Papers and preprints

This is a list of published and yet-to-be-published articles.

• Computing with necklaces on elliptic curves joint with Marusia Rebolledo
  in preparation (2025)
  The code for the computations is available in one file here, but it needs these files created by Drew Sutherland.
• Counting Points on Elliptic Curves
  joint with Thomas Oliver
  LMS Newsletter November 2023, 31–35.
  Preprint (2023).
• Mordell-Weil groups as Galois modules
  joint with Thomas Vavasour
  Preprint (2023).
• Integrality of twisted L-values of elliptic curves
  joint with Hanneke Wiersema
  Documenta Mathematica 27 (2022), 2041–2066.
  Published version, Preprint or on arxiv.
• The sub-leading coefficient of the L-function of an elliptic curve
  Publ. Math. Besançon Algèbre Théorie Lab. Math. Besançon, Besançon, (2016), 95-96.
  Preprint or on arxiv or published.
• Numerical modular symbols for elliptic curves
  Mathematics of Computation 87 (2018), no. 313, 2393–2423.
  Published version, preprint or on arxiv.
  The code for the computations is available here.
• Vanishing of some Galois cohomology groups for elliptic curves
  joint with Tyler Lawson
  in Elliptic Curves, Modular Forms and Iwasawa Theory edited by David Loeffler and Sarah Livia Zerbes, Springer Proceedings in Mathematics & Statistics, Volume 188, Springer, (2017), 373 - 399
  Preprint or on the arxiv or the published version.
• Book review: "Fermat's Last Theorem" by T. Saito (two volumes).
  Bull. London Math. Soc. (2015) 47 (5): 892-894.
  draft or the published version.
• A moduli interpretation for the non-split Cartan modular curve
  joint with Marusia Rebolledo
  Glasgow Mathematical Journal 60 (2018), no. 2, 411–434.
  Published version, preprint or on the arxiv.
• On the Galois structures of Selmer groups
  joint with David Burns and Daniel Macias Castillo.
  International Mathematics Research Notices 22 (2015), 11909 - 11933.
  Preprint or the published version
• On Mordell-Weil groups and congruences between derivatives of twisted Hasse-Weil L-functions
  joint with David Burns and Daniel Macias Castillo.
  Journal für die reine und angewandte Mathematik 734 (2018), 187–228.
  Published version or preprint.
  The sage code to compute that was used to compute numerical examples. etnc.py
• Overview of some Iwasawa theory
  in Iwasawa Theory 2012; State of the Art and Recent Advances edited by Athanasios Bouganis and Otmar Venjakob, Contributions in Mathematical and Computational Sciences, Volume 7, Springer, (2014).
  preprint or the published version
• On the integrality of modular symbols and Kato's Euler system for elliptic curves
  Documenta Mathematica 19 (2014), 381 - 402.
  Preprint or the published version
• The class group pairing and p-descent on elliptic curves
  joint with Jean Gillibert.
  Proceedings of the London Mathematical Society (3) 106 (2013), no. 2, 345–374.
  Preprint (pdf) or on the arXiv. The two Sage files class_group_pairing.py and pdescent.py can be used to do numerical computations using the logarithmic class group pairing and isogeny via a p-isogeny.
• Algorithms for the Arithmetic of Elliptic Curves using Iwasawa Theory
  joint with William Stein.
  Mathematics of Computation 82 (2013), 1757-1792.
  See here for a preprint version and here for the published version.
• Parity conjectures for elliptic curves over global fields of positive characteristic
  joint with Fabien Trihan. Compositio Mathemtica, 147 (2011), issue 04, 1105-1128.
  Available at compositio or on the arXiv.
• Some remarks on self-points on elliptic curves
  joint with Christophe Delaunay.
  Actes de la Conférence "Fonctions L et Arithmétique", 69-84, Publ. Math. Besançon Algèbre Théorie Nr., Lab. Math. Besançon, Besançon, 2010.
  (pdf)
• Self-points on elliptic curves
  Algebra and Number Theory 3 (2009), 283-315.
  (pdf)
• Self-points on elliptic curves of prime conductor
  joint with Christophe Delaunay.
  International Journal of Number Theory 5 (2009), 1-22.
  (pdf)
• Self-points on an elliptic curve of conductor 14
  RIMS Kôkyûroku Bessatsu 4 (2007), 189-195.
  (pdf)
• On p-adic elliptic logarithms and p-adic approximation lattices
  no longer intended for publication.
  (pdf)
• Extending Kato's result to curves with p-isogenies
  Mathematical Research Letters 13 (2006), no. 5, p. 713-718
  (pdf)
  Tadashi Ochiai found an important error in the paper. The paper "On the integrality of modular symbols ... " contains, among other things, the correction and generalisation of the main result of this paper.
• Iwasawa theory of the fine Selmer group
  Journal of Algebraic Geometry 16 (2007), p. 83-108.
  (pdf)
• The fine Tate-Shafarevich group
  Mathematical Proceedings of the Cambridge Philosophical Society 142 (2007), no. 1, p. 1-12.
  (pdf)
• On p-adic heights in families of elliptic curves
  J. London Math. Soc. (2) 70 (2004) p. 23-40
  (pdf)
• Sur la hauteur p-adique dans une famille de courbes elliptiques
  Ann. Sci. Math. Québec 28 (2004), no 1-2, 219-223
  (pdf)
• Une quintique de genre 1 qui contredit le principe de Hasse
  Enseign. Math. (2) 47 (2001), p. 161-172
  (pdf) Errata

Here my PhD and two essays I wrote before.

• The fine Selmer group and height pairings
  PhD-thesis in Cambridge, 2004, under the supervision of John Coates.
• The Hare and the Wolf
  An essay on p-adic heights on elliptic curves.
  Smith-prize essay in Cambridge, 2003.
• Letters to an Egyptian princess
  An essay on the Distribution of Prime Numbers, Including the Prime Number Theorem and Dirichlet's Theorem on Primes in Arithmetic Progressions.
  Third-year essay at the University of Warwick, 1999.

Teaching

These are lecture notes and other material for the courses i am or was teaching. if interested in the LaTeX source or problem sheets, please email me.

• Local fields

  Here are the lecture notes for the magic module on Local Fields. These notes are based on an earlier version by Chris Williams.

  

• Selmer groups

  These are notes to three hours of lectures at the summer school at Baskerville on elliptic curves.

• Elliptic curves

  I started to teach a course on elliptic curves in 2017. It is aimed at third year student here at Nottingham that have never seen any algebraic geometry. In that it is rather different than other courses on elliptic curves.

  Nottingham students can find all information on this module on its moodle page.

  Here are the complete lecture notes on this module as I give it in 2018. The students were given a version without the pictures.

• Elementary number theory

  Since 2015 I am teaching a course on number theory as the second part of a year 2 module called algebra and number theory.

  Again Nottingham students can find all information on this module on its moodle page.

  Here are the complete lecture notes on this module as I gave it in 2017.

• Group theory

  Between 2012 and 2016 I was teaching a course on group theory.

  Here are the complete lecture notes on this module as I gave it in 2016.

• Coding and cryptography

  For many years I was teaching a course on coding and cryptography.

  The lecture notes with all problem sheets are here.

• Further number theory

  I was teaching the course called further number theory for a number of years. It contained the following chapters:

  Prime numbers, Quadratic reciprocity, Bernoulli numbers, L-functions, Gaussian integers , and p-adic numbers.

• Algebraic number theory

  In 2005/6 I was teaching a course on algebraic number theory in the masters program at the ÉPFL. The course covered both the classical approach via ideals as well as an introduction to the theory of valuations and adèles.

  Here are the lecture notesin french: I am indebted to Sébastien Nobs for having written up the notes of the full course. These notes are not corrected, and probably never will.