Spacetime Geometry from String Theory
• Geometric Kac-Moody Modularity
• Revisiting arithmetic solutions to the $W=0$ condition
• Complex Multiplication of Exactly Solvable Calabi-Yau Varieties
• String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication
• A modularity test for elliptic mirror symmetry
• Emergent spacetime and black hole probes from automorphic forms
• K-Rational D-Brane Crystals