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Representations of p-adic groups and arithmetic (STEVENSSU20SF)


Project Description

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.

MORE INFORMATION

Project supervisor: https://people.uea.ac.uk/shaun_stevens
Mode of study: Full time
Start date: October 2020
Entry requirements: First degree (2:1 or above) in Mathematics.

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.

References

“The local Langlands conjecture for GL(2),” Colin Bushnell and Guy Henniart, Grundlehren der Mathematischen Wissenschaften 335, Springer-Verlag, Berlin, 2006.

"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.

“Eisenstein series and automorphic L-functions,” Freydoon Shahidi, American Mathematical Society Colloquium Publications, 58. American Mathematical Society, Providence, RI, 2010.

“The supercuspidal representations of p-adic classical groups,” Shaun Stevens, Invent. Math. 172(2) (2008) 289-352.

“Jordan blocks of cuspidal representations of symplectic groups,” Corinne Blondel, Guy Henniart and Shaun Stevens, Algebra Number Theory 12 (2018), no. 10, 2327–2386.

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