Diophantine equations after Fermat’s last theorem
Samir Siksek1
1Mathematics Institute University of Warwick Coventry, CV4 7AL, United Kingdom
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 423-434

These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions:

  • (i) Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem?
  • (ii) Is it useful to combine this approach with traditional approaches to Diophantine equations: Diophantine approximation, arithmetic geometry, ...?

Cet article reprend les notes de mon exposé aux 25-ièmes Journées Arithmétiques, du 2 au 6 juillet 2007 à Edimbourg en Écosse. J’ai pour but d’apporter un peu de lumière sur les deux questions suivantes :

  • (i) Étant donnée une équation diophantienne, quelle information peut-on obtenir en suivant la stratégie de Wiles pour sa preuve du théorème de Fermat ?
  • (ii) Est-il utile de combiner cette approche avec les approches traditionnelles des équations diophantiennes : approximation diophantienne, géométrie arithmétique, ... ?
Published online:
DOI: 10.5802/jtnb.679

Samir Siksek  1

1 Mathematics Institute University of Warwick Coventry, CV4 7AL, United Kingdom 
Samir Siksek. Diophantine equations after Fermat’s last theorem. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 423-434. doi: 10.5802/jtnb.679
@article{JTNB_2009__21_2_423_0,
     author = {Samir Siksek},
     title = {Diophantine equations after {Fermat{\textquoteright}s} last theorem},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {423--434},
     year = {2009},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {2},
     doi = {10.5802/jtnb.679},
     zbl = {pre05620659},
     mrnumber = {2541434},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.679/}
}
TY  - JOUR
AU  - Samir Siksek
TI  - Diophantine equations after Fermat’s last theorem
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2009
SP  - 423
EP  - 434
VL  - 21
IS  - 2
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.679/
DO  - 10.5802/jtnb.679
LA  - en
ID  - JTNB_2009__21_2_423_0
ER  - 
%0 Journal Article
%A Samir Siksek
%T Diophantine equations after Fermat’s last theorem
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 423-434
%V 21
%N 2
%I Université Bordeaux 1
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.679/
%R 10.5802/jtnb.679
%G en
%F JTNB_2009__21_2_423_0

[1] C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, User’s guide to PARI-GP, version 2.3.2. (See also http://pari.math.u-bordeaux.fr/)

[2] M. A. Bennett, Powers in recurrence sequences : Pell equations. Trans. Amer. Math. Soc. 357 (2005), 1675–1691. | MR | Zbl

[3] N. Bruin, On powers as sums of two cubes. Pages 169–184 of Algorithmic number theory (edited by W. Bosma), Lecture Notes in Comput. Sci. 1838, Springer, Berlin, 2000. | MR | Zbl

[4] W. Bosma, J. Cannon and C. Playoust, The Magma Algebra System I: The User Language. J. Symb. Comp. 24 (1997), 235–265. (See also http://magma.maths.usyd.edu.au/magma/) | MR | Zbl

[5] C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over ℚ: wild 3-adic exercises. J. Amer. Math. Soc. 14 No.4 (2001), 843–939. | MR | Zbl

[6] Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Annals of Mathematics 163 (2006), 969–1018. | MR | Zbl

[7] Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell Equation. Compositio Mathematica 142 (2006), 31–62. | MR | Zbl

[8] Y. Bugeaud, M. Mignotte and S. Siksek, A multi-Frey approach to some multi-parameter families of Diophantine equations. Canadian Journal of Mathematics 60 (2008), no. 3, 491–519. | MR | Zbl

[9] I. Chen and S. Siksek, Perfect powers expressible as sums of two cubes. Journal of Algebra, to appear.

[10] J. H. E. Cohn, On square Fibonacci numbers; J. London Math. Soc. 39 (1964), 537–540. | MR | Zbl

[11] J. H. E. Cohn, The Diophantine equation x 2 +C=y n . Acta Arith. LXV.4 (1993), 367–381. | MR | Zbl

[12] J. E. Cremona, Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, 1997. | MR | Zbl

[13] H. Cohen, Number Theory, Vol. I: Tools and Diophantine Equations and Vol. II: Analytic and Modern Tools. Springer-Verlag, GTM 239, 240, 2007. | MR | Zbl

[14] S. R. Dahmen, Classical and modular methods applied to Diophantine equations. University of Utrecht Ph.D. thesis, 2008.

[15] A. Kraus, Sur l’équation a 3 +b 3 =c p . Experimental Mathematics 7 (1998), No. 1, 1–13. | MR | Zbl

[16] B. Poonen, E. F. Schaefer and M. Stoll, Twists of X(7) and primitive solutions to x 2 +y 3 =z 7 . Duke Math. J. 137 (2007), 103–158. | MR | Zbl

[17] K. Ribet, On modular representations of Gal (ℚ ¯/ℚ) arising from modular forms. Invent. Math. 100 (1990), 431–476. | MR | Zbl

[18] S. Siksek and J. E. Cremona, On the Diophantine equation x 2 +7=y m . Acta Arith. 109.2 (2003), 143–149. | MR | Zbl

[19] W. A. Stein, Modular Forms: A Computational Approach. American Mathematical Society, Graduate Studies in Mathematics 79, 2007. | MR | Zbl

[20] R. L. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras. Annals of Math. 141 (1995), 553–572. | MR | Zbl

[21] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem. Annals of Math. 141 (1995), 443–551. | MR | Zbl

Cited by Sources: