Lecture 4. Global £-functions
Once again, we let k be a global field, A. its ring of adeles, and fix a non-trivial
continuous additive character '¢ = ©'l/Jv of A. trivial on k.
Let ( 7f, V11") be an cuspidal representation of G Ln (A.) (see Lecture 1 for all the
implied assumptions in this terminology) and ( 7f^1 , V11"') a cuspidal representation of
GLm(A.). Since they are irreducible we have restricted tensor product decomposi-
tions 7f '.:::::'. ©^1 1fv and 7f^1 '.:::::'. ©^1 7r~ with (7rv, V71"J and (7r~, V71"J irreducible admissible
smooth generic unitary representations of GLn(kv) and GLm(kv) [19, 26]. Let
w = ©' Wv and w' = ©' w~ be their central characters. These are both continuous
characters of k x \A. x.
Let S be a finite set of places of k, containing the archimedean places 800 , such
that for all v rf. S we have that 1fv, 1f~, and '¢v are unramified.
For each place v of k we have defined the local factors L(s, 1fv x 7r~) and
t: ( s, 7f v x 7f~, '¢v). Then we can at least formally defineL(s, 7f x 7r
1
) =II L(s, 1fv x 7r~) and t:(s, 7f x 7r^1 ) =II t:(s, 1fv x 7f~, '¢v)·
v v
We need to discuss convergence of these products. Let us first consider the
convergence of L(s, 7f x 7r^1 ). For those v rf. S, so 1fv, 1f~, and 7/Jv are unramified, we
know that L(s, 1fv x 7r~) = det(J -q;;s A11"v @A71"J-^1 and that the eigenvalues of A11"v
and A11"~ are all of absolute value less than q;,12. Thus the partial (or incomplete)
£-functionL^8 (s,1f X 7r^1 ) =II L(s,1fv X 7r~) =II det(I-q-sA11"v ©A71"J-^1
v(/:.S v(/:.S
is absolutely convergent for Re(s) >> 0. Thus the same is true for L(s,1f x 7r^1 ).
For the t:-factor, we have seen that t:(s, 1fv x 7f~, '¢v) = 1 for v rf. S so that the
product is in fact a finite product and there is no problem with convergence. The
fact that t:(s, 7f x 7r^1 ) is independent of 7/J can either be checked by analyzing how
the local t:-factors vary as you vary'¢, as is done in [9, Lemma 2.1], or it will follow
from the global functional equation presented below.
4 .1. The basic analytic properties
Our first goal is to show that these £-functions have nice analytic properties.137