The arithmetic of p-adic automorphic forms and Galois representations

点击翻译按钮获取中文摘要

One of the fundamental problems in number theory is to understand linear representations of the Galois groups of number fields. These encode structure and symmetries of the number fields (for example the rational numbers).The Langlands programme seeks to study these Galois representations using automorphic forms - these are special kinds of analytic functions which are conjecturally (and sometimes provably) related to Galois representations. Initiated in the 1970s, the work of Langlands and others has been extremely influential, playing a crucial part in Wiles and Taylor's celebrated proof of Fermat's last theorem, as well as more recent results on conjectures of Serre and Sato-Tate.In recent years number theorists have been to develop an extension of the Langlands programme which seeks to understand the finer p-adic structure of automorphic forms and Galois representations (particularly the way they move in p-adic families). This proposal focuses on developing this `p-adic Langlands programme', which is currently only understood in special cases.Specific aims of the proposal include proving some cases of p-adic Langlands functoriality, allowing one to move between p-adic automorphic forms on different groups, by studying the arithmetic of Shimura varieties for unitary groups. Secondly, studying p-adic Banach space representations of GL_2(K), where K is a finite extension of Q_p, using tools from arithmetic geometry and the completed cohomology of Shimura curves.