2.7. EISENSTEIN SERIES AND THEIR L-FUNCTIONS 21
2.7.3. The only singularity of Eχ,c(ν; −) lying on Re(ν) 0 is a possible simple
pole at ν = 1. Indeed,
Lemma 2.13. The residue at ν = 1 of Eχ,c(ν; g) is
eχ,c,−1(g) =
δ(χ2
= 1, c = oF )
2 vol(F
×\A1)
vol(ZAGF \GA)
χ−1(det
g), g GA.
We have
2 vol(F
×\A1)
vol(ZAGF \GA)
=
DF
−1/2
RF
ζF (2)
.
with RF the residue of ζF (s) at s = 1.
Proof. (cf. [20, Proposition 6.13]) Let us examine the function m(ν) =
DF
−1/2
Aχ,c(ν) L(ν,
χ2)/L(ν
+
1,χ2),
which controls the singularity of the constant
term Eχ,c(ν; −) at ν = 1 by Lemma 2.10. The L-function L(ν, χ2) has a possible
simple pole at ν = 1 which occurs if and only if
χ2
= 1. When
χ2
= 1, the factor
Aχ,c(ν) has a zero at ν = 1 unless S(c) = ∅. Thus, the possible simple pole of
Eχ,c(ν; −) at ν = 1 occurs only when
χ2
= 1 and c = oF .
The Maass-Selberg relation applied to our Eisenstein series takes the form (cf.
[12, Formula (5.13)]) :
∧T
Eχ,c(σ)
2
vol(F
×\A1)−1
=
T
σ
σ
fχ,c) 2

T
−σ
σ
M(σ)fχ,c) 2
(2.28)
+ 2
δ(χ2
= 1)
{fχ,c)|M(σ)fχ,c)
log T
fχ,c)|M (σ)fχ,c)},σ(
where
∧T
is the truncation operator and M(σ) : I(χ||A
σ/2
)
I(χ−1||A
−σ/2
) is the
global intertwining operator and M (σ) its derivative at a point σ R {1}.
On the left-hand side, the norm is the
L2-norm
of
L2(ZAGF
\GA), while, on the
right-hand side, the norm (or the hermitian pairing) is considered for elements of
I(χ||A
σ/2
)
I(χ−1||A
−σ/2
) by the formula (2.23). Suppose
χ2
= 1, c = oF and write
m(ν) =
R
ν−1
+ O(ν 1). Then, by
M(ν)fχ,c)
= m(ν) f
(−ν)
χ−1,c
, the formula (2.28)
allows us to compute
lim
σ→1

1)2 ∧T
Eχ,c(σ)
2
vol(F
×\A1)−1
= 2R + O(T
−1)
on the one hand. On the other hand, from Lemma 2.10, Eχ,c(ν; g) =
fχ,c)(g)
+
m(ν) f
(−ν)
χ−1,c
(g), which yields eχ,c,−1(g) = R f
(−1)
χ−1,c
(g) = R χ−1(det g). By this, the
same limit is computed as R2 vol(ZAGF \GA). Thus,
R = 2 vol(F
×\A1)/vol(ZAGF
\GA).
This completes the first assertion. Since
R = Resν=1m(ν)
=
δ(χ2
= 1) DF
−1/2
A1,c(1)RF /ζF (2)
=
δ(χ2
= 1, c = oF ) DF
−1/2
RF /ζ(2),
we also have the second statement.
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