Teaching:

Rings and Modules,
lecture notes

Lockdown Abstract Algebra
Handout 1,
Handout 2,
Handout 3,
Handout 4
 Explicit Methods for Modular Curves,
CMIHIMR 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, 914 July 2017.
Lecture notes will appear here soon.
 Galois Representations and Diophantine Equations,
Building Bridges Summer School, Sarajevo, 1115 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, SallinslesBains, 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
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:

(with Khawaja) The modular approach to Diophantine equations over totally real fields
on arXiv

Sums of integer cubes,
Proceedings of the National Academy of Sciences 118 (2021),
No. 16,
paper.
 (with Ozman)
Sunit equations and the asymptotic Fermat
conjecture over number fields,
on arXiv

(with Bennett and Mihailescu)
The Generalized Fermat Equation,
pages 173205 of
Open Problems in Mathematics,
(J. F. Nash, Jr. and M. Th. Rassias eds),
Springer, New York, 2016.
pdf

Chabauty and the MordellWeil Sieve
, pages 194224 of
Advances on Superelliptic Curves and Their Applications,
(L. Beshaj, T. Shaska, E. Zhupa eds), IOS Press, 2015.
pdf
 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
 Diophantine equations after Fermat's Last Theorem,
Journal de Théorie des Nombres de Bordeaux
21 (2009), no. 2, 425436.
pdf
Publications:
(these may differ from final versions)
 (with Khawaja)
A single source theorem for primitive points on curves
on arXiv
 (with Nowroozi)
Perfect powers in elliptic divisibility sequences, to appear in Bulletin of the LMS,
on arXiv
 (with Khawaja)
Primitive algebraic points on curves,
Resesearch in Number Theory 10 (2024), no.3, Paper No. 57.
on arXiv
 (with Visser)
Curves with few bad primes over cyclotomic \( \mathbb{Z}_\ell \)extensions,
to appear in Algebra & Number Theory,
on arXiv
 (with Gherga)
Efficient resolution of ThueMahler equations,
to appear in Algebra & Number Theory,
on arXiv
 (with Bennett and MichaudJacobs)
\( \mathbb{Q} \)curves and the LebesgueNagell equation,
Journal de Théorie des Nombres de Bordeaux
35 (2023), 335370.
on arXiv
 (with Bennett)
Differences between perfect powers : prime power gaps
, Algebra & Number Theory 17 (2023), 17891846.
on arXiv
 (with Bennett)
Differences between perfect powers : the LebesgueNagell Equation
, Transactions of the AMS 376 (2023), 225370. on arXiv
 Integral points on punctured abelian varieties,
European Journal of Mathematics 8 (2022), 687703.
on arXiv
 (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
 (with Bennett, Gherga and Patel)
Odd values of the Ramanujan tau function,
Math. Ann. 382 (2022), 203238.
on arXiv
 (with Freitas and Kraus)
The unit equation over cyclic number fields of prime degree ,
Algebra & Number Theory 15 (2021), 26472653.
on arXiv
 (with Freitas and Kraus)
Chevalley's class number formula, unit equations and the asymptotic Fermat's Last Theorem
on arXiv
 (with Freitas and Kraus)
On Asymptotic Fermat over \( \mathbb{Z}_p \) extensions of \( \mathbb{Q} \), Algebra & Number Theory 14 (2020), 25712574.
on arXiv
 (with Bennett)
A conjecture of Erdos, supersingular primes and short character sums,
Annals of Mathematics 191 (2020), 355392.
on arXiv
 (with Freitas and Kraus)
Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem, Advances in Mathematics 363 (2020), 137.
on arXiv
 (with Derickx and Najman)
Elliptic curves over totally real cubic fields are modular,
Algebra & Number Theory 14 (2020), 17911800.
on arXiv,
programs
 (with Bennett and Patel)
Shifted Powers in LucasLehmer Sequences,
Research in Number Theory (2019), 5:15.
on arXiv
 (with Ozman)
Quadratic Points on Modular Curves,
Mathematics of Computation
88 (2019), 24612484.
on arXiv
 (with Freitas and Kraus)
On asymptotic Fermat over the \( \mathbb{Z}_2\) extension
of \( \mathbb{Q} \) ,
Ann. Math. Blaise Pascal 28 (2021), 16.
on arXiv

Quadratic Chabauty for modular curves,
on arXiv
 (with Sengun)
On the asymptotic Fermat's Last Theorem over numbers fields,
Commentarii Mathematici Helvetici,
93 (2018), 359375.
on arXiv

(with Papanikolopoulos)
A MordellWeil theorem for cubic hypersurfaces of high dimension,
Algebra & Number Theory 118 (2017), 19531965.
on arXiv

Every integer greater than 454 is the sum of at most seven positive cubes,
Algebra & Number Theory 10 (2016), no. 10, 20932119.
on arXiv,
programs

(with Patel)
On powers that are sums of consecutive like powers,
Research in Number Theory (2017), 3:2.
on arXiv
open access
 (with Anni)
Modular elliptic curves over real abelian fields
and the generalized Fermat equation \( x^{2\ell}+y^{2m}=z^p \),
Algebra & Number Theory,
106 (2016), 11471172.
on arXiv
programs
 (with Bennett)
Rational points on ErdosSelfridge superelliptic curves,
Compositio Mathematica 152 (2016), no. 11, 22492254.
on arXiv
 (with Bennett and Patel)
Perfect powers that are sums of consecutive cubes,
Mathematika 63 (2016), 230249.
on arXiv
 (with Bennett and Patel)
Superelliptic equations arising from sums of consecutive powers,
Acta Arithmetica
172 (2016), no. 4, 377393.
on arXiv
 (with Bremner)
Squares in arithmetic progression over cubic fields,
International Journal of Number Theory 12 (2016), 14091414.
on arXiv
 (with Anni and Lemos)
Residual representations of semistable principally polarized
abelian varieties
Research in Number Theory (2016) 2:1.
on arXiv
open access
 (with Anni)
On Serre's uniformity conjecture for semistable elliptic curves over
totally real fields,
Mathematische Zeitschrift 281 (2015), 193199.
on arXiv
 (with Freitas and Le Hung)
Elliptic Curves over Real Quadratic Fields are Modular,
Inventiones Mathematicae 201 (2015), 159206.
on arXiv
programs
 (with Freitas) The Asymptotic Fermat's Last
Theorem for FiveSixths of Real Quadratic Fields,
Compositio Mathematica 151 (2015), 13951415.
on arXiv
 (with Freitas) Fermat's Last Theorem for some
Small Real Quadratic Fields,
Algebra & Number Theory 9 (2015), 875895.
on arXiv
 (with Freitas) Criteria for irreducibility of mod p representations of Frey curves,
Journal de Théorie des Nombres de Bordeaux 27 (2015), 6776.
on arXiv
 (with Bennett, Dahmen and Mignotte)
Shifted powers in binary recurrence sequences,
Mathematical Proceedings
Cambridge Philosophical Society,
158 (2015), 305329.
on arXiv.
 (with Dahmen) Perfect powers expressible as sums of two fifth or seventh powers,
Acta Arith. 164 (2014), 65100.
on arXiv
 (with Stoll) The generalized Fermat equation \(x^2+y^3=z^{15}\),
Archiv der Mathematik 102 (2014), issue 5, 411421.
on arXiv
 Explicit Chabauty over number fields,
Algebra & Number Theory 7 (2013), issue 4, 765793.
on arXiv,
programs
 On the number of MordellWeil generators for cubic surfaces,
Journal of Number Theory 132 (2012), 26102629.
on arXiv
 (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), 151166.
on arXiv
 (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), 745749.
 (with Stoll)
On a problem of Hajdu and Tengely,
pages 316330 of Algorithmic Number Theory,
Proceedings of the 9th International Symposium,
Nancy, France, July 1923, 2010, editors
Guillaume Hanrot, François Morain and Emmanuel Thomé,
LNCS 6197,
SpringerVerlag.
on arXiv
 (with Hargreaves)
Cycles and fixed points of happy functions.
Journal of Combinatorics and Number Theory 2 (2010),
no. 3, 13 pages.
pdf
 (with Luca)
Factorials expressible as sums of at most three
Fibonacci numbers,
Proceedings of the Edinburgh
Mathematical Society 53 (2010), no. 3, 679729.
pdf
 (with Abu Muriefah, Luca and Tengely)
On the Diophantine equation \(x^2+C=2y^n\),
International Journal of Number Theory 5 (2009),
11171128.
pdf
 (with Chen)
Perfect powers expressible as sums of two cubes,
Journal of Algebra 322 (2009), 638656.
pdf
 Chabauty for Symmetric Powers of Curves,
Algebra & Number Theory 3 (2009), No. 2, 209236.
pdf
 (with Bugeaud and Reutenauer)
A Sturmian sequence related to the
uniqueness conjecture for Markoff numbers,
Theoretical Computer Science 410 (2009),
28642869.
 Diophantine equations after Fermat's Last Theorem,
Journal de Théorie des Nombres de Bordeaux
21 (2009), no. 2, 425436.
pdf
 (with Bugeaud, Mignotte, Stoll, Tengely)
Integral Points on Hyperelliptic Curves,
Algebra & Number Theory 2 (2008),
No. 8, 859885.
on arXiv
programs
 (with Bugeaud, Luca and Mignotte)
Almost powers in the Lucas sequences,
Journal de Théorie des Nombres de Bordeaux
20 (2008), 555600.
pdf
 (with Martin Bright)
Functions, Reciprocity and the Obstruction to
Divisors on Curves,
Journal of the LMS 77 (2008), 789807.
pdf
 (with Bugeaud and Mignotte)
A multiFrey approach to some multiparameter families
of Diophantine equations,
Canadian Journal of Mathematics 60 (2008), 491519.
pdf
 (with Bugeaud, Luca and Mignotte)
Fibonacci Numbers at most one away from a perfect power,
Elemente der Mathematik 63
(2008), no. 2, 6575.
pdf
dvi
 (with Bugeaud, Luca and Mignotte)
Perfect powers from products of
terms in Lucas sequences,
Journal für die reine und angewandte Mathematik,
611 (2007), 109129.
dvi
pdf
 (with Bugeaud and Mignotte)
Classical and modular approaches to exponential
Diophantine equations II: The LebesgueNagell equation,
Compositio Mathematica 142 (2006), 3162.
pdf
 (with Bugeaud and Mignotte)
Classical and modular approaches to exponential
Diophantine equations I. Fibonacci and Lucas perfect powers,
Annals of Mathematics 163 (2006), 9691018.
dvi
pdf
 (with Cremona and Prickett)
Height difference bounds for elliptic curves over number fields,
Journal of Number Theory 116 (2006), 4268.
dvi
pdf
 (with Cremona)
Computing a Lower Bound for the
Canonical Height on Elliptic Curves over Q,
in Algorithmic Number Theory, 7th International Symposium,
ANTSVII, pages 275286, Springer Verlag, LNCS 4076, July 2006
pdf
 (with Bugeaud, Luca and Mignotte)
On perfect powers in Lucas sequences,
International Journal of Number Theory 1 (2005), 309332.
dvi
pdf
 (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

On standardized models for elliptic curves,
Math. Comp. 74 (2005), 949951.
dvi

The modular approach to Diophantine equations,
notes for lectures given at the Institut Henri
Poincaré (October 2004).
pdf

On the BrauerManin obstruction for curves with split Jacobian,
Journal de Théorie des Nombres de Bordeaux 16 (2004), 773777.
dvi
pdf
 (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), 327330.
pdf
 (with ElSedy)
On points of nondifferentiability of convex functions,
Applied Mathematics and Computation 148 (2004), 725728.
 (with Cremona)
On the Diophantine equation \( x^2+7=y^m \)
,
Acta Arithmetica 109.2 (2003), 143149.
dvi
pdf
 (with Skorobogatov)
On a Shimura curve that is a counterexample to the Hasse principle,
Bulletin of the London Mathematical Society 35 (2003), 143149.
pdf

On the Diophantine equation \( x^2=y^p+2^k z^p\)
Journal de Théorie des Nombres de Bordeaux 15 (2003), 839846.
dvi
pdf
 (with ElSedy)
On the irrationality of certain rapidly convergent series,
International Mathematical Journal 3 (2003), 11931195.

Descent of Picard groups using functions on curves,
Bulletin of the Australian Mathematical Society 66 (2002), 119124.
dvi
pdf

Sieving for rational points on hyperelliptic curves,
Mathematics of Computation 70 (2001), 16611674.
dvi
 (with ElSedy)
On happy numbers,
Rocky Mountain Journal of Mathematics 30 (2000), 565570.
dvi
pdf

4descent, pages 418422, appendix A to
``Beyond the Manin obstruction''
by A. N. Skorobogatov, Inventiones Mathematicae 135
(1999), 399424.

The height pairing on an elliptic curve with complex multiplication,
Arab Journal of Mathematical Sciences 5 (1999), 4348.
dvi
pdf
 (with N. Smart)
A fast DiffieHellman protocol in genus 2,
Journal of Cryptology 12 (1999), 6773.
 (with N. Smart)
On the complexity of computing the 2Selmer group of an elliptic curve,
Glasgow Mathematical Journal 39 (1997), 251258.
 (with J. Merriman and N. Smart)
Explicit 4descents on an elliptic curve,
Acta Arithmetica LXXVII.4 (1996), 358404.
dvi
pdf

Infinite descent on elliptic curves,
Rocky Mountain Journal of Mathematics 25 (1995), 15011538.
dvi
pdf

Descents on curves of genus 1,
PhD thesis, University of Exeter, 1995.
dvi
pdf
Some of my talks:
 Sunits and curves with few bad primes over cyclotomic Z_lextensions
video
 The Fermat Equation and the Unit Equation
video
 Integral points on punctured curves and
punctured abelian varieties
video,
slides
 On the Generalized Fermat Equation \( x^{2\ell}+y^{2m}=z^p\),
Schney, June 2015.
pdf
 Sums of Seven Cubes,
Oberwolfach, Simon Fraser University, Oxford, ..., 2015.
pdf
 Elliptic Curves over Totally Real Fields are Modular,
(many versions of this talk given, 20132015).
pdf
 The Fermat Equation over Totally Real Fields,
Building Bridges, Bristol, March 2014.
pdf
 The Fermat Equation over Totally Real Fields
(version intended for a broad audience),
Maringa, July 2014.
pdf
 Chabauty over Number Fields
Leiden, 20 April 2009.
pdf (scan)
 Integral Points on Curves of Higher Genus
(a more uptodate version) MAGICMuggle Number
Theory Seminar, 7 November 2008.
pdf (scan)
 Integral Points on Curves of Higher Genus
Bordeaux, 15 October 2007. pdf (scan)
 Chabauty for Symmetric Powers on Curves
Banff, 7 February 2007. pdf (scan)
 Chabauty for Symmetric Powers on Curves
(a version of the above talk aimed at those new to Chabauty)
Rome, 13 April 2007. pdf (scan)
 Computing a lower bound for the canonical height on elliptic curves,
ANTS VII conference, Berlin, 24 July 2006.
pdf (scan)
 The LebesgueNagellRamanujan equation, University
of Warwick, 24 August 2005. pdf (scan)
 Functions, reciprocity and the obstruction to divisors
on curves, Bremen and Oberwolfach, July 2005. pdf (scan)
 Beyond Fermat's Last Theorem, University of
Warwick, January 2005. pdf (scan)
 A multiexponential diophantine spree, Institute
Henri Poincare, Paris, December 2004. pdf (scan)
 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, 1718 April, 2012.

Contemporary Methods for Solving Diophantine Equations, Banff, 1017 June, 2012.

Arithmetic Geometry minisymposium,
European Congress of Mathematics,
Krakow, Poland, 27 July 2012.

EPSRC Warwick Number Theory Symposium
, University of Warwick, 20122013:

Selmer Groups, Descent and the Distribution of Ranks, 2428 September 2012.

Explicit Methods for Modular Forms
, 1822 March 2013.

Iwasawa Theory and Galois Representations,
812 April 2013.

Higher rank automorphic forms and Lfunctions,
29 April3 May 2013

Rational Points: Geometric, Analytic and Explicit Approaches, 2731 May 2013.

Summer School: Number Theory for Cryptography, 2428 June 2013.

Number Theory, Geometry and Cryptography,
15 July 2013.

Explicit Methods in Number Theory: Conference
in Honour of John Cremona's 60th Birthday,
University of Warwick,
48 April 2016.

Automorphic Forms: Theory and Computation,
Kings College London, 59 September 2016.

Arithmetic
Aspects of Explicit Moduli Problems,
28 May2 June 2017, BIRS, Banff.

Modern Breakthroughs in Diophantine Problems,
30 August4 September 2020, BIRS, Banff.

Modern Breakthroughs in Diophantine Problems,
19 June24 June 2022, BIRS, Banff.
 Modular curves and their arithmetic,
68 December, University of Warwick