1549380232-Automorphic_Forms_and_Applications__Sarnak_

Lecture 2. Eulerian integrals for GLn

Let f ( T) again be a holomorphic cusp form of weight k on fJ for the full modular

group with Fourier expansion

Then Hecke [34] associated to f an £-function

and analyzed its analytic properties, namely continuation, order of growth, and

functional equation, by writing it as the Mellin transform of f

An application of the modular transformation law for f ( T) under the transformation

T f--> -1 / T gives the functional equation

A(s,f) = (-l)kf^2 A(k-s,f).

Moreover, if f was an eigenfunction of all Hecke operators then L(s, f) had an Euler

product expansion

L(s, !) =IT (l _ app-s + Pk-1-2s)-1.

p

We will present a similar theory for cuspidal representations (7r, Vn) of GLn(A).
For applications to functoriality via the Converse Theorem (see Lecture 6) we will
need not only the standard £-functions L(s, 7r) but the twisted £-functions L(s, 7r x
7r^1 ) for ( 7r^1 , Vn') a cuspidal automorphic representation of GLm (A) form < n as well.
One point to notice from the outset is that we want to associate a single £-function
to an infinite dimensional representation (or pair of representations). The approach
we will take will be that of integral representations, but it will broadened in the
sense of Tate's thesis [91].
The basic references for the material in this section are Jacquet-Langlands [39],
Jacquet, Piatetski-Shapiro, and Shalika [40], and Jacquet and Shalika [45].

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