High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics the modularity theorem (previously known as the Taniyama–Shimura–Weil conjecture and by several related names) establishes a connection between elliptic curves over the field of rational numbers and modular forms. It was fully proved jointly by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor in 2001, borrowing many of the techniques used in Andrew Wiles' proof of Fermat's Last Theorem. The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved.

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