On locally analytic vectors of completed cohomology

This project falls within the EPSRC Number Theory research area.

There are many conjectural relations between Galois representations, Motives, and automorphic forms: the connection between the latter two is intimately related to the Langlands programme and an enormous amount of current research. The Fontaine-Mazur conjecture proposes the relation between Galois representations and motives, namely that all "reasonable" Galois representations are geometric.

Being more explicit (my exposition will mostly follow Richard Taylor's article, "Galois representations"), a standard approach to constructing p-adic representations is as follows. Firstly, G_Q has a natural action on the points of any algebraic variety (over -Q), which leads to an action on the etale cohomology groups (with coefficients in -Q_p). This action can then be twisted through the Tate character (which describes how G_Q acts on p-power roots of unity) any number of times. The subquotients of representations arising in this way are said to come "from geometry". In 1995, Fontaine and Mazur conjectured that any irreducible representation G_G to GL_n (-Q I ) which is unramified at all except finitely many primes p, and "de Rham" when restricted to G_(Q_I ), arises from geometry. These geometric representations have been studied in algebraic geometry, and conjecturally have a decomposition into "pure motives", in turn leading to further conjectural properties of Galois representations, namely that they come in compatible families as I varies, with associated invariants called Hodge-Tate numbers.

Even partial results in resolving these connections between L-functions, automorphic forms, Galois representations, and motives, can have enormous impact, for instance their use in Wiles' proof of Fermat's last theorem. The goal of this project would be to understand the approach taken by Emerton to attack the case n=2 of the Fontaine-Mazur conjecture using completed cohomology, as well as recent progress by Lue Pan in his paper "On locally analytic vectors of completed cohomology". A novel aspect of the methodology would be to use a p-adic analogue of the Beilinson-Bernstein localisation theorem, as developed by Ardakov, to establish a possible connection a sheaf of locally analytic vectors playing a central role in Pan's paper.