Mathematical page

Description of research interests

My field of research is algebraic number theory and algebraic geometry; in particular, I am interested in the theory of modular forms and Galois representations.

Given a modular form which is an eigenform for a certain algebra of Hecke operators, it is a theorem of Deligne (following work of Eichler, Shimura and Igusa, and part of the result is jointly with Serre) that one may associate to it a system of 2-dimensional Galois representations. For modular forms over totally real fields, the analogous result is due to Carayol (with other notable contributions from Ohta, Blasius, Rogawski, Tunnell, and Taylor). These representations occur in the cohomology of certain Shimura varieties.

Serre conjectured a converse to Deligne's result: he suggested a criterion for the reduction of a Galois representation to come from a modular form, and even made precise guesses as to the weight and level of the form. Although Serre's criterion remains highly conjectural, it is known that if it does come from a modular form, then it comes from one of the weight and level predicted by Serre. These are due, respectively, to Edixhoven (with work by Gross, Coleman and Voloch) and Ribet (with work by Carayol, Mazur and Diamond).

For totally real fields, one can make analogous conjectures, and my current work involves trying to solve the case of the weight, jointly with Kevin Buzzard and Fred Diamond. The question of the correct level seems to be more or less resolved (papers [2] and [3] below, unpublished work of Fujiwara and Rajaei, and other papers below).

The results of Ribet and Edixhoven are crucial to Wiles's proof of Fermat's Last Theorem. One might hope to use analogous statements over totally real fields to deduce some similar results - what can one say, for example, about solutions to the Fermat equation over real quadratic fields? Fujiwara has proven the analogue of Wiles's result, and one might hope that the study of Galois representations associated to Hilbert modular forms might give some interesting results.

Recently, these questions have begun to be the main focus of my research. With my former PhD student, Paul Meekin, who successfully gained his PhD in 2003, we have examined the methods of Ribet to deduce an implication of the form "Modularity implies Fermat" for a (very) small number of real quadratic fields (in fact, in [8], we show that only Q(sqrt(2)) will work!). More recently, I have worked with Jayanta Manoharmayum, to generalise some theorems on the modularity of elliptic curves over totally real fields. In particular, it looks as if we can prove the modularity of semistable elliptic curves over some real quadratic fields, and, in conjunction with the results of Paul Meekin, mentioned above, leads to the result that the Fermat equation xn+yn=zn has no non-trivial solution in Z(sqrt(2)) if n>3.

Graduate students

I also jointly supervised Rajender Adibhatla (2005–2009) for the first year of his thesis.

Books

In about 2006, Springer suggested that I might like to write a book on Algebraic Number Theory for their Undergraduate Mathematics Series (SUMS). So I did, but it took me quite a long time; thanks to various Springer editors for their patience. It was eventually published in July 2014. Already, kind readers are sending suggestions for improvements, and I'll make a page dedicated to the book, containing these suggestions, in due course.

Publications and other submitted papers

Hilbert modular forms etc.

  On Galois representations associated to Hilbert modular forms
PhD thesis, University of Cambridge (supervisor: Richard Taylor)
[1] On Galois representations associated to Hilbert modular forms of low weight
Crelle 491 (1997) 199-216
[2] Mazur's Principle for totally real fields of odd degree
Compositio Mathematica 116 (1999) 39-79
[3] Level lowering for modular mod l Galois representations over totally real fields
Mathematische Annalen 313 (1999) 141-160
[7] Correspondences on Shimura curves and Mazur's Principle above p
Pacific J. Math. 213 (2004) 267-280
[8] The Fermat equation over Q(sqrt(2)) (with Paul Meekin)
Journal of Number Theory 109 (2004) 182-196
[13] Optimal levels for modular mod 2 representations over totally real fields
Documenta Math., Extra Volume Coates (2006), 533-550
[16] On the modularity of supersingular elliptic curves over certain totally real number fields
(with Jayanta Manoharmayum)
Journal of Number Theory 128 (2008) 589-618
[18] On Serre's conjecture for mod l Galois representations over totally real fields
(with Kevin Buzzard, Fred Diamond)
Duke Math. J. 155 (2010) 105-161
[20] On a pairing between symmetric power modules
Glasgow Math. J. 55 (2013) 309-312

K-theory of elliptic curves

[4] A distribution relation for elliptic curves
Bull. LMS 32 (2000) 146-154
[5] An elementary proof of a distribution relation for elliptic curves
manuscripta mathematica 103 (2000) 329-337

Combinatorial number theory

[6] Applications of the AGM of Gauss: some new properties of the Catalan-Larcombe-French sequence (with Peter Larcombe, David French)
Congressus Numerantium 161 (2003) 151-162
[9] Linear recurrences between two recent integer sequences
(with Peter Larcombe, David French)
Congressus Numerantium 169 (2004) 79-99
[10] Power series identities generated by two recent integer sequences
(with Peter Larcombe, David French)
Bull. Inst. Combinatorics and Applications 43 (2005) 85-95
[11] On small prime divisibility of the Catalan-Larcombe-French sequence
(with Peter Larcombe, David French)
Indian Journal of Mathematics 47 (2005) 159-181
[12] A short proof of the 2-adic valuation of the Catalan-Larcombe-French number
(with Peter Larcombe, David French)
Indian Journal of Mathematics 48 (2006) 135-138
[17] Higher genus arithmetic-geometric means
The Ramanujan Journal 17 (2008) 1-17
[19] Supercongruences for the Catalan-Larcombe-French numbers
(with Helena Verrill)
The Ramanujan Journal 22 (2010) 171-186

School/undergraduate journals

  13, 31 and the 3x+1 problem
Eureka 49 (1989) 22-25
  The generating function for the Catalan numbers
Mathematical Spectrum 36 (2003) 9-12
  Square roots by subtraction
Mathematical Spectrum 37 (2005) 119-122
[14] Mathematics of Sudoku I
(with Bertram Felgenhauer)
Mathematical Spectrum 39 (2006) 15-22
See my Sudoku page for more details
[15] Mathematics of Sudoku II
(with Ed Russell)
Mathematical Spectrum 39 (2006) 54-58
See my Sudoku page for more details