Samir Siksek Professor at the Mathematics Institute, University of Warwick. Email: firstname dot lastname at gmail.com Postal Address: Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom

Teaching:

• Rings and Modules, lecture notes
• Lockdown Abstract Algebra Handout 1, Handout 2, Handout 3, Handout 4
• Explicit Methods for Modular Curves, CMI-HIMR Summer School, lecture notes, Lecture I, Exercises I, Lecture II, Exercises II, Lecture III, Exercises III, Lecture IV, Exercises IV.
• Algebraic Number Theory lecture notes, HW 1, HW 2, HW 3, HW 4.
• Galois Representations of Elliptic Curves and Diophantine Equations, Alpbach, 9-14 July 2017. Lecture notes will appear here soon.
• Galois Representations and Diophantine Equations, Building Bridges Summer School, Sarajevo, 11-15 July 2016. lecture notes, talk 1, talk 2, talk 3, talk 4, sage instructions, exercises 1, exercises 2.
• Arithmetic of Hyperelliptic Curves, Ohrid, Macedonia, 2014, Lecture I , Lecture II , Lecture III , Lecture Notes, Exercises
• Rational Points on Curves, Sallins-les-Bains, 2013, Lectures I, Lectures II, Lectures III, Lectures IV, Exercises (complete)
• The Modular Approach to Diophantine Equations, Banff, 2012, videos, slides, notes and exercises
• Introduction to Abstract Algebra lecture notes, HW 1, HW 2, HW 3, HW 4, HW 5.
• Galois Theory lecture notes, HW 1, HW 2, HW 3, HW 4.
• Topics in Number Theory lecture notes, HW 1, HW 2, HW 3, HW 4, HW 5, HW 6.
• Vectors and Matrices lecture notes, HW 1, HW 2, HW 3, HW 4, HW 5.

Miscellaneous

• Information for prospective PhD students
• Postdocs and PhD Students working with me
• Number Theory Group at Warwick
• Warwick Number Theory Seminar
• L-Functions and Modular Forms Project
• Fermat's Last Theorem on "In Our Time", Radio 4.
• Travel

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 Khawaja) The modular approach to Diophantine equations over totally real fields on arXiv
2. Sums of integer cubes, Proceedings of the National Academy of Sciences 118 (2021), No. 16, paper.
3. (with Ozman) S-unit equations and the asymptotic Fermat conjecture over number fields, on arXiv
4. (with Bennett and Mihailescu) The Generalized Fermat Equation, pages 173-205 of Open Problems in Mathematics, (J. F. Nash, Jr. and M. Th. Rassias eds), Springer, New York, 2016.   pdf
5. 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
6. 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
7. Diophantine equations after Fermat's Last Theorem, Journal de Théorie des Nombres de Bordeaux 21 (2009), no. 2, 425--436.  pdf

Publications:  (these may differ from final versions)

1. (with Khawaja) A single source theorem for primitive points on curves on arXiv
2. (with Nowroozi) Perfect powers in elliptic divisibility sequences, to appear in Bulletin of the LMS, on arXiv
3. (with Khawaja) Primitive algebraic points on curves, Resesearch in Number Theory 10 (2024), no.3, Paper No. 57. on arXiv
4. (with Visser) Curves with few bad primes over cyclotomic \( \mathbb{Z}_\ell \)-extensions, to appear in Algebra & Number Theory, on arXiv
5. (with Gherga) Efficient resolution of Thue-Mahler equations, to appear in Algebra & Number Theory, on arXiv
6. (with Bennett and Michaud-Jacobs) \( \mathbb{Q} \)-curves and the Lebesgue-Nagell equation, Journal de Théorie des Nombres de Bordeaux 35 (2023), 335-370. on arXiv
7. (with Bennett) Differences between perfect powers : prime power gaps , Algebra & Number Theory 17 (2023), 1789-1846. on arXiv
8. (with Bennett) Differences between perfect powers : the Lebesgue-Nagell Equation , Transactions of the AMS 376 (2023), 225-370. on arXiv
9. Integral points on punctured abelian varieties, European Journal of Mathematics 8 (2022), 687-703. on arXiv
10. (with Freitas and Kraus) Local criteria for the unit equation and the asymptotic Fermat's Last Theorem , Proceedings of the National Academy of Sciences 118 (2021), No. 12, on arXiv
11. (with Bennett, Gherga and Patel) Odd values of the Ramanujan tau function, Math. Ann. 382 (2022), 203-238. on arXiv
12. (with Freitas and Kraus) The unit equation over cyclic number fields of prime degree , Algebra & Number Theory 15 (2021), 2647-2653. on arXiv
13. (with Freitas and Kraus) Chevalley's class number formula, unit equations and the asymptotic Fermat's Last Theorem on arXiv
14. (with Freitas and Kraus) On Asymptotic Fermat over \( \mathbb{Z}_p \) extensions of \( \mathbb{Q} \), Algebra & Number Theory 14 (2020), 2571-2574. on arXiv
15. (with Bennett) A conjecture of Erdos, supersingular primes and short character sums, Annals of Mathematics 191 (2020), 355--392. on arXiv
16. (with Freitas and Kraus) Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem, Advances in Mathematics 363 (2020), 1--37. on arXiv
17. (with Derickx and Najman) Elliptic curves over totally real cubic fields are modular, Algebra & Number Theory 14 (2020), 1791-1800. on arXiv, programs
18. (with Bennett and Patel) Shifted Powers in Lucas-Lehmer Sequences, Research in Number Theory (2019), 5:15. on arXiv
19. (with Ozman) Quadratic Points on Modular Curves, Mathematics of Computation 88 (2019), 2461--2484. on arXiv
20. (with Freitas and Kraus) On asymptotic Fermat over the \( \mathbb{Z}_2\) extension of \( \mathbb{Q} \) , Ann. Math. Blaise Pascal 28 (2021), 1-6. on arXiv
21. Quadratic Chabauty for modular curves, on arXiv
22. (with Sengun) On the asymptotic Fermat's Last Theorem over numbers fields, Commentarii Mathematici Helvetici, 93 (2018), 359-375. on arXiv
23. (with Papanikolopoulos) A Mordell-Weil theorem for cubic hypersurfaces of high dimension, Algebra & Number Theory 11-8 (2017), 1953-1965. on arXiv
24. Every integer greater than 454 is the sum of at most seven positive cubes, Algebra & Number Theory 10 (2016), no. 10, 2093-2119.   on arXiv, programs
25. (with Patel) On powers that are sums of consecutive like powers, Research in Number Theory (2017), 3:2.   on arXiv   open access
26. (with Anni) Modular elliptic curves over real abelian fields and the generalized Fermat equation \( x^{2\ell}+y^{2m}=z^p \), Algebra & Number Theory, 10-6 (2016), 1147--1172.   on arXiv   programs
27. (with Bennett) Rational points on Erdos-Selfridge superelliptic curves, Compositio Mathematica 152 (2016), no. 11, 2249--2254.   on arXiv
28. (with Bennett and Patel) Perfect powers that are sums of consecutive cubes, Mathematika 63 (2016), 230--249.   on arXiv
29. (with Bennett and Patel) Superelliptic equations arising from sums of consecutive powers, Acta Arithmetica 172 (2016), no. 4, 377-393.   on arXiv
30. (with Bremner) Squares in arithmetic progression over cubic fields, International Journal of Number Theory 12 (2016), 1409-1414.   on arXiv
31. (with Anni and Lemos) Residual representations of semistable principally polarized abelian varieties Research in Number Theory (2016) 2:1.   on arXiv   open access
32. (with Anni) On Serre's uniformity conjecture for semistable elliptic curves over totally real fields,   Mathematische Zeitschrift 281 (2015), 193-199.   on arXiv
33. (with Freitas and Le Hung) Elliptic Curves over Real Quadratic Fields are Modular,   Inventiones Mathematicae 201 (2015), 159-206.   on arXiv   programs
34. (with Freitas) The Asymptotic Fermat's Last Theorem for Five-Sixths of Real Quadratic Fields,   Compositio Mathematica 151 (2015), 1395-1415.   on arXiv
35. (with Freitas) Fermat's Last Theorem for some Small Real Quadratic Fields,   Algebra & Number Theory 9 (2015), 875-895.   on arXiv
36. (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
37. (with Bennett, Dahmen and Mignotte) Shifted powers in binary recurrence sequences, Mathematical Proceedings Cambridge Philosophical Society, 158 (2015), 305-329.   on arXiv.
38. (with Dahmen) Perfect powers expressible as sums of two fifth or seventh powers, Acta Arith. 164 (2014), 65-100.   on arXiv
39. (with Stoll) The generalized Fermat equation \(x^2+y^3=z^{15}\), Archiv der Mathematik 102 (2014), issue 5, 411-421.   on arXiv
40. Explicit Chabauty over number fields, Algebra & Number Theory 7 (2013), issue 4, 765-793.   on arXiv,   programs
41. On the number of Mordell-Weil generators for cubic surfaces, Journal of Number Theory 132 (2012), 2610-2629.   on arXiv
42. (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
43. (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.
44. (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
45. (with Hargreaves) Cycles and fixed points of happy functions. Journal of Combinatorics and Number Theory 2 (2010), no. 3, 13 pages.   pdf
46. (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
47. (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
48. (with Chen) Perfect powers expressible as sums of two cubes, Journal of Algebra 322 (2009), 638-656.   pdf
49. Chabauty for Symmetric Powers of Curves, Algebra & Number Theory 3 (2009), No. 2, 209-236.   pdf
50. (with Bugeaud and Reutenauer) A Sturmian sequence related to the uniqueness conjecture for Markoff numbers, Theoretical Computer Science 410 (2009), 2864-2869.
51. Diophantine equations after Fermat's Last Theorem, Journal de Théorie des Nombres de Bordeaux 21 (2009), no. 2, 425--436.  pdf
52. (with Bugeaud, Mignotte, Stoll, Tengely) Integral Points on Hyperelliptic Curves, Algebra & Number Theory 2 (2008), No. 8, 859-885.   on arXiv programs
53. (with Bugeaud, Luca and Mignotte) Almost powers in the Lucas sequences, Journal de Théorie des Nombres de Bordeaux 20 (2008), 555--600.   pdf
54. (with Martin Bright) Functions, Reciprocity and the Obstruction to Divisors on Curves, Journal of the LMS 77 (2008), 789-807.   pdf
55. (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
56. (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
57. (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
58. (with Bugeaud and Mignotte) Classical and modular approaches to exponential Diophantine equations II: The Lebesgue-Nagell equation, Compositio Mathematica 142 (2006), 31-62.  pdf
59. (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
60. (with Cremona and Prickett) Height difference bounds for elliptic curves over number fields, Journal of Number Theory 116 (2006), 42-68.  dvi  pdf
61. (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
62. (with Bugeaud, Luca and Mignotte) On perfect powers in Lucas sequences, International Journal of Number Theory 1 (2005), 309-332.  dvi  pdf
    
63. (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
64. On standardized models for elliptic curves, Math. Comp. 74 (2005), 949-951.  dvi
65. The modular approach to Diophantine equations, notes for lectures given at the Institut Henri Poincaré (October 2004).  pdf
66. On the Brauer-Manin obstruction for curves with split Jacobian, Journal de Théorie des Nombres de Bordeaux 16 (2004), 773-777.  dvi  pdf
67. (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
68. (with El-Sedy) On points of non-differentiability of convex functions, Applied Mathematics and Computation 148 (2004), 725-728.
69. (with Cremona) On the Diophantine equation \( x^2+7=y^m \) , Acta Arithmetica 109.2 (2003), 143-149.  dvi  pdf
70. (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
71. 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
72. (with El-Sedy) On the irrationality of certain rapidly convergent series, International Mathematical Journal 3 (2003), 1193-1195.
73. Descent of Picard groups using functions on curves, Bulletin of the Australian Mathematical Society 66 (2002), 119-124.  dvi  pdf
74. Sieving for rational points on hyperelliptic curves, Mathematics of Computation 70 (2001), 1661-1674.  dvi
75. (with El-Sedy) On happy numbers, Rocky Mountain Journal of Mathematics 30 (2000), 565-570.  dvi  pdf
76. 4-descent, pages 418-422, appendix A to ``Beyond the Manin obstruction'' by A. N. Skorobogatov, Inventiones Mathematicae 135 (1999), 399-424.
77. The height pairing on an elliptic curve with complex multiplication, Arab Journal of Mathematical Sciences 5 (1999), 43-48.  dvi  pdf
78. (with N. Smart) A fast Diffie-Hellman protocol in genus 2, Journal of Cryptology 12 (1999), 67-73.
79. (with N. Smart) On the complexity of computing the 2-Selmer group of an elliptic curve, Glasgow Mathematical Journal 39 (1997), 251-258.
80. (with J. Merriman and N. Smart) Explicit 4-descents on an elliptic curve, Acta Arithmetica LXXVII.4 (1996), 358-404.  dvi  pdf
81. Infinite descent on elliptic curves, Rocky Mountain Journal of Mathematics 25 (1995), 1501-1538.  dvi  pdf
82. Descents on curves of genus 1, PhD thesis, University of Exeter, 1995.   dvi   pdf

Some of my talks:

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

Events I'm involved in organizing:

• Number Theory and Algebraic Geometry , British Mathematical Colloquium, University of Kent, 17-18 April, 2012.
• Contemporary Methods for Solving Diophantine Equations, Banff, 10-17 June, 2012.
• Arithmetic Geometry minisymposium, European Congress of Mathematics, Krakow, Poland, 2-7 July 2012.
• EPSRC Warwick Number Theory Symposium , University of Warwick, 2012-2013:
  • Selmer Groups, Descent and the Distribution of Ranks, 24-28 September 2012.
  • Explicit Methods for Modular Forms , 18-22 March 2013.
  • Iwasawa Theory and Galois Representations, 8-12 April 2013.
  • Higher rank automorphic forms and L-functions, 29 April-3 May 2013
  • Rational Points: Geometric, Analytic and Explicit Approaches, 27-31 May 2013.
  • Summer School: Number Theory for Cryptography, 24-28 June 2013.
  • Number Theory, Geometry and Cryptography, 1-5 July 2013.
• Explicit Methods in Number Theory: Conference in Honour of John Cremona's 60th Birthday, University of Warwick, 4-8 April 2016.
• Automorphic Forms: Theory and Computation, Kings College London, 5-9 September 2016.
• Arithmetic Aspects of Explicit Moduli Problems, 28 May-2 June 2017, BIRS, Banff.
• Modern Breakthroughs in Diophantine Problems, 30 August-4 September 2020, BIRS, Banff.
• Modern Breakthroughs in Diophantine Problems, 19 June-24 June 2022, BIRS, Banff.
• Modular curves and their arithmetic, 6-8 December, University of Warwick