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Representations of p-adic groups and arithmetic (STEVENSSU21SF)

School of Mathematics

About the Project

The Local Langlands Correspondence has motivated a great deal of Number Theory and Representation Theory over the last fifty years. It connects representations of (roughly) the absolute galois group of a p-adic field to the representations of a matrix group G over the field. In the original formulation (now proved for several families of matrix groups G, including general linear groups) it considered only complex representations but, more recently, representations with coefficients in other rings or fields have been considered also. The correspondence results in a partition of the irreducible representations of G into L-packets (which are singletons for general linear groups), each of which is conjecturally characterised by an equality of local factors (L-functions and -factors) – though a definition for such factors is only known for “generic” representations, of which there is (at least conjecturally) precisely one in each packet.

One of the difficulties in studying this is that the representations of G are (almost all) infinite-dimensional, so they are quite hard to get a handle on. One way for doing so, which has been very successful, is to restrict them to compact open subgroups K: we then look for representations of K whose presence in this restriction characterizes some property which representations of G might have; this property might be being “unramified”, ”tame”, “generic”,… Similarly, this restriction can be used to give very explicit descriptions of the representations of G, using arithmetic data. Such results also have interpretations via the Langlands correspondence, so consequences for the absolute galois group.

This PhD project will be in the area of the local Langlands correspondence, exploring some of the problems raised above, or related questions.

For more information on the supervisor for this project, please go here

This is a PhD programme.

The start date is 1st October 2021.

The mode of study is full time.

Funding Notes

This PhD project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found at View Website

A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. Applicants should contact the primary supervisor for further information about the fee associated with the project.

Entry Requirements

Acceptable first degree in Mathematics. The standard minimum entry requirement is 2:1.


i) "Représentations l-modulaires d'un groupe réductif p-adique avec l≠p," Marie-France Vignéras, Progress in Mathematics, 137, Birkhäuser Boston, Inc., Boston, MA, 1996.
ii) “The local Langlands conjecture for GL(2),” Colin Bushnell and Guy Henniart, Grundlehren der Mathematischen Wissenschaften 335, Springer-Verlag, Berlin, 2006.
iii) “The supercuspidal representations of p-adic classical groups,” Shaun Stevens, Invent. Math. 172(2) (2008) 289-352.
iv) "Endo-classes for p-adic classical groups," Robert Kurinczuk, Daniel Skodlerack and Shaun Stevens, Invent. Math. (2020).
v) “Generic smooth representations,” Alexandre Pyvovarov, 2018 arXiv:1803.02693

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