The Langlands programme is a vast web of conjectures (some vague, some less so, some proved) which link, for example, the representations of the absolute Galois group Gal(Q/Q) to families of representations of matrix groups over local fields (R, C and the p-adic numbers Q_p). The implications of these links to Number Theory could be enormous: for example, the Taniyama-Shimura conjecture (proved by Wiles, Taylor et al.) follows from the case of two-dimensional representations of the Galois group.

As with many problems in Number Theory, the Langlands programme splits up according to primes; so, for each prime p, there is a local Langlands programme connecting representations of the absolute Galois group Gal(Q_p/Q_p) to representations of matrix groups over Q_p. The existence of a suitable correspondence has now been proved when the matrix group is the full group of invertible matrices GL_n, and also when it is a symplectic or orthogonal group. In all these cases, there are explicit descriptions of the representations on both sides of the correspondence so one can ask how these match up, in order to use the correspondence to gain arithmetic information. In the case of GL_n, when n is coprime to p, there is indeed an "explicit" description of the correspondence, but this is not currently true in other cases.

Another question which remains incompletely answered is: exactly what is need to characterize the correspondences? In some cases, it is possible to define L-functions and e-factors for the representations on both sides of the correspondence and these are often used as a way of making the correspondence unique. In special cases, it is even possible to compute these explicitly. However, when one passes to symplectic and orthogonal groups, some impotant representations (called non-generic) do not allow L-functions to be defined in the usual way. Then one can ask: which representations are non-generic and how should L-functions be defined, and calculated, for them?

Finally, the proof of Wiles, Taylor et al. actually worked not just by looking at representations, but by looking at congruences between them. This leads to seeking an understanding of the modular representations of p-adic groups, and of a modular local Langlands correspondence. Again here, very little is known beyond GL_n; indeed in some cases, not even GL₂ is fully understood!

The project would be in the area of the local Langlands programme, looking at one (or more) of the questions raised above, or related questions.

Bushnell, C., Henniart, G., The local Langlands conjecture for GL(2). Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335. Springer-Verlag, Berlin, 2006. ISBN: 978-3-540-31486-8. MR2234120 (2007m:22013)

Shahidi, F., Eisenstein series and automorphic L-functions, American Mathematical Society Colloquium Publications, 58. American Mathematical Society, Providence, RI, 2010. ISBN: 978-0-8218-4989-7. MR2683009 (2012d:11119)