Arithmetic Study Group: Abelian and non-abelian p-adic L-functions
22 October 2013 16:00 in CM219
p-adic L-functions play a central role in classical (abelian) Iwasawa Theory since they constitute the analytic input in the so-called Main Conjectures. Conjecturally one can attach such a p-adic L-function to any critical motive. Known examples are the p-adic L-function
of an elliptic curve, of a Hecke character of a CM or totally real field, or of an elliptic modular form. The p-adic L-function, say of an elliptic curve E defined over the rationals, encodes information about the critical values of the L-function of E as we base change E over the p-cyclotomic tower. In the context of non-abelian Main Conjectures, as they were formalized in the work of Coates, Fukaya, Kato, Sujatha and Venjakob, one is interested in replacing the p-cyclotomic tower with a Galois extension whose Galois group is a p-adic Lie group. This leads to the notion of non-abelian p-adic L-functions, which are higly conjectural and only a few examples are known.
In this talk I will start by explaining the notion of p-adic L-functions in the abelian setting and their role in the Main Conjectures. Then I will discuss the non-abelian setting and the recent progress made by various people in this direction with respect to the existence of non-abelian p-adic L-functions.