A multi-Frey approach to Fermat equations of signature $(r,r,p)$
HTML articles powered by AMS MathViewer
- by Nicolas Billerey, Imin Chen, Luis Dieulefait and Nuno Freitas PDF
- Trans. Amer. Math. Soc. 371 (2019), 8651-8677 Request permission
Abstract:
In this paper, we give a resolution of the generalized Fermat equations \begin{equation*} x^5 + y^5 = 3 z^n\quad \text {and}\quad x^{13} + y^{13} = 3 z^n, \end{equation*} for all integers $n \ge 2$ and all integers $n \ge 2$ which are not a power of $7$, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents $n$.
We also give a number of results for the equations $x^5 + y^5 = d z^n$, where $d = 1, 2$, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat’s Last Theorem and which uses a new application of level raising at $p$ modulo $p$.
References
- Michael A. Bennett, Imin Chen, Sander R. Dahmen, and Soroosh Yazdani, On the equation $a^3+b^{3n}=c^2$, Acta Arith. 163 (2014), no. 4, 327–343. MR 3217670, DOI 10.4064/aa163-4-3
- Michael A. Bennett and Chris M. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math. 56 (2004), no. 1, 23–54. MR 2031121, DOI 10.4153/CJM-2004-002-2
- Michael A. Bennett, Vinayak Vatsal, and Soroosh Yazdani, Ternary Diophantine equations of signature $(p,p,3)$, Compos. Math. 140 (2004), no. 6, 1399–1416. MR 2098394, DOI 10.1112/S0010437X04000983
- Nicolas Billerey, Équations de Fermat de type $(5,5,p)$, Bull. Austral. Math. Soc. 76 (2007), no. 2, 161–194 (French, with French summary). MR 2353205, DOI 10.1017/S0004972700039575
- Nicolas Billerey, Imin Chen, Luis Dieulefait, and Nuno Freitas, supporting Magma program files for this paper, http://people.math.sfu.ca/~ichen/xrrdp.
- Nicolas Billerey and Luis V. Dieulefait, Solving Fermat-type equations $x^5+y^5=dz^p$, Math. Comp. 79 (2010), no. 269, 535–544. MR 2552239, DOI 10.1090/S0025-5718-09-02294-7
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- Christophe Breuil, Sur quelques représentations modulaires et $p$-adiques de $\textrm {GL}_2(\mathbf Q_p)$. II, J. Inst. Math. Jussieu 2 (2003), no. 1, 23–58 (French, with French summary). MR 1955206, DOI 10.1017/S1474748003000021
- Christophe Breuil and Ariane Mézard, Multiplicités modulaires et représentations de $\textrm {GL}_2(\textbf {Z}_p)$ et de $\textrm {Gal}(\overline \textbf {Q}_p/\textbf {Q}_p)$ en $l=p$, Duke Math. J. 115 (2002), no. 2, 205–310 (French, with English and French summaries). With an appendix by Guy Henniart. MR 1944572, DOI 10.1215/S0012-7094-02-11522-1
- Peter Bruin and Filip Najman, A criterion to rule out torsion groups for elliptic curves over number fields, Res. Number Theory 2 (2016), Paper No. 3, 13. MR 3501016, DOI 10.1007/s40993-015-0031-5
- Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, Almost powers in the Lucas sequence, J. Théor. Nombres Bordeaux 20 (2008), no. 3, 555–600 (English, with English and French summaries). MR 2523309
- Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), no. 3, 969–1018. MR 2215137, DOI 10.4007/annals.2006.163.969
- Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations. II. The Lebesgue-Nagell equation, Compos. Math. 142 (2006), no. 1, 31–62. MR 2196761, DOI 10.1112/S0010437X05001739
- Yann Bugeaud, Maurice Mignotte, and Samir Siksek, A multi-Frey approach to some multi-parameter families of Diophantine equations, Canad. J. Math. 60 (2008), no. 3, 491–519. MR 2414954, DOI 10.4153/CJM-2008-024-9
- Jose Ignacio Burgos Gil and Ariel Pacetti, Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields, Math. Comp. 86 (2017), no. 306, 1949–1978. MR 3626544, DOI 10.1090/mcom/3187
- J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
- Sander R. Dahmen, A refined modular approach to the Diophantine equation $x^2+y^{2n}=z^3$, Int. J. Number Theory 7 (2011), no. 5, 1303–1316. MR 2825973, DOI 10.1142/S1793042111004472
- Sander R. Dahmen and Samir Siksek, Perfect powers expressible as sums of two fifth or seventh powers, Acta Arith. 164 (2014), no. 1, 65–100. MR 3223319, DOI 10.4064/aa164-1-5
- Henri Darmon and Andrew Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, Bull. London Math. Soc. 27 (1995), no. 6, 513–543. MR 1348707, DOI 10.1112/blms/27.6.513
- Henri Darmon and Loïc Merel, Winding quotients and some variants of Fermat’s last theorem, J. Reine Angew. Math. 490 (1997), 81–100. MR 1468926
- Martin Derickx, Sheldon Kamienny, William Stein, and Michael Stoll, Torsion points on elliptic curves over number fields of small degree, arXiv e-prints, July 2017.
- Luis Dieulefait and Nuno Freitas, Fermat-type equations of signature $(13,13,p)$ via Hilbert cuspforms, Math. Ann. 357 (2013), no. 3, 987–1004. MR 3118622, DOI 10.1007/s00208-013-0920-7
- Luis Dieulefait and Nuno Freitas, The Fermat-type equations $x^5+y^5=2z^p$ or $3z^p$ solved through $\Bbb {Q}$-curves, Math. Comp. 83 (2014), no. 286, 917–933. MR 3143698, DOI 10.1090/S0025-5718-2013-02731-7
- G. Lejeune Dirichlet, Mémoire sur l’impossibilité de quelques équations indéterminées du cinquième degré, J. Reine Angew. Math. 3 (1828), 354–375 (French). MR 1577706, DOI 10.1515/crll.1828.3.354
- Jordan S. Ellenberg, Galois representations attached to $\Bbb Q$-curves and the generalized Fermat equation $A^4+B^2=C^p$, Amer. J. Math. 126 (2004), no. 4, 763–787. MR 2075481
- Nuno Freitas, Recipes to Fermat-type equations of the form $x^r+y^r=Cz^p$, Math. Z. 279 (2015), no. 3-4, 605–639. MR 3318242, DOI 10.1007/s00209-014-1384-5
- Nuno Freitas, Bao V. Le Hung, and Samir Siksek, Elliptic curves over real quadratic fields are modular, Invent. Math. 201 (2015), no. 1, 159–206. MR 3359051, DOI 10.1007/s00222-014-0550-z
- Nuno Freitas and Samir Siksek, Criteria for irreducibility of $\textrm {mod}\, p$ representations of Frey curves, J. Théor. Nombres Bordeaux 27 (2015), no. 1, 67–76 (English, with English and French summaries). MR 3346965
- Nuno Freitas and Samir Siksek, Fermat’s last theorem over some small real quadratic fields, Algebra Number Theory 9 (2015), no. 4, 875–895. MR 3352822, DOI 10.2140/ant.2015.9.875
- Kazuhiro Fujiwara, Level optimization in the totally real case, arXiv mathematics e-prints, February 2006.
- Frazer Jarvis, Level lowering for modular mod $l$ representations over totally real fields, Math. Ann. 313 (1999), no. 1, 141–160. MR 1666809, DOI 10.1007/s002080050255
- Frazer Jarvis, Correspondences on Shimura curves and Mazur’s principle at $p$, Pacific J. Math. 213 (2004), no. 2, 267–280. MR 2036920, DOI 10.2140/pjm.2004.213.267
- Alain Kraus, Courbes elliptiques semi-stables et corps quadratiques, J. Number Theory 60 (1996), no. 2, 245–253 (French, with French summary). MR 1412962, DOI 10.1006/jnth.1996.0122
- Alain Kraus, Majorations effectives pour l’équation de Fermat généralisée, Canad. J. Math. 49 (1997), no. 6, 1139–1161 (French, with French summary). MR 1611640, DOI 10.4153/CJM-1997-056-2
- Alain Kraus, Sur l’équation $a^3+b^3=c^p$, Experiment. Math. 7 (1998), no. 1, 1–13 (French, with English and French summaries). MR 1618290
- A. Kraus and J. Oesterlé, Sur une question de B. Mazur, Math. Ann. 293 (1992), no. 2, 259–275 (French). MR 1166121, DOI 10.1007/BF01444715
- The LMFDB Collaboration, The L-functions and modular forms database, http://www.lmfdb.org, 2013.
- Ioannis Papadopoulos, Sur la classification de Néron des courbes elliptiques en caractéristique résiduelle $2$ et $3$, J. Number Theory 44 (1993), no. 2, 119–152 (French, with French summary). MR 1225948, DOI 10.1006/jnth.1993.1040
- Ali Rajaei, On the levels of mod $l$ Hilbert modular forms, J. Reine Angew. Math. 537 (2001), 33–65. MR 1856257, DOI 10.1515/crll.2001.058
- Kenneth A. Ribet, On the equation $a^p+2^\alpha b^p+c^p=0$, Acta Arith. 79 (1997), no. 1, 7–16. MR 1438112, DOI 10.4064/aa-79-1-7-16
- Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
- Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI 10.1215/S0012-7094-87-05413-5
- Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
Additional Information
- Nicolas Billerey
- Affiliation: Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France
- MR Author ID: 823614
- Email: nicolas.billerey@uca.fr
- Imin Chen
- Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
- MR Author ID: 609304
- Email: ichen@sfu.ca
- Luis Dieulefait
- Affiliation: Departament d’Algebra i Geometria, Universitat de Barcelona, G.V. de les Corts Catalanes 585, 08007 Barcelona, Spain
- MR Author ID: 671876
- Email: ldieulefait@ub.edu
- Nuno Freitas
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2 Canada
- MR Author ID: 1044711
- Email: nunobfreitas@gmail.com
- Received by editor(s): October 8, 2017
- Received by editor(s) in revised form: November 30, 2017
- Published electronically: March 7, 2019
- Additional Notes: The first author acknowledges the financial support of ANR-14-CE-25-0015 Gardio.
The fourth author was partly supported by the grant Proyecto RSME-FBBVA $2015$ José Luis Rubio de Francia. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 8651-8677
- MSC (2010): Primary 11D41; Secondary 11F80, 11G05
- DOI: https://doi.org/10.1090/tran/7477
- MathSciNet review: 3955559