1549380232-Automorphic_Forms_and_Applications__Sarnak_

112 J.W. COGDELL, £-FUNCTIONS FOR GLn

2.1. Eulerian integrals for GL2

Let us first consider the L-functions for cuspidal representations (rr, V1T) of G L 2 (A)

with twists by an idele class character x, or what is the same, a (cuspidal) auto-

morphic representation of GL 1 (A), as in Jacquet-Langlands [39].

Following Jacquet and Langlands, who were following Hecke, for each <p E V7r

we consider the integral

I(s; <p, x) = ( l.fJ (a 1) x(a)ials-1/2 dxa.

JkX\J;,.X

Since a cusp form on GL 2 (A) is rapidly decreasing upon restriction to Ax as in

the integral, it follows that the integral is absolutely convergent for alls, uniformly

for Re(s) in an interval. Thus I(s; <p, x) is an entire function of s, bounded in any

vertical strip a~ Re(s) ~ b. Moreover, if we let cp(g) = <p(tg-^1 ) = <p(wn tg-^1 ) then

cp E V7r and the simple change of variables a f---7 a-^1 in the integral shows that each

integral satisfies a functional equation of the form

I(s;<p,x) = I(l - s;cp,x-^1 ).

So these integrals individually enjoy rather nice analytic properties.

If we replace <p by its Fourier expansion from Lecture 1 and unfold, we find

where we have used the fact that the function x(a)lals-^1 /^2 is invariant under kx. By

standard gauge estimates on Whittaker functions [40] this converges for Re(s) >> 0

after the unfolding. As we have seen in Lecture 1, if W'P E W(7r, 'l/J) corresponds to

a decomposable vector <p E V7r '.::::'. 0 'V1Tv then the Whittaker function factors into a

product of lo cal Whittaker functions

v

Since the character x and the adelic absolute value factor into local components and

the domain of integration Ax also factors we find that our global integral naturally
factors into a product of local integrals

with the infinite product still convergent for Re(s) >> 0, or

v

with the obvious definition of the local integrals

'1fv(s;W'Pv,Xv) = 1: W'Pv (av 1 ) Xv(av)lavls-l/^2 dxav.

Thus each of our global integrals is Eulerian.