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Final Answers
© 2000-2017   Gérard P. Michon, Ph.D.

Linking Elliptic Curves & Modular Forms

Cuius rei demonstrationem mirabilem sane detexi. 
Hanc marginis exiguitas non caperet.  

Pierre de Fermat  (1601-1665)   
  • Fermat's Last Theorem (FLT)  is a consequence of the  modularity theorem.
  • Elliptic curves.
  • Modular forms.
  • Modularity theorem:  Every  elliptic curve  is  modular.

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Elliptic Curves  by  J.S. Milnes  (238 pages, 2006).

The Modularity Theorem
Every Elliptic Curve is Modular

 Pierre de Fermat  
 (1601-1665) (2017-07-06)   The Proof of Fermat's Last Theorem   (FLT)
Ken Ribet  showed the  modularity conjecture  implied  FLT.

The modern story which utimately led to a proof of  Fermat's Last Theorem  had its origin with a wonderful conjecture made by the Japanese mathematician  Yutaka Taniyama (1927-1958) which was later refined by his friend  Goro Shimura (b.1930)  and by the Frenchman  André Weil (1906-1998).  It was known that some  elliptic curves  are  associated with a  modular form.  The Taniyama–Shimura–Weil conjecture  (or  modularity conjecture)  said that all of them were.  This is now known as the  Modularity Theorem,  which can be stated very concisely:

Every elliptic curve is modular.

In 1982,  Gerhard Frey  (b.1944)  connected that to FLT by considering an hypothetical counterexample, for a prime exponent  p :

a p  +  b p   =   c p

He suspected that the following elliptic curve  (the  Frey curve)  wouldn't be modular.  That would make  FLT  follow from the modularity conjecture.

y2   =   x  (x - a p ) (x + b p )

In 1985,  Jean-Pierre Serre (b.1926)  analyzed the situation and showed that the validity of Frey's reduction of FLT to the modularity conjecture was only missing very little  (he called that  epsilon  and that's how the puzzle became known for a while).  In 1986, Ken Ribet (b.1948)  found a restricted proof of that missing bit, which he was soon able to generalize with some help from  Barry Mazur (b.1937).  From then on,  a proof of the modularity conjecture would constitute  de facto  a proof of FLT...

This made Andrew Wiles (b.1953)  embark on a lonely 7-year journey to prove the modularity theorem, at least for  semistable elliptic curves,  which is enough to prove FLT.  Surprisingly enough, he was successful.

The full  Modularity theorem  was subsequently proved,  using Wiles' own techniques,  by  Christophe Olivier Breuil (b.1968, X1989) and three former doctoral students of  Andrew Wiles,  namely:  Richard Taylor (b.1962),  Fred Diamond (b.1964),  and  Brian Conrad (b.1970).

The Bridges to Fermat's Last Theorem (27:52)   by  Ken Ribet   (Numberphile by Brady Haran, 2015-03-11).
Introducing the proof of Fermat's last theorem (16:06)   by  Randell Hyman   (2016-09-10).

(2017-07-06)   Elliptic Curves

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 still working on this one...

Wikipedia :   Elliptic curves

(2017-07-06)   Modular Forms

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 still working on this one...

Wikipedia :   Modular forms

(2017-07-06)   Modularity Theorem
Formerly known as the  Taniyama–Shimura–Weil conjecture.

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 still working on this one...

Wikipedia :   Modularity theorem

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