The Number Theory Group at Royal Holloway has a wide range of interests. Research is carried out in many areas such as: additive and multiplicative number theory, sieve methods, Diophantine approximation, Diophantine equations, metric number theory, the circle method, exponential sums, Salem and Pisot numbers, Mahler measure, totally real algebraic integers, the Riemann Zeta-function, zeta functions of groups and rings, arithmetic groups, p-adic Lie groups and their representations.

Recent work of the group has included representations by quadratic forms, probabilistic Galois theory, random Diophantine problems, the distribution of Gaussian primes in narrow sectors or small circles, prime values of the integer parts of points on algebraic curves, prime values of polynomials, the prime k-tuple problem, work on pairing-friendly abelian varieties, Lehmer’s problem for reciprocal polynomials of integer symmetric matrices, the Schur-Siegel-Smyth trace problem, graph Salem numbers, and non-linear Diophantine approximation to complex numbers. There are regular seminars in these and related fields - contact Dr Rainer Dietmann for details.

For opportunities for postgraduate research in number theory please contact Dr Dietmann, Dr Klopsch or Dr McKee.

### Group members

### Research students

- Jonathan Cooley
- Gary Greaves
- Lee Gumbrell
- Asim Islam
- Nektarios Kyriakou

**Dr Dietmann** works in Analytic Number Theory, focusing on applications of exponential sums and the circle method to Diophantine problems in many variables. He recently also obtained new results on a classical problem in Probabilistic Galois Theory, and jointly with Jörg Brüdern started a research project on Random Diophantine problems. Recent key publications include

1. (with T. D. Browning) On the representation of integers by quadratic forms, Proc. London Math. Soc. 96 (2008), 389--416,

2. Systems of cubic forms, J. London Math. Soc. 77 (2008), 666--686,

3. On the distribution of Galois groups, Mathematika 58 (2012), 35--44,

4. (with Jörg Brüdern) Random Diophantine equations of additive type, arXiv:1110.3496v1.

**Dr Klopsch**'s research activities are mainly directed towards the study of inifinite groups from an arithmetic point of view. He has investigated the structure and properties of automorphism groups of local fields, p-adic Lie groups, arithmetic groups, and related objects. Currently, he has a keen interest in zeta functions associated to the representations of compact p-adic Lie groups and arithmetic groups. This topic touches upon the theory of Igusa local zeta functions. Dr Klopsch is also intrigued by problems in Additive Combinatorics and has co-authored two short articles in this area. He is currently supervising three PhD students.

Among his key publications are:

1. Automorphisms of the Nottingham group, J. Algebra 223 (2000), 37-56.

2. Pro-p groups with linear subgroup growth, Mat. Z. 245 (2003), 335-370.

3. Index-subgroups of the Nottingham group, joint work with Yiftach Barnea, Adv. Math. 180 (2003), 187-221.

4. Analytic groups over general pro-p domains, joint work with Andrei Jaikin-Zapirain, J. London Math. Soc. 76 (2007), 365-383.

5. Igusa-type functions associated to finite formed spaces and their functional equations, joint work with Christopher Voll, Trans. Amer. Math. Soc. 361 (2009), 4405-4436

For a full list of publications and other contributions, visit Dr Klopsch's personal website. You can also read about his involvement within the Algebra Group.

**Dr McKee, **jointly with Chris Smyth (Edinburgh), has applied combinatorial techniques to prove striking theorems concerning Salem numbers and Mahler measure. With Steven Galbraith and Paula Valenca he pioneered a new approach in the study of pairing-friendly abelian varieties. His recent key publications include:

1. Salem numbers of trace -2, and traces of totally positive algebraic integers, with C.J. Smyth, ANTS VI, Lecture Notes in Computer Science 3076, 2004, pages 327-337

2. There are Salem numbers of every trace, with C.J. Smyth, Bulletin of the London Mathematical Society, 37, 2005, pages 25-36.

3. Pisot numbers, Salem numbers, graphs, and Mahler measure, with C.J. Smyth. Experimental Mathematics, 14, 2005, no. 2, pages 211-229.

4. Ordinary abelian varieties having small embedding degree, with S.D. Galbraith and P.Valença. Finite Fields and Their Applications, volume 13, issue 4, 2007, pages 800-814.

5. Integer symmetric matrices having all their eigenvalues in [-2,2], with C.J. Smyth. Journal of Algebra, volume 317, issue 1, 2007, pages 260-290.

6. Integer symmetric matrices of small spectral radius and Mahler measure, with C.J. Smyth. International Mathematics Research Notices, 2011, doi:10.1093/imrn/rnr011.

### Recently completed PhD's

Dr Christos Christopoulos

Dr David Mireles

Dr Laurence Rackham

**Former members of the group**

Dr Yvonne Buttkewitz

Dr Ben Smith