Representations of GLn over p-adic fields, and topics in analytic number theory
The goal of this course to show how the local representation theory can be used to study many analytic number theory problems.
In the first half of this course, we will review the theories on the representations of the general linear groups over p-adic fields. The main focus will be on and the compact induction theory for the supercuspidal representations.
In the second half of the course, we will discuss the applications of the local representation theory to some problems in the analytic number theory. If time allows, we will cover the following research-level topics: period integrals, relative trace formulae, sup norm problem, subconvexity bound and moments of L-functions.
Basic representation theory, complex analysis. Some knowledges on the p-adic field and modular forms/automorphic forms will also be helpful.
 C.Bushnell and G.Henniart, The Local Langlands Conjecture for GL(2). Springer -Verlag, Berlin, 2006. (This is the most important reference for the first half);
 D.Bump, Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics,vol.55,Cambridge University Press, Cambridge, 1997. (For backgrouds on modular forms and automorphic forms);
 H.Iwaniec and E.Kowalski, Analytic number theory. AMS Colloquium Publications, 53. AMS, Providence, RI, 2004. (If you have plenty of time and want to know how some other people do analytic number theory)
Further references, especially those on research-level topics, will be given as the course goes on or at request.