Diophantine equations after Fermat’s last theorem

Siksek, Samir

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 2, pp. 423-434.

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, ... ?

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, ...?

[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 $\mathbb{Q}$: 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 $X\left(7\right)$ 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 $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ 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