Editing Ribet's theorem (section)

== History ==
In his thesis, {{Interlanguage link multi|Yves Hellegouarch|fr}} originated the idea of associating solutions (''a'',''b'',''c'') of Fermat's equation with a different mathematical object: an elliptic curve.<ref>{{cite journal|last=Hellegouarch|first=Yves|title=Courbes elliptiques et equation de Fermat|journal=Doctoral Dissertation|year=1972|id={{BNF|359121326}}}}</ref> If ''p'' is an odd prime and ''a'', ''b'', and ''c'' are positive integers such that

:<math>a^p + b^p = c^p,</math>

then a corresponding [[Frey curve]] is an algebraic curve given by the equation

:<math>y^2 = x(x - a^p)(x + b^p).</math>

This is a nonsingular algebraic curve of genus one defined over <math>\mathbb{Q}</math>, and its projective completion is an elliptic curve over <math>\mathbb{Q}</math>.

In 1982 [[Gerhard Frey]] called attention to the unusual properties of the same curve, now called a [[Frey curve]].<ref>{{Citation | last1=Frey | first1=Gerhard | title=Rationale Punkte auf Fermatkurven und getwisteten Modulkurven| trans-title=Rational points on Fermat curves and twisted modular curves | language=de | year=1982 | journal=[[J. Reine Angew. Math.]] | volume=1982 | issue=331 | pages=185–191 | mr=0647382 | doi=10.1515/crll.1982.331.185| s2cid=118263144 }}</ref> This provided a bridge between [[Pierre de Fermat|Fermat]] and [[Yutaka Taniyama|Taniyama]] by showing that a counterexample to FLT would create a curve that would not be modular. The conjecture attracted considerable interest when [[Gerhard Frey|Frey]] suggested that the Taniyama–Shimura conjecture implies FLT. However, his argument was not complete.<ref>{{Citation | last1=Frey | first1=Gerhard | title=Links between stable elliptic curves and certain Diophantine equations | mr=853387 | year=1986 | journal=Annales Universitatis Saraviensis. Series Mathematicae | issn=0933-8268 | volume=1 | issue=1 | pages=iv+40}}</ref> In 1985 [[Jean-Pierre Serre]] proposed that a Frey curve could not be modular and provided a partial proof.<ref>{{citation
 | last = Serre | first = J.-P. | authorlink = Jean-Pierre Serre
 | contribution = Lettre à J.-F. Mestre [Letter to J.-F. Mestre]
 | doi = 10.1090/conm/067/902597
 | language = French
 | mr = 902597
 | pages = 263–268
 | publisher = American Mathematical Society | location = Providence, RI
 | series = Contemporary Mathematics
 | title = Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985)
 | volume = 67
 | year = 1987| isbn = 9780821850749 }}</ref><ref>{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Sur les représentations modulaires de degré 2 de Gal({{overline|'''Q'''}}/'''Q''') | doi=10.1215/S0012-7094-87-05413-5 | mr=885783 | year=1987 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=54 | issue=1 | pages=179–230}}</ref> This showed that a proof of the semistable case of the Taniyama–Shimura conjecture would imply FLT. Serre did not provide a complete proof and the missing bit became known as the epsilon conjecture or ε-conjecture. In the summer of 1986, [[Kenneth Alan Ribet]] proved the epsilon conjecture, thereby proving that the [[Modularity theorem]] implied FLT.<ref name="ribet">{{cite journal|last=Ribet|first=Ken|authorlink=Ken Ribet|title=On modular representations of Gal({{overline|'''Q'''}}/'''Q''') arising from modular forms|journal=Inventiones Mathematicae|volume=100|year=1990|issue=2|pages=431–476|doi=10.1007/BF01231195|mr=1047143|url=http://math.berkeley.edu/~ribet/Articles/invent_100.pdf|bibcode=1990InMat.100..431R|s2cid=120614740 }}</ref>

The origin of the name is from the ε part of "Taniyama-Shimura conjecture + ε ⇒ Fermat's last theorem".