Ribet's theorem

<h2 id="statement">Statement</h2>
Let f be a weight 2 <a href="/facts/Atkin%25E2%2580%2593Lehner_theory/KgNk1KCW">newform</a> on Γ0(qN) – i.e. of level qN where q does not divide N – with absolutely irreducible 2-dimensional mod p Galois representation ρf,p 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

ρ
          
            f
            ,
            p
          
        
        ≃
        
          ρ
          
            g
            ,
            p
          
        
        .
      
    
    {\displaystyle \rho _{f,p}\simeq \rho _{g,p}.}

In particular, if E is an <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curve</a> over 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 with <a href="/facts/Conductor_of_an_elliptic_curve/qMI4S3Cb">conductor</a> qN, then the <a href="/facts/Modularity_theorem/aI3HRxwj">modularity theorem</a> guarantees that there exists a weight 2 newform f of level qN such that the 2-dimensional mod p Galois representation ρf, p of f is isomorphic to the 2-dimensional mod p Galois representation ρE, p of E. To apply Ribet's Theorem to ρE, p, it suffices to check the irreducibility and ramification of ρE, p. Using the theory of the <a href="/facts/Tate_curve/211hY4TK">Tate curve</a>, one can prove that ρE, p is unramified at q ≠ p and finite flat at q = p if p divides the power to which q appears in the minimal discriminant ΔE. Then Ribet's theorem implies that there exists a weight 2 newform g of level N such that ρg, p ≈ ρE, p.

<h2 id="level-lowering">Level lowering</h2>
Ribet's theorem states that beginning with an elliptic curve E of conductor qN does not guarantee the existence of an elliptic curve E′ of level N such that ρE, p ≈ ρE′, p. The newform g of level N may not have rational <a href="/facts/Fourier_series/s3fsxPAT">Fourier</a> coefficients, and hence may be associated to a higher-dimensional <a href="/facts/Abelian_variety/dGSDVruG">abelian variety</a>, not an elliptic curve. For example, elliptic curve 4171a1 in the Cremona database given by the equation

E
        :
        
          y
          
            2
          
        
        +
        x
        y
        +
        y
        =
        
          x
          
            3
          
        
        −
        663204
        x
        +
        206441595
      
    
    {\displaystyle E:y^{2}+xy+y=x^{3}-663204x+206441595}

with conductor 43 × 97 and discriminant 437 × 973 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.
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 ≫ NN1+ε, the mod p Galois representation of a rational newform cannot be isomorphic to an irrational newform of level N.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
Similarly, the Frey-<a href="/facts/Barry_Mazur/4tGbAnCX">Mazur</a> conjecture predicts that for large enough p (independent of the conductor N), elliptic curves with isomorphic mod p Galois representations are in fact <a href="/facts/Isogeny/ngeefttF">isogenous</a>, and hence have the same conductor. Thus non-trivial level-lowering between rational newforms is not predicted to occur for large p (p > 17).

<h2 id="history">History</h2>
In his thesis, Yves Hellegouarch [fr] originated the idea of associating solutions (a,b,c) of Fermat's equation with a different mathematical object: an elliptic curve.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> If p is an odd prime and a, b, and c are positive integers such that

a
          
            p
          
        
        +
        
          b
          
            p
          
        
        =
        
          c
          
            p
          
        
        ,
      
    
    {\displaystyle a^{p}+b^{p}=c^{p},}

then a corresponding <a href="/facts/Frey_curve/cUQD50a1">Frey curve</a> is an algebraic curve given by the equation

y
          
            2
          
        
        =
        x
        (
        x
        −
        
          a
          
            p
          
        
        )
        (
        x
        +
        
          b
          
            p
          
        
        )
        .
      
    
    {\displaystyle y^{2}=x(x-a^{p})(x+b^{p}).}

This is a nonsingular algebraic curve of genus one defined over 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
, and its projective completion is an elliptic curve over 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
.
In 1982 <a href="/facts/Gerhard_Frey/MBQcRzPm">Gerhard Frey</a> called attention to the unusual properties of the same curve, now called a <a href="/facts/Frey_curve/cUQD50a1">Frey curve</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> This provided a bridge between <a href="/facts/Pierre_de_Fermat/FLbUh75u">Fermat</a> and <a href="/facts/Yutaka_Taniyama/1EGULkp1">Taniyama</a> by showing that a counterexample to FLT would create a curve that would not be modular. The conjecture attracted considerable interest when <a href="/facts/Gerhard_Frey/MBQcRzPm">Frey</a> suggested that the Taniyama–Shimura conjecture implies FLT. However, his argument was not complete.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> In 1985 <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Jean-Pierre Serre</a> proposed that a Frey curve could not be modular and provided a partial proof.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> 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, <a href="/facts/Kenneth_Alan_Ribet/6hU9bzw8">Kenneth Alan Ribet</a> proved the epsilon conjecture, thereby proving that the <a href="/facts/Modularity_theorem/aI3HRxwj">Modularity theorem</a> implied FLT.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
The origin of the name is from the ε part of "Taniyama-Shimura conjecture + ε ⇒ Fermat's last theorem".

<h2 id="implications">Implications</h2>
Suppose that the Fermat equation with exponent p ≥ 5<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> had a solution in non-zero integers a, b, c. The corresponding Frey curve Eap,bp,cp is an elliptic curve whose <a href="/facts/Discriminant/0SmQtrvU">minimal discriminant</a> Δ is equal to 2−8 (abc)2p and whose conductor N is the <a href="/facts/Radical_of_an_integer/xyoLHVNs">radical</a> of abc, i.e. the product of all distinct primes dividing abc. An elementary consideration of the equation ap + bp = cp, makes it 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 pth power in the minimal discriminant Δ, by Ribet's theorem repetitive <a href="/facts/Atkin%25E2%2580%2593Lehner_theory/KgNk1KCW">level</a> <a href="/facts/Descent_(mathematics)/YTMNz2vH">descent</a> modulo p strips all odd primes from the conductor. However, no newforms of level 2 remain because the genus of the modular curve X0(2) is zero (and newforms of level N are differentials on X0(N)).

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Abc_conjecture/yJeHld4Q">ABC conjecture</a></li>
<li><a href="/facts/Wiles%2527_proof_of_Fermat%2527s_Last_Theorem/Xgr6wAf2">Wiles' proof of Fermat's Last Theorem</a></li></ul>
<h2 id="notes">Notes</h2>
<ol>
<li>^ <a href="https://web.archive.org/web/20081210102243/http://cgd.best.vwh.net/home/flt/flt08.htm">"The Proof of Fermat's Last Theorem"</a>. 2008-12-10. Archived from <a href="http://cgd.best.vwh.net/home/flt/flt08.htm">the original</a> on 2008-12-10.
</li>
<li>^ Silliman, Jesse; Vogt, Isabel (2015). "Powers in Lucas Sequences via Galois Representations". <a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a>. 143 (3): 1027–1041. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1307.5078">1307.5078</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.742.7591">10.1.1.742.7591</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9939-2014-12316-1">10.1090/S0002-9939-2014-12316-1</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3293720">3293720</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:16892383">16892383</a>.
</li>
<li>^ Hellegouarch, Yves (1972). "Courbes elliptiques et equation de Fermat". Doctoral Dissertation. <a href="/facts/BNF_(identifier)/4TNyWsxG">BnF</a> <a href="http://catalogue.bnf.fr/ark:/12148/cb359121326.public">359121326</a>.
</li>
<li>^ Frey, Gerhard (1982), "Rationale Punkte auf Fermatkurven und getwisteten Modulkurven" [Rational points on Fermat curves and twisted modular curves], <a href="/facts/J._Reine_Angew._Math./YCcRSaAQ">J. Reine Angew. Math.</a> (in German), 1982 (331): 185–191, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1982.331.185">10.1515/crll.1982.331.185</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0647382">0647382</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:118263144">118263144</a>
</li>
<li>^ Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae, 1 (1): iv+40, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0933-8268">0933-8268</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0853387">0853387</a>
</li>
<li>^ <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, J.-P.</a> (1987), "Lettre à J.-F. Mestre [Letter to J.-F. Mestre]", Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemporary Mathematics (in French), vol. 67, Providence, RI: American Mathematical Society, pp. 263–268, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fconm%2F067%2F902597">10.1090/conm/067/902597</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780821850749, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0902597">0902597</a>
</li>
<li>^ <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1987), "Sur les représentations modulaires de degré 2 de Gal(Q/Q)", <a href="/facts/Duke_Mathematical_Journal/S6wn8HYp">Duke Mathematical Journal</a>, 54 (1): 179–230, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1215%2FS0012-7094-87-05413-5">10.1215/S0012-7094-87-05413-5</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0012-7094">0012-7094</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0885783">0885783</a>
</li>
<li>^ a b <a href="/facts/Ken_Ribet/6hU9bzw8">Ribet, Ken</a> (1990). <a href="http://math.berkeley.edu/~ribet/Articles/invent_100.pdf">"On modular representations of Gal(Q/Q) arising from modular forms"</a> (PDF). Inventiones Mathematicae. 100 (2): 431–476. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1990InMat.100..431R">1990InMat.100..431R</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01231195">10.1007/BF01231195</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1047143">1047143</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120614740">120614740</a>.
</li>
</ol>

<ul><li>Kenneth Ribet, <a href="http://www.numdam.org/item?id=AFST_1990_5_11_1_116_0">From the Taniyama-Shimura conjecture to Fermat's last theorem.</a> Annales de la faculté des sciences de Toulouse Sér. 5, 11 no. 1 (1990), p. 116–139.</li>
<li><a href="/facts/Andrew_Wiles/bdc3k3pQ">Andrew Wiles</a> (May 1995). <a href="http://math.stanford.edu/~lekheng/flt/wiles.pdf">"Modular elliptic curves and Fermat's Last Theorem"</a> (PDF). Annals of Mathematics. 141 (3): 443–551. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.169.9076">10.1.1.169.9076</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2118559">10.2307/2118559</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2118559">2118559</a>.</li>
<li><a href="/facts/Richard_Taylor_(mathematician)/crBWfG0r">Richard Taylor</a> and Andrew Wiles (May 1995). <a href="http://math.stanford.edu/~lekheng/flt/taylor-wiles.pdf">"Ring-theoretic properties of certain Hecke algebras"</a> (PDF). Annals of Mathematics. 141 (3): 553–572. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.128.531">10.1.1.128.531</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2118560">10.2307/2118560</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2118560">2118560</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/37032255">37032255</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0823.11030">0823.11030</a>.</li>
<li><a href="http://mathworld.wolfram.com/FreyCurve.html">Frey Curve</a> and <a href="http://mathworld.wolfram.com/RibetsTheorem.html">Ribet's Theorem</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/3830/show_video">Ken Ribet and Fermat's Last Theorem</a> by <a href="/facts/Kevin_Buzzard/axgXTFR5">Kevin Buzzard</a> June 28, 2008</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"The Proof of Fermat's Last Theorem". 2008-12-10. Archived from the original on 2008-12-10. <a href="https://web.archive.org/web/20081210102243/http://cgd.best.vwh.net/home/flt/flt08.htm" target="_blank">https://web.archive.org/web/20081210102243/http://cgd.best.vwh.net/home/flt/flt08.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Silliman, Jesse; Vogt, Isabel (2015). "Powers in Lucas Sequences via Galois Representations". Proceedings of the American Mathematical Society. 143 (3): 1027–1041. arXiv:1307.5078. CiteSeerX 10.1.1.742.7591. doi:10.1090/S0002-9939-2014-12316-1. MR 3293720. S2CID 16892383. <a href="/wiki/Proceedings_of_the_American_Mathematical_Society" target="_blank">/wiki/Proceedings_of_the_American_Mathematical_Society</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Hellegouarch, Yves (1972). "Courbes elliptiques et equation de Fermat". Doctoral Dissertation. BnF 359121326. <a href="/wiki/BNF_(identifier)" target="_blank">/wiki/BNF_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Frey, Gerhard (1982), "Rationale Punkte auf Fermatkurven und getwisteten Modulkurven" [Rational points on Fermat curves and twisted modular curves], J. Reine Angew. Math. (in German), 1982 (331): 185–191, doi:10.1515/crll.1982.331.185, MR 0647382, S2CID 118263144 <a href="/wiki/J._Reine_Angew._Math." target="_blank">/wiki/J._Reine_Angew._Math.</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae, 1 (1): iv+40, ISSN 0933-8268, MR 0853387 <a href="/wiki/ISSN_(identifier)" target="_blank">/wiki/ISSN_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Serre, J.-P. (1987), "Lettre à J.-F. Mestre [Letter to J.-F. Mestre]", Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemporary Mathematics (in French), vol. 67, Providence, RI: American Mathematical Society, pp. 263–268, doi:10.1090/conm/067/902597, ISBN 9780821850749, MR 0902597 <a href="9780821850749" target="_blank">9780821850749</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Serre, Jean-Pierre (1987), "Sur les représentations modulaires de degré 2 de Gal(Q/Q)", Duke Mathematical Journal, 54 (1): 179–230, doi:10.1215/S0012-7094-87-05413-5, ISSN 0012-7094, MR 0885783 <a href="/wiki/Jean-Pierre_Serre" target="_blank">/wiki/Jean-Pierre_Serre</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Ribet, Ken (1990). "On modular representations of Gal(Q/Q) arising from modular forms" (PDF). Inventiones Mathematicae. 100 (2): 431–476. Bibcode:1990InMat.100..431R. doi:10.1007/BF01231195. MR 1047143. S2CID 120614740. <a href="/wiki/Ken_Ribet" target="_blank">/wiki/Ken_Ribet</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Ribet, Ken (1990). "On modular representations of Gal(Q/Q) arising from modular forms" (PDF). Inventiones Mathematicae. 100 (2): 431–476. Bibcode:1990InMat.100..431R. doi:10.1007/BF01231195. MR 1047143. S2CID 120614740. <a href="/wiki/Ken_Ribet" target="_blank">/wiki/Ken_Ribet</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Ribet's theorem open-in-new

Ribet's theorem