Modularity theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Modularity_theorem


00:01:10 1 Statement
00:05:17 2 History



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SUMMARY
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In mathematics, the modularity theorem (formerly called the Taniyama–Shimura conjecture or the Taniyama–Shimura–Weil conjecture) states that elliptic curves over the field of rational numbers are related to  modular forms.  Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's last theorem. Later, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor extended Wiles' techniques to prove the full modularity theorem in 2001. 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.  However,  Freitas, Le Hung & Siksek (2015) proved that elliptic curves defined over real quadratic fields are modular.