Editing Ribet's theorem

{{Technical|date=February 2022}}
{{more citations needed|date=January 2021}}
{{short description|Result concerning properties of Galois representations associated with modular forms}}
'''Ribet's theorem''' (earlier called the '''epsilon conjecture''' or '''ε-conjecture''') is part of [[number theory]]. It concerns properties of [[Galois representation]]s associated with [[modular form]]s. It was proposed by [[Jean-Pierre Serre]] and [[mathematical proof|proven]] by [[Ken Ribet]]. The proof was a significant step towards the proof of [[Fermat's Last Theorem]] (FLT). As shown by Serre and Ribet, the [[Taniyama–Shimura conjecture]] (whose status was unresolved at the time) and the epsilon conjecture together imply that FLT is true.

In mathematical terms, Ribet's theorem shows that if the Galois representation associated with an [[elliptic curve]] has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form that gives rise to the same representation).<ref>{{cite web | url=http://cgd.best.vwh.net/home/flt/flt08.htm | archive-url=https://web.archive.org/web/20081210102243/http://cgd.best.vwh.net/home/flt/flt08.htm | url-status=dead | archive-date=2008-12-10 | title=The Proof of Fermat's Last Theorem| date=2008-12-10}}</ref>

== Statement ==

Let {{math|''f''}} be a weight 2 [[Atkin–Lehner theory|newform]] on {{math|Γ<sub>0</sub>(''qN'')}} &ndash; i.e. of level {{math|''qN''}} where {{math|''q''}} does not divide {{math|''N''}} &ndash; with absolutely irreducible 2-dimensional mod {{math|''p''}} Galois representation {{math|''ρ<sub>f,p</sub>''}} unramified at {{math|''q''}} if {{math|''q'' ≠ ''p''}} and finite flat at {{math|''q'' {{=}} ''p''}}. Then there exists a weight 2 newform {{math|''g''}} of level {{math|''N''}} such that
:<math> \rho_{f,p} \simeq \rho_{g,p}. </math>

In particular, if {{math|''E''}} is an [[elliptic curve]] over <math>\mathbb{Q}</math> with [[conductor of an elliptic curve|conductor]] {{math|''qN''}}, then the [[modularity theorem]] guarantees that there exists a weight 2 newform {{math|''f''}} of level {{math|''qN''}} such that the 2-dimensional mod {{math|''p''}} Galois representation {{math|''ρ<sub>f, p</sub>''}} of {{math|''f''}} is isomorphic to the 2-dimensional mod {{math|''p''}} Galois representation {{math|''ρ<sub>E, p</sub>''}} of {{math|''E''}}. To apply Ribet's Theorem to {{math|''ρ''<sub>''E'', ''p''</sub>}}, it suffices to check the irreducibility and ramification of {{math|''ρ<sub>E, p</sub>''}}. Using the theory of the [[Tate curve]], one can prove that {{math|''ρ<sub>E, p</sub>''}} is unramified at {{math|''q'' ≠ ''p''}} and finite flat at {{math|''q'' {{=}} ''p''}} if {{math|''p''}} divides the power to which {{math|''q''}} appears in the minimal discriminant {{math|Δ<sub>''E''</sub>}}. Then Ribet's theorem implies that there exists a weight 2 newform {{math|''g''}} of level {{math|''N''}} such that {{math|''ρ''<sub>''g'', ''p''</sub> ≈ ''ρ''<sub>''E'', ''p''</sub>}}.

== Level lowering ==

Ribet's theorem states that beginning with an elliptic curve {{math|''E''}} of conductor {{math|''qN''}} does not guarantee the existence of an elliptic curve {{math|''E{{prime}}''}} of level {{math|''N''}} such that {{math|''ρ''<sub>''E, p''</sub> ≈ ''ρ''<sub>''E{{prime}}'', ''p''</sub>}}. The newform {{math|''g''}} of level {{math|''N''}} may not have rational [[Fourier series|Fourier]] coefficients, and hence may be associated to a higher-dimensional [[abelian variety]], not an elliptic curve. For example, elliptic curve 4171a1 in the Cremona database given by the equation

:<math>E: y^2 + xy + y = x^3 - 663204x + 206441595</math>

with conductor {{math|43 × 97}} and discriminant {{math|43<sup>7</sup> × 97<sup>3</sup>}} does not level-lower mod 7 to an elliptic curve of conductor 97. Rather, the mod {{math|''p''}} Galois representation is isomorphic to the mod {{math|''p''}} Galois representation of an irrational newform {{math|''g''}} of level 97.

However, for {{math|''p''}} large enough compared to the level {{math|''N''}} of the level-lowered newform, a rational newform (e.g. an elliptic curve) must level-lower to another rational newform (e.g. elliptic curve). In particular for {{math|''p'' ≫ ''N''<sup>''N''<sup>1+''ε''</sup></sup>}}, the mod {{math|''p''}} Galois representation of a rational newform cannot be isomorphic to an irrational newform of level {{math|''N''}}.<ref>{{cite journal | last1=Silliman | first1 = Jesse| last2=Vogt| first2=Isabel | arxiv=1307.5078 |title=Powers in Lucas Sequences via Galois Representations |year=2015|mr=3293720|journal=[[Proceedings of the American Mathematical Society]]|volume=143|issue=3|pages=1027–1041|doi=10.1090/S0002-9939-2014-12316-1| citeseerx = 10.1.1.742.7591| s2cid = 16892383}}</ref>

Similarly, the Frey-[[Barry_Mazur|Mazur]] conjecture predicts that for large enough {{math|''p''}} (independent of the conductor {{math|''N''}}), elliptic curves with isomorphic mod {{math|''p''}} Galois representations are in fact [[Isogeny|isogenous]], and hence have the same conductor. Thus non-trivial level-lowering between rational newforms is not predicted to occur for large {{math|''p'' (''p'' > 17)}}.

== 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".

== Implications ==
Suppose that the Fermat equation with exponent {{math|''p'' ≥ 5}}<ref name="ribet"/> had a solution in non-zero integers {{math|''a'', ''b'', ''c''}}. The corresponding Frey curve {{math|''E''<sub>''a''<sup>''p''</sup>,''b''<sup>''p''</sup>,''c''<sup>''p''</sup></sub>}} is an elliptic curve whose [[Discriminant|minimal discriminant]] {{math|Δ}} is equal to {{math|2<sup>−8</sup> (''abc'')<sup>2''p''</sup>}} and whose conductor {{math|''N''}} is the [[radical of an integer|radical]] of {{math|''abc''}}, i.e. the product of all distinct primes dividing {{math|''abc''}}. An elementary consideration of the equation {{math|''a''<sup>''p''</sup> + ''b''<sup>''p''</sup> {{=}} ''c''<sup>''p''</sup>}}, makes it clear that one of {{math|''a'', ''b'', ''c''}} is even and hence so is ''N''. By the Taniyama–Shimura conjecture, {{math|''E''}} is a modular elliptic curve. Since all odd primes dividing {{math|''a'', ''b'', ''c''}} in {{math|''N''}} appear to a {{math|''p''th}} power in the minimal discriminant {{math|Δ}}, by Ribet's theorem repetitive [[Atkin–Lehner theory|level]] [[Descent (mathematics)|descent]] modulo {{math|''p''}} strips all odd primes from the conductor. However, no newforms of level 2 remain because the genus of the modular curve {{math|''X''<sub>0</sub>(2)}} is zero (and newforms of level ''N'' are differentials on {{math|''X''<sub>0</sub>(''N''))}}.

== See also ==
* [[abc conjecture|ABC conjecture]]
* [[Wiles' proof of Fermat's Last Theorem]]

== Notes ==
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== References ==
{{Reflist}}
* Kenneth Ribet, [http://www.numdam.org/item?id=AFST_1990_5_11_1_116_0 ''From the Taniyama-Shimura conjecture to Fermat's last theorem''.] Annales de la faculté des sciences de Toulouse Sér. 5, 11 no. 1 (1990), p.&nbsp;116–139.
* {{cite journal | author = Andrew Wiles | authorlink=Andrew Wiles |date=May 1995 | title = Modular elliptic curves and Fermat's Last Theorem | journal = Annals of Mathematics | volume = 141 | issue = 3 | pages = 443–551 | url = http://math.stanford.edu/~lekheng/flt/wiles.pdf | doi = 10.2307/2118559 | jstor=2118559 | citeseerx=10.1.1.169.9076 }}
* {{cite journal | author = [[Richard Taylor (mathematician)|Richard Taylor]] and Andrew Wiles |date=May 1995 | title = Ring-theoretic properties of certain Hecke algebras | journal = Annals of Mathematics | volume = 141 | issue = 3 | pages = 553–572 | url = http://math.stanford.edu/~lekheng/flt/taylor-wiles.pdf | doi = 10.2307/2118560 | issn=0003-486X|oclc=37032255 | jstor = 2118560 | zbl = 0823.11030|citeseerx=10.1.1.128.531 }}
* [http://mathworld.wolfram.com/FreyCurve.html Frey Curve] and [http://mathworld.wolfram.com/RibetsTheorem.html Ribet's Theorem]

== External links ==
*[http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/3830/show_video Ken Ribet and Fermat's Last Theorem] by [[Kevin Buzzard]] June 28, 2008

[[Category:Algebraic curves]]
[[Category:Riemann surfaces]]
[[Category:Modular forms]]
[[Category:Theorems in number theory]]
[[Category:Theorems in algebraic geometry]]
[[Category:Fermat's Last Theorem]]
[[Category:Abc conjecture]]