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| In mathematics, '''Ribet's theorem''' (earlier called the '''epsilon conjecture''' or '''ε-conjecture''') is a statement in [[number theory]] concerning properties of [[Galois representation]]s associated with [[modular form]]s. It was proposed by [[Jean-Pierre Serre]] and proved by [[Ken Ribet]]. The proof of epsilon conjecture was a significant step towards the proof of [[Fermat's Last Theorem]]. As shown by Serre and Ribet, the [[Taniyama–Shimura conjecture]] (whose status was unresolved at the time) and the epsilon conjecture together imply that [[Fermat's Last Theorem]] is true.
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| == Statement ==
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| Let ''f'' be a weight 2 newform on Γ<sub>0</sub>(''qN'') -- i.e. of level ''qN'' where ''q'' does not divide ''N''—with absolutely irreducible 2-dimensional mod ''p'' Galois representation ''ρ<sub>f,p</sub>'' unramified at ''q'' if ''q ≠ p'' and finite flat at ''q = p''. Then there exists a weight 2 newform ''g'' of level ''N'' such that
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| :<math> \rho_{f,p} \simeq \rho_{g,p}. </math>
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| In particular, if ''E'' is an [[elliptic curve]] over <math>\mathbb{Q}</math> with conductor ''qN'', then the [[Modularity theorem]] guarantees that there exists a weight 2 newform ''f'' of level ''qN'' such that the 2-dimensional mod ''p'' Galois representation ''ρ<sub>f, p</sub>'' of ''f'' is isomorphic to the 2-dimensional mod ''p'' Galois representation ''ρ<sub>E, p</sub>'' of ''E''. To apply Ribet's Theorem to ''ρ<sub>E, p</sub>'', it suffices to check the irreducibility and ramification of ''ρ<sub>E, p</sub>''. Using the theory of the [[Tate curve]], one can prove that ''ρ<sub>E, p</sub>'' is unramified at ''q ≠ p'' and finite flat at ''q = p'' if ''p'' divides the power to which ''q'' appears in the minimal discriminant ''Δ<sub>E</sub>''. Then Ribet's theorem implies that there exists a weight 2 newform ''g'' of level ''N'' such that ''ρ<sub>g, p</sub> ≈ ρ<sub>E, p</sub>''.
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| == The Result of Level Lowering ==
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| Note that Ribet's theorem '''does not''' guarantee that if you begin with an elliptic curve ''E'' of conductor ''qN'', there exists an elliptic curve ''E' '' of level ''N'' such that ''ρ<sub>E, p</sub> ≈ ρ<sub>E', p</sub>''. The newform ''g'' of level ''N'' may not have rational 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
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| :<math> E: y^2 +xy +y = x^3 - 663204x + 206441595</math>
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| with conductor 43*97 and discriminant 43<sup>7</sup> * 97<sup>3</sup> does not level-lower mod 7 to an elliptic curve of conductor 97. Rather, the mod ''p'' Galois representation is isomorphic to the mod ''p'' Galois representation of an irrational newform ''g'' of level 97.
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| However, for ''p'' large enough compared to the level ''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 ''p''>>''N<sup>N<sup>1+ε</sup></sup>'', the mod ''p'' Galois representation of a rational newform cannot be isomorphic to that of an irrational newform of level ''N''.<ref>{{cite arXiv | last1=Silliman | last2=Vogt| eprint=1307.5078v2 |title=Powers in Lucas Sequences via Galois Representations |class= math.NT |year=2013}}</ref>
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| Similarly, the Frey-Mazur conjecture predicts that for ''p'' large enough (independent of the conductor ''N''), elliptic curves with isomorphic mod ''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 ''p'' (in particular ''p'' > 17).
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| == History ==
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| In his thesis, {{ill|de|Yves Hellegouarch}} came up with the idea of associating solutions (''a'',''b'',''c'') of Fermat's equation with a completely 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}}</ref>
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| If ''p'' is an odd prime and ''a'', ''b'', and ''c'' are positive integers such that | |
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| :<math>a^p + b^p = c^p,\ </math>
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| then a corresponding [[Frey curve]] is an algebraic curve given by the equation
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| :<math>y^2 = x(x - a^p)(x + b^p).\ </math> | |
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| 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>.
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| In 1982 [[Gerhard Frey]] called attention to the unusual properties of the same curve as Hellegouarch, now called a [[Frey curve]].<ref>{{Citation | last1=Frey | first1=Gerhard | title=Rationale Punkte auf Fermatkurven und getwisteten Modulkurven| year=1982 | journal=[[Crelle's Journal|J. reine u. angew. Math.]] | volume=331 | pages=185–191}}</ref> This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular. The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem. 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 of this.<ref>{{cite journal | author = J.F Mestre and J.P Serre | year = 1987 | title = Current Trends in Arithmetical Algebraic Geometry | journal = Contemporary Mathematics | volume = 67 | pages = 263–268 }}</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 Fermat's Last Theorem. Serre did not provide a complete proof and what was missing 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 Taniyama–Shimura–Weil conjecture implied Fermat's Last Theorem.<ref>{{cite journal|author=Ken Ribet|authorlink=Ken Ribet|title=On modular representations of Gal({{overline|'''Q'''}}/'''Q''') arising from modular forms|journal=Inventiones mathematicae 100|year=1990|issue=2|pages=431–471|url=http://math.berkeley.edu/~ribet/Articles/invent_100.pdf|format=PDF}}</ref>
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| == Implication of Fermat's Last Theorem ==
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| Suppose that the Fermat equation with exponent ''p'' ≥ 3 had a solution in non-zero integers ''a'', ''b'', ''c''. Let us form the corresponding Frey curve ''E<sub>a<sup>p</sup>,b<sup>p</sup>,c<sup>p</sup></sub>''. It is an elliptic curve and one can show that its [[Discriminant|minimal discriminant]] Δ is equal to 2<sup>−8</sup> (''abc'')<sup>2p</sup> and its conductor ''N'' is the [[radical of an integer|radical]] of ''abc'', i.e. the product of all distinct primes dividing ''abc''. By an elementary consideration of the equation ''a<sup>p</sup> + b<sup>p</sup> = c<sup>p</sup>'', it is clear that one of ''a, b, c'' is even and hence so is ''N''. By the Taniyama–Shimura conjecture, ''E'' is a modular elliptic curve. Since all odd primes dividing ''a,b,c'' in ''N'' appear to a ''p''th power in the minimal discriminant Δ, by Ribet's theorem one can perform level descent modulo ''p'' repetitively to strip off all odd primes from the conductor. However, there are no newforms of level 2 as the genus of the modular curve ''X''<sub>0</sub>(2) is zero (and newforms of level ''N'' are differentials on ''X''<sub>0</sub>(''N'')).
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| == See also ==
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| * [[abc conjecture]]
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| * [[Modularity theorem]]
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| * [[Wiles' proof of Fermat's Last Theorem]]
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| == References ==
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| {{Reflist}}
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| * 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. 116–139.
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| * {{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 |format=PDF| doi = 10.2307/2118559 }}
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| * {{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 |format=PDF | issn=0003486X|oclc=37032255 | publisher = Annals of Mathematics | jstor = 2118560 | zbl = 0823.11030}}
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| * [http://mathworld.wolfram.com/FreyCurve.html Frey Curve] and [http://mathworld.wolfram.com/RibetsTheorem.html Ribet's Theorem]
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| == External links ==
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| *[http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/3830/show_video Ken Ribet and Fermat's Last Theorem] by [[Kevin Buzzard]] June 28, 2008
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| [[Category:Algebraic curves]]
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| [[Category:Riemann surfaces]]
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| [[Category:Modular forms]]
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| [[Category:Theorems in number theory]]
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| [[Category:Theorems in algebraic geometry]]
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