# Samir Siksek

## Research Area:

Galois representations, modularity, explicit arithmetic geometry, arithmetic of curves and surfaces, Diophantine equations (and the generalized Fermat equation in particular), Waring's problem.

## Survey Articles and Expository Papers:

1. (with Bennett and Mihailescu) The Generalized Fermat Equation, in Open Problems in Mathematics, (J. F. Nash, Jr. and M. Th. Rassias eds), Springer, New York, to appear.   pdf
2. Chabauty and the Mordell-Weil Sieve , pages 194-224 of Advances on Superelliptic Curves and Their Applications, (L. Beshaj, T. Shaska, E. Zhupa eds), IOS Press, 2015.   pdf
3. The Modular Approach to Diophantine Equations, in Explicit Methods in Number Theory: Rational Points and Diophantine Equations , (Belabas, Lenstra, Gaudry, Stoll, Watkins, McCallum, Poonen, Beukers, Siksek), Panoramas et synthèses 36 (2012). link   pdf

## Selected Publications: (these may differ from final versions)

1. A Mordell-Weil theorem for cubic hypersurfaces of high dimension on arXiv.
2. Every integer greater than 454 is the sum of at most seven positive cubes. on arXiv, programs
3. (with Anni) Modular elliptic curves over real abelian fields and the generalized Fermat equation $$x^{2\ell}+y^{2m}=z^p$$, Algebra & Number Theory (to appear).   on arXiv   programs
4. (with Bennett) Rational points on Erdos-Selfridge superelliptic curves, Compositio Mathematica (to appear).   on arXiv
5. (with Bennett and Patel) Perfect powers that are sums of consecutive cubes.   on arXiv
6. (with Bennett and Patel) Superelliptic equations arising from sums of consecutive powers, Acta Arithmetica (to appear).   on arXiv
7. (with Bremner) Squares in arithmetic progression over cubic fields, International Journal of Number Theory 12 (2016), 1409-1414.   on arXiv
8. (with Anni and Lemos) Residual representations of semistable principally polarized abelian varieties Research in Number Theory (2016) 2:1.   on arXiv   open access
9. (with Anni) On Serre's uniformity conjecture for semistable elliptic curves over totally real fields,   Mathematische Zeitschrift 281 (2015), 193-199.   on arXiv
10. (with Freitas and Le Hung) Elliptic Curves over Real Quadratic Fields are Modular,   Inventiones Mathematicae 201 (2015), 159-206.   on arXiv   programs
11. (with Freitas) An Asymptotic Fermat's Last Theorem for Five-Sixths of Real Quadratic Fields,   Compositio Mathematica 151 (2015), 1395-1415.   on arXiv
12. (with Freitas) Fermat's Last Theorem for some Small Real Quadratic Fields,   Algebra & Number Theory 9 (2015), 875-895.   on arXiv
13. (with Freitas) Criteria for irreducibility of mod p representations of Frey curves, Journal de Théorie des Nombres de Bordeaux 27 (2015), 67-76.   on arXiv
14. (with Bennett, Dahmen and Mignotte) Shifted powers in binary recurrence sequences, Mathematical Proceedings Cambridge Philosophical Society, 158 (2015), 305-329.   on arXiv.
15. (with Dahmen) Perfect powers expressible as sums of two fifth or seventh powers, Acta Arith. 164 (2014), 65-100.   on arXiv
16. (with Stoll) The generalized Fermat equation $$x^2+y^3=z^{15}$$, Archiv der Mathematik 102 (2014), issue 5, 411-421.   on arXiv
17. Explicit Chabauty over number fields, Algebra & Number Theory 7 (2013), issue 4, 765-793.   on arXiv
18. On the number of Mordell-Weil generators for cubic surfaces, Journal of Number Theory 132 (2012), 2610-2629.   on arXiv
19. (with Stoll) Partial descent on hyperelliptic curves and the generalized Fermat equation $$x^3+y^4+z^5=0$$, Bulletin of the LMS 44 (2012), 151-166.   on arXiv
20. (with Deajim) On the $$\mathbb{Q}$$-linear independence of the sums $$\sum_{n=1}^\infty \sigma_k(n)/n!$$ , Journal of Number Theory 131 (2011), 745-749.
21. (with Stoll) On a problem of Hajdu and Tengely, pages 316--330 of Algorithmic Number Theory, Proceedings of the 9th International Symposium, Nancy, France, July 19--23, 2010, editors Guillaume Hanrot, François Morain and Emmanuel Thomé, LNCS 6197, Springer-Verlag.   on arXiv
22. (with Hargreaves) Cycles and fixed points of happy functions. Journal of Combinatorics and Number Theory 2 (2010), no. 3, 13 pages.   pdf
23. (with Luca) Factorials expressible as sums of at most three Fibonacci numbers, Proceedings of the Edinburgh Mathematical Society 53 (2010), no. 3, 679-729.   pdf
24. (with Abu Muriefah, Luca and Tengely) On the Diophantine equation $$x^2+C=2y^n$$, International Journal of Number Theory 5 (2009), 1117-1128.   pdf
25. (with Chen) Perfect powers expressible as sums of two cubes, Journal of Algebra 322 (2009), 638-656.   pdf
26. Chabauty for Symmetric Powers of Curves, Algebra & Number Theory 3 (2009), No. 2, 209-236.   pdf
27. (with Bugeaud and Reutenauer) A Sturmian sequence related to the uniqueness conjecture for Markoff numbers, Theoretical Computer Science 410 (2009), 2864-2869.
28. Diophantine equations after Fermat's Last Theorem, Journal de Théorie des Nombres de Bordeaux 21 (2009), no. 2, 425--436.  pdf
29. (with Bugeaud, Mignotte, Stoll, Tengely) Integral Points on Hyperelliptic Curves, Algebra & Number Theory 2 (2008), No. 8, 859-885.   on arXiv programs
30. (with Bugeaud, Luca and Mignotte) Almost powers in the Lucas sequences, Journal de Théorie des Nombres de Bordeaux 20 (2008), 555--600.   pdf
31. (with Martin Bright) Functions, Reciprocity and the Obstruction to Divisors on Curves, Journal of the LMS 77 (2008), 789-807.   pdf
32. (with Bugeaud and Mignotte) A multi-Frey approach to some multi-parameter families of Diophantine equations, Canadian Journal of Mathematics 60 (2008), 491-519.  pdf
33. (with Bugeaud, Luca and Mignotte) Fibonacci Numbers at most one away from a perfect power, Elemente der Mathematik 63 (2008), no. 2, 65-75.  pdf  dvi
34. (with Bugeaud, Luca and Mignotte) Perfect powers from products of terms in Lucas sequences, Journal für die reine und angewandte Mathematik, 611 (2007), 109-129.  dvi  pdf
35. (with Bugeaud and Mignotte) Classical and modular approaches to exponential Diophantine equations II: The Lebesgue-Nagell equation, Compositio Mathematica 142 (2006), 31-62.  pdf
36. (with Bugeaud and Mignotte) Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Annals of Mathematics 163 (2006), 969-1018.  dvi  pdf
37. (with Cremona and Prickett) Height difference bounds for elliptic curves over number fields, Journal of Number Theory 116 (2006), 42-68.  dvi  pdf
38. (with Cremona) Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q, in Algorithmic Number Theory, 7th International Symposium, ANTS-VII, pages 275-286, Springer Verlag, LNCS 4076, July 2006  pdf
39. (with Bugeaud, Luca and Mignotte) On perfect powers in Lucas sequences, International Journal of Number Theory 1 (2005), 309-332.  dvi  pdf
40. (with Bugeaud, Luca and Mignotte) On Fibonacci numbers with few prime divisors, Proceedings of the Japan Academy 81, Ser. A, No. 2 (2005).  dvi  pdf
41. On standardized models for elliptic curves, Math. Comp. 74 (2005), 949-951.  dvi
42. The modular approach to Diophantine equations, notes for lectures given at the Institut Henri Poincaré (October 2004).  pdf
43. On the Brauer-Manin obstruction for curves with split Jacobian, Journal de Théorie des Nombres de Bordeaux 16 (2004), 773-777.  dvi  pdf
44. (with Bugeaud and Mignotte) Sur les nombres de Fibonacci de la forme $$q^k y^p$$ , C. R. Acad. Sci. Paris, Ser. I 339 (2004), 327-330.  pdf
45. (with El-Sedy) On points of non-differentiability of convex functions, Applied Mathematics and Computation 148 (2004), 725-728.
46. (with Cremona) On the Diophantine equation $$x^2+7=y^m$$ , Acta Arithmetica 109.2 (2003), 143-149.  dvi  pdf
47. (with Skorobogatov) On a Shimura curve that is a counterexample to the Hasse principle, Bulletin of the London Mathematical Society 35 (2003), 143-149.  pdf
48. On the Diophantine equation $$x^2=y^p+2^k z^p$$ Journal de Théorie des Nombres de Bordeaux 15 (2003), 839-846.  dvi  pdf
49. (with El-Sedy) On the irrationality of certain rapidly convergent series, International Mathematical Journal 3 (2003), 1193-1195.
50. Descent of Picard groups using functions on curves, Bulletin of the Australian Mathematical Society 66 (2002), 119-124.  dvi  pdf
51. Sieving for rational points on hyperelliptic curves, Mathematics of Computation 70 (2001), 1661-1674.  dvi
52. (with El-Sedy) On happy numbers, Rocky Mountain Journal of Mathematics 30 (2000), 565-570.  dvi  pdf
53. 4-descent, pages 418-422, appendix A to Beyond the Manin obstruction'' by A. N. Skorobogatov, Inventiones Mathematicae 135 (1999), 399-424.
54. The height pairing on an elliptic curve with complex multiplication, Arab Journal of Mathematical Sciences 5 (1999), 43-48.  dvi  pdf
55. (with N. Smart) A fast Diffie-Hellman protocol in genus 2, Journal of Cryptology 12 (1999), 67-73.
56. (with N. Smart) On the complexity of computing the 2-Selmer group of an elliptic curve, Glasgow Mathematical Journal 39 (1997), 251-258.
57. (with J. Merriman and N. Smart) Explicit 4-descents on an elliptic curve, Acta Arithmetica LXXVII.4 (1996), 358-404.  dvi  pdf
58. Infinite descent on elliptic curves, Rocky Mountain Journal of Mathematics 25 (1995), 1501-1538.  dvi  pdf
59. Descents on curves of genus 1, PhD thesis, University of Exeter, 1995.   dvi   pdf

## Some of my talks:

1. On the Generalized Fermat Equation $$x^{2\ell}+y^{2m}=z^p$$, Schney, June 2015. pdf
2. Sums of Seven Cubes, Oberwolfach, Simon Fraser University, Oxford, ..., 2015. pdf
3. Elliptic Curves over Totally Real Fields are Modular, (many versions of this talk given, 2013-2015). pdf
4. The Fermat Equation over Totally Real Fields, Building Bridges, Bristol, March 2014. pdf
5. The Fermat Equation over Totally Real Fields (version intended for a broad audience), Maringa, July 2014. pdf
6. Chabauty over Number Fields Leiden, 20 April 2009. pdf (scan)
7. Integral Points on Curves of Higher Genus (a more up-to-date version) MAGIC-Muggle Number Theory Seminar, 7 November 2008. pdf (scan)
8. Integral Points on Curves of Higher Genus Bordeaux, 15 October 2007. pdf (scan)
9. Chabauty for Symmetric Powers on Curves Banff, 7 February 2007. pdf (scan)
10. Chabauty for Symmetric Powers on Curves (a version of the above talk aimed at those new to Chabauty) Rome, 13 April 2007. pdf (scan)
11. Computing a lower bound for the canonical height on elliptic curves, ANTS VII conference, Berlin, 24 July 2006. pdf (scan)
12. The Lebesgue-Nagell-Ramanujan equation, University of Warwick, 24 August 2005. pdf (scan)
13. Functions, reciprocity and the obstruction to divisors on curves, Bremen and Oberwolfach, July 2005. pdf (scan)
14. Beyond Fermat's Last Theorem, University of Warwick, January 2005. pdf (scan)
15. A multi-exponential diophantine spree, Institute Henri Poincare, Paris, December 2004. pdf (scan)
16. Giving Baker's Theory a modular helping hand, Dagstuhl, Germany, May 2004. pdf (scan)