Research interests and projects
Pure Mathematics research at Exeter is mostly concerned with Number Theory, Arithmetic and Algebraic Geometry and related areas. An important theme in modern mathematics is that number theory and geometry are closely interrelated, so that, for example, the function fields of algebraic curves behave in a very similar way to algebraic number fields. Arithmetic geometry combines and unifies the areas of Number Theory and Geometry. One often looks for invariants associated with objects of geometric and number-theoretic interest like curves or equations defined over an arithmetic field (a number field, a local field or a finite field). In many cases these invariants are elements in K-theory or in a (co)homology theory.
Particular areas of interest at Exeter include p-adic cohomology and the classification of p-divisible formal groups, fundamental groups of curves, lattices and codes, combinatorics and additive number theory, Galois module structure of rings of integers, and Hopf-Galois structures on field extensions. See also our publications.
In a recent joint work with Christopher Davis and Thomas Zink, funded by ESPRC, we constructed an overconvergent de Rham-Witt complex suitable to compute the rigid cohomology of a quasiprojective, smooth variety X over k. The work solved an old and important problem in the study of p-adic cohomology theories. It remained an open question whether the image of overconvergent de Rham-Witt cohomology indeed defines a lattice in rigid cohomology, hence an integral structure. This is the content of the next research project. Another project is to generalize the above formula to semistable schemes.
Realisable Galois Module Classes for Tame Extensions
Dr Nigel Byott
If a Galois extension of number fields N/K is at most tamely ramified, the ring of algebraic integers in N is locally free as a module over the group ring of the Galois group with coefficients in the ring of algebraic integers of K. The aim of this ongoing collaboration between Nigel Byott (Exeter) and Bouchaib Sodaigui (Valenciennes) is to determine what module structures can occur for various nonabelian Galois groups.
Wild Galois Module Structure and Ramification Theory
Dr Nigel Byott
For a wildly ramified Galois extension L/K of local fields of residue characteristic p>0, an important question is whether the valuation ring S of L is free as a module over its associated order. This joint project of Nigel Byott (Exeter) and Griff Elder (Omaha) seeks to relate this question to the ramification filtration of the Galois group of L/K.
Hopf-Galois Theory, Hopf Orders and Galois Module Structure
Dr Nigel Byott
Hopf algebras provide a generalisation of classical Galois theory. At the same time, Hopf orders give rise to freeness results in Galois module structure. This ongoing project seeks to investigate various aspects of these connections, by determining the Hopf-Galois structures on various classes of field extensions, investigating the occurrence of Hopf orders as associated orders, and extending existing results on Hopf orders in characteristic 0 to the characteristic p setting.
Iwasawa theory of modular forms
Dr Sarah Zerbes
In ongoing joint work with Antonio Lei and David Loeffler, we study the Selmer group of modular forms which are ordinary or supersingular at a prime p. The Selmer group of a modular form has a simpler structure when the modular form is ordinary at p; in the non-ordinary (supersingular) case, the behaviour of both the L-function and the Selmer group is more subtle, and so far only isolated results were known in the Iwasawa theory of supersingular modular forms. In my papers with Lei and Loeffler, we develop a general approach to Iwasawa theory that applies in both the ordinary and supersingular cases, allowing us to unify existing results in the Iwasawa theory of modular forms and substantially extend them. The main new tools are methods from p-adic analysis, notably from the theory of p-adic differential equations, combining Iwasawa theory with p-adic Hodge theory.
