# James W. Cogdell (Ohio State University, Columbus): L-functions and Functoriality

The principle of functoriality is one of the central tenets of the Langlands program; it is a purely automorphic avater of Langlands vision of a non-abelian class field theory. There are two main approaches to functoriality. The one envisioned by Langlands is through the Arthur-Selberg trace formula, and with the recent work of Ngo, Arthur, and others this is now becoming available. The second method is that of L-functions as envisioned by Piatetski-Shapiro and is based on the converse theorem for *GL(n)*. In this series of lectures I would like to explain the L-function approach to functoriality and how it has been applied.

I will begin with some basic material on automorphic forms and representations, primarily for *GL(n)*. Then I will spend a number of lectures developing the theory of integral representations for Rankin-Selberg L-functions for *GL(n)xGL(m)*, up to and including the converse theorems for *GL(n)*. The converse theorem in this context gives a way of telling when a representation of *GL(n)* is automorphic in terms of the analytic applications of its twisted L-functions.

To apply the converse theorem one must control the analytic properties of L-functions. There are two principal ways to do this. One is the method of integral representations, as we will have discussed for *GL(n)*. The other is the Langlands-Shahidi method, which understands L-functions through the Fourier coefficients of Eisenstein series. As we will need this for our applications, I will spend a few lectures surveying this theory.

Finally, I will explain the local and global Langlands conjectures and the formulation of Langlands' principal of functoriality. I will discuss how one can use the converse theorem for *GL(n)* as a vehicle to obtain functoriality to *GL(n)* and then implement this for the liftings from classical groups to *GL(n)* and also the symmetric power liftings for *GL(2)*. These symmetric power liftings and their variants give the best general bounds towards the Ramanjan conjectures for *GL(2)*, and I will end by explaining this.

Lectures: Wednesday, 9:00 -10:30 and 11:00 - 11:45