Some of Mark's research interests.
Most of my current research centres on p-adic Hodge theory and its
applications to the arithmetic of modular forms and modularity of
p-adic Hodge theory is a fairly young branch of number theory which
seeks to study the p-adic analogue of the relationship between Betti
cohomology and de Rham cohomology over the complex numbers. The p-adic
analogue of Betti cohomology is p-adic etale cohomology, and it is
related to other cohomology theories using the p-adic period rings
introduced by J.M Fontaine. Since p-adic etale cohomology carries a
Galois action, the theory contains a lot of rich arithmetic
p-adic Hodge theory has found many number theoretic applications. The
most spectacular of these is perhaps Wiles' proof of the
Taniyama-Shimura-Weil conjecture, and Mochizuki's work on
Grothendieck's anabelian conjecture. It also plays an important role in
Faltings' proof of the Mordell and Tate conjectures.
A recent exciting development in p-adic Hodge theory is Breuil's p-adic
Langlands program, which relates (2-dimensional) p-adic Galois
and p-adic unitary representations of GL_2(Q_p). This nascent theory
has already found applications in proving instances of modularity
of global Galois representations (the Fontaine-Mazur) conjecture.
p-adic modular forms
p-adic modular forms may be thought of as p-adic deformations or
interpolations of classical modular forms. They may also be viewed of
avatars of the unitary representations of Breuil. One can often attach
Galois representations (of the absolute Galois group of Q) to p-adic
Using p-adic Hodge theory one can show that the representations
attached to finite slope, overconvergent eigenforms satisfy the
Studying the p-adic periods of these representations yields a lot of
information about the geometry of the space of such eigenforms (the
"Eigencurve" of Coleman-Mazur).