# Transition: Eisenstein series on adele groups

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Transition: Eisenstein series on adele groups
```(May 18, 2016)
Transition: Eisenstein series on adele groups
Paul Garrett [email protected]
http://www.math.umn.edu/egarrett/
[This document is http://www.math.umn.edu/˜garrett/m/mfms/notes 2013-14/12 2 transition Eis.pdf]
1.
2.
3.
4.
5.
6.
Moving automorphic forms from domains to groups
Iwasawa decompositions of GL2 (R) and GL2 (Qp )
Rewriting the GL2 Eisenstein series
Bruhat decomposition for GL2
Application: constant term of GL2 Eisenstein series
Application: Hecke operators on GL2 Eisenstein series
A simple Eisenstein series, for z ∈ H, Γ = SL2 (Z), is
Es (z) =
X
s
(Im γz) =
Γ∞ \Γ
X
1
2
c,d coprime
ys
|cz + d|2s
(with γ =
a
c
b
∗
, Γ∞ = {
d
0
∗
∗
∈ Γ})
This Eisenstein series can be rewritten to exhibit the role of p-adic groups SL2 (Qp ), thereby exhibiting
Hecke operators as integral operators, parallel to the relevance of the representation theory of SL2 (R) to
invariant differential operators. [1] The explicit nature of Eisenstein series allows vivid illustration of some
basic mechanisms.
For SL2 (Q), one can survive without this viewpoint. However, for SL2 over number fields with non-trivial
class groups, for SLn with n > 2, and for more general groups, the neo-classical elementary ideas sufficient
for SL2 (Q) fall short. Even in the simplest case of SL2 (Z), understanding the role of the p-adic groups
SL2 (Qp ) greatly simplifies and clarifies many classical issues.
That is, this shift in viewpoint is helpful, not merely a stylistic choice.
1. Moving automorphic forms from domains to groups
Automorphic forms on domains such as the upper half-plane H are usually introduced first because it is
easier to discuss them. However, this picture hides important features about the representation theory of
SL2 (R). It is easy to convert automorphic forms on H, both waveforms and holomorphic modular forms, to
automorphic forms on SL2 (R). [2]
[1.1] Γ-invariant functions and SO2 (R)-invariance Choice of (maximal) compact subgroup SO(2, R)
in SL2 (R) amounts to choice of basepoint i ∈ H, as SO2 (R) is the isotropy subgroup of i. Via the isomorphism
SL( R)/SO2 (R) → H by gSO2 (R) → g(i), and function f on H can be converted to a function F on SL2 (R)
by
F (g) = f (g(i))
[1] Despite contrary assertions in the literature, rewriting Eisenstein series, as opposed to more general automorphic
forms, on adele groups does not use Strong Approximation. Strong Approximation does make precise the relation
between general automorphic forms on adele groups and automorphic forms on SL2 and even on SLn , but rewriting
these Eisenstein series does not need this comparison. Indeed, Strong Approximation does not hold in the simplest
form for general semi-simple or reductive groups, but this does harm anything. Strong approximation does show that
there can be no other extension of the Eisenstein series to the adele group beyond that described here.
[2] This is an example of conversion to automorphic forms on reductive or semi-simple real Lie groups, of which
SL2 (R) is a small example.
1
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
The function F is right SO2 (R)-invariant:
(for g ∈ SL2 (R) and k ∈ SO2 (R))
F (gk) = f (gk(i)) = f (g(i)) = F (g)
Oppositely, any such right-SO( R)-invariant function F on SL2 (R) descends to a function f on the quotient
H by
f (z) = F (gz )
(for any gz ∈ SL2 (R) with gz (i) = z)
For f also left invariant by Γ = SL2 (Z) on H, the corrersponding function F on SL2 (R) is left Γ-invariant,
and vice-versa:
F (γg) = f (γg(i)) = f (g(i)) = F (g)
(for γ ∈ Γ and g ∈ SL2 (R))
[1.2] Automorphy condition and SO2 (R)-equivariance To motivate extension of the treatment of left
Γ-invariant function on H, of course holomorphic modular forms f on H are not quite left Γ-invariant, but
satisfy an automorphy condition
f (γz) = j(γ, z) · f (z)
(where j(αβ, z) = j(α, βz) · j(β, z))
2k
Apart
fromthe trivial cocycle j(g, z) = 1, the simplest example of cocycle is j(g, z) = (cz + d) for
a b
g=
. This extends to be a cocycle defined not merely on Γ × H but on SL2 (R) × H. Associate the
c d
function F on SL2 (R) given by
F (g) = j(g, i)−1 · f (g(i))
[1.2.1] Claim: The function F is left Γ-invariant on SL2 (R), and right SO2 (R)-equivariant by
F (gk) = j(k, i)−1 · F (g)
and k → j(k, i)−1 is a group homomorphism SO2 (R) → C× .
Proof: This is a direct computation. For γ ∈ Γ,
F (γg) = j(γg, i)−1 · f (γg(i)) =
−1
j(γ, g(i)) j(g, i)
j(γ, g(i)) · f (g(i))
= j(g, i)−1 · j(γ, g(i))−1 · j(γ, g(i)) · f (g(i)) = j(g, i)−1 · f (g(i)) = F (g)
For k ∈ SO2 (R),
F (gk) = j(gk, i)−1 · f (gk(i)) =
−1
−1
j(g, k(i)) j(k, i)
· f (g(i)) = j(g, i) j(k, i)
· f (g(i))
= j(k, i)−1 · j(g, i)−1 · f (g(i)) = j(k, i)−1 · F (g)
Finally, for k, h ∈ SO2 (R),
j(hk, i) = j(h, k(i)) · j(k, i) = j(h, i) · j(k, i)
proving that k → j(k, i) is a group homomorphism on SO2 (R).
///
[1.2.2] Remark: Half-integral weight automorphic forms have a cocycle on Γ × H which does not extend to
SL2 (R) × H. The obstruction to this extension defines a two-fold covering group of SL2 (R), the metaplectic
group, the group where half-integral weight automorphic forms live. This and other complications account
for our emphasis on integral-weight holomorphic modular forms, and on waveforms, in these notes.
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Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
[1.2.3] Remark: The seemingly different analytic conditions, holomorphy and being an eigenfunction for
the invariant Laplacian, both become eigenfunction conditions on the group SL2 (R). We will return to this
a little later.
[1.3] Conversion to automorphic forms on GL2 (R) To eventually accommodate Hecke operators, and
for many other reasons, the group GL2 is better than SL2 (R).
GL+
2 (R) = {g ∈ GL2 (R) : det g > 0}
preserves the upper half-plane H, and
GL+
2 (Z) = {g ∈ GL2 (Z) : det g > 0} = SL2 (Z)
since g ∈ GL2 (Z) has determinant ±1. Now the isotropy group of i ∈ H is
a
Z = {
0
0
a
: a ∈ R× }
As with SL2 (R), we have
+
+
+
H ≈ GL+
2 (R)/(isotropy subgroup of i) = GL2 (R)/ZR GL2 (Z)
+
For left GL+
2 (Z)-invariant functions f on H, the associated function F on GL2 (R) is
F (g) = f (g(i))
and F is left GL+
2 (Z)-invariant, right SO2 (R)-invariant, and Z-invariant. Since Z is the center of GL2 (R),
the function F is both right and left Z-invariant.
For f not left GL+
2 (Z)-invariant, but only meeting a cocycle condition f (γz) = j(γ, z) · f (z) with a cocycle
extending to GL+
2 (R) × H, such as
2k
j(g, z) = (cz + d)
−k
· (det g)
(with g =
a
c
b
d
∈ GL+
2 (R))
for weight 2k holomorphic modular forms, the corresponding function F on GL+
2 (R) is again
F (g) = j(g, i)−1 · f (g(i))
as for SL2 (R). Some cocycles are trivial on Z, some are not.
Further, the positive-determinant condition can be removed when the function f on H is extended to be a
function on H ∪ H, since GL2 (R) stabilizes the union of both half-planes, producing functions F on GL2 (R)
that are left GL2 (Z)-invariant, right O2 (R)-equivariant by some representation of O2 (R), and Z-equivariant
by some character Z → C× .
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Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
2. Iwasawa decompositions of GL2(R) and GL2(Qp)
That g ∈ GL2 (R) can be written as a product g = pk of upper-triangular p and orthogonal matrices k was
known for a long time before K. Iwasawa’s work in the 1940s, but after Iwasawa’s work this and other related
classical facts are understood as instances of a very general pattern that applies, not only to real or complex
matrix groups, but also to related p-adic matrix groups.
The pattern of an Iwasawa decomposition in a real or complex or p-adic matrix group is essentially [3]
whole group = (upper-triangular elements) · (maximal compact subgroup)
The subgroup P of upper-triangular matrices is a parabolic subgroup. The general definition of parabolic is
not essential here, nor is determination or certification that various subgroups are maximal compact. Rather,
we are acknowledging that our present examples fit into a larger pattern.
a b
[2.1] Iwasawa decomposition for GL2 (R) Let G∞ = GL2 (R). Given g = c d ∈ G∞ , choose
the element k ∈ K∞ = O2 (R) to right multiply by to put gk −1 into the group P∞ of upper-triangular real
matrices, by rotating the lower half (c d) of g in R2 into the form (0 ∗). Indeed, realizing that matrices in
K∞ are of the form
cos θ
sin θ
(for θ ∈ R)
− sin θ cos θ
in particular with the squares of the left column summing to 1, choose
k
−1
√ d
c2 +d2
√ −c
c2 +d2
=
∗
∗
!
with the right column of course completely determined by the left, so that
gk
−1
=
a
c
b
d
√ d
c2 +d2
√ −c
c2 +d2
∗
∗
!
=
c·
∗
+d·
√ d
c2 +d2
√ −c
c2 +d2
∗
∗
=
∗
0
∗
∗
as desired. For application to Eisenstein series, we only need the diagonal entries of gk −1 , and we easily
further compute the lower-right entry
gk
−1
=
a
c
b
d
√ d
c2 +d2
√ −c
c2 +d2
√ c
c2 +d2
√ d
c2 +d2
!
=
∗
0
√ ∗
c2 + d2
For g ∈ SL2 (R), the determinant-one condition gives
a
c
b
d
∈
√ 1
c2 +d2
0
√
∗
c2 + d2
· K∞ ⊂ P∞ · K∞
[3] Probably the group should be reductive or semi-simple, whose formal definitions do not concern us for the moment.
Rather, we note that this class of groups includes important examples: GLn (R), GLn (R), GLn (C), GLn (C), as well
as orthogonal groups, unitary groups, and other matrix groups defined by preservation of additional structure. The
class does not include upper-triangular matrices in GLn (R), although this subgroup is important in its own right.
This class of groups does also include p-adic versions of groups over R and C.
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Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
a b
For general g =
∈ G∞ ,
c d
a
c
b
d
∈
√det g
c2 +d2
0
√
∗
c2 + d2
· K∞ ⊂ P∞ · K∞
[2.2] Iwasawa decomposition for GL2 (Qp )
Even though the substance of a p-adic Iwasawa
decomposition is quite different from that of archimedean Iwasawa decompositions, the commonalities are
very useful.
The standard maximal compact subgroup Kv of Gv = GL2 (Qv ) with v a finite prime, in sharp contrast to
the orthogonal group as maximal compact subgroup of GL2 (R), is
Kv = GL2 (Zv ) = {p-adic integral matrices with determinants in Z×
v}
(v is finite prime p)
It is not so hard to prove that this is compact, but a little harder to prove that it is maximal compact. But,
for the moment, we don’t use either property, especially not the maximality.
The essential aspect of Qp is that, for any two x, y ∈ Q×
p , either x/y ∈ Zp or y/x ∈ Zp , since
Zp = {z ∈ Q : |z|p ≤ 1}
This contrasts violently with the classical situation of rational numbers x, y, where the classical
(=archimedean!)
notion of ”size” is very far from a sufficient determiner of divisibility. Thus, given
a b
g=
∈ GL2 (Qp ),
c d

a


 c


 a
c
1 0
∗ ∗
·
=
(for |c|v ≤ |d|v )
−c 1
0 ∗
d d
∗ ∗
b
1 −c
=
(for |c|v ≥ |d|v )
·
0 1
∗ 0
d
b
d
In both cases, the right-multiplying matrix is in Kv = GL2 (Zv ): in the first case |c/d|v ≤ 1 implies c/d ∈ Zv ,
and in the second |d/c|v ≤ 1 implies d/c ∈ Zv . In the first case, the resulting product is already in the group
Pv ofupper-triangular
matrices in Gv . In the second, further right-multiplication by the long Weyl element
0 1
w=
puts the product into Pv . This proves that the p-adic Iwasawa decomposition succeeds.
1 0
series:

a


 c


 a
c
entries in the Iwasawa decomposition will be needed in rewriting Eisenstein
b
1 0
a − bc
∗
d
=
(for |c|v ≤ |d|v )
·
0
d
d
−c 1
dd
b
−c 1
− c +b ∗
·
=
(for |c|v ≥ |d|v )
d
1 0
0
c
That is,

a



c


 a
c
b
d
b
d
∈
∈
a − bc
d
0
∗
d
c +b
0
· Kv
∗
· Kv
c
(for |c|v ≤ |d|v )
(for |c|v ≥ |d|v )
[2.2.1] Remark: It is very convenient that the shape of the parabolic Pv is the same for both archimedean
and non-archimedean v, while the shape of the (maximal) compact subgroup Kv is significantly different.
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Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
3. Rewriting the GL2 Eisenstein series
∗
∈ Γ}, the simplest waveform-type Eisenstein series on the domain H
∗
X
Es (z) =
(Im γz)s
(for z ∈ H and Re (s) > 1)
With Γ = SL2 (Z) and Γ∞ = {
is the usual
∗
0
Γ∞ \Γ
This Eisenstein series on the domain H is easily converted to a left Γ-invariant, right SO2 (R)-invariant
function on the group GL+
2 (R), still called an Eisenstein series, by
(for g ∈ GL+
2 (R))
Esgroup (g) = Es (g(i))
The basepoint i ∈ H is chosen since K∞ = SO2 (R) is its isotropy group, which immediately explains the
right SO2 (R)-invariance.
In fact, Es can be extended to an automorphic form on the union of upper and lower half-planes by
Es (−z) = Es (z), noting that
−1 0
(z) = −z
0 1
∗ ∗
Thus, now letting Γ = GL2 (Z) and Γ∞ = {
∈ Γ}, we have an expression for the extended Eisenstein
0 ∗
series that is of the same form:
X Im γz s
Es (z) =
(for z ∈ H ∪ H)
Γ∞ \Γ
The Eisenstein series on the whole group G∞ = GL2 (R) is
Esgroup (g) = Es (g(i))
(for g ∈ GL2 (R))
As above, let v be an index for completions [4] of Q, with Qv the v th
completion, and Zv the v-adic integers for v a prime, with Q∞ = R, and | · |∞ the usual real absolute value.
The latter is distinguished from genuine primes by the convention of calling it the infinite prime, in analogy
with a more legitimate use of this in the function-field setting.
[3.1] The localized rewrite
For each place v, finite and infinite, using the v th Iwasawa decomposition Gv = Pv Kv , define a function ϕv
on Gv by

for k ∈ GL2 (Zv )
(for v non-archimedean)

a s

a b
ϕv
·k = cos θ sin θ
0 d

d v
 for k =
∈ O(2) (for v archimedean)
− sin θ cos θ
where in all cases a, d ∈ Q×
v and b ∈ Qv . By design, each ϕv is invariant under the center Zv of Gv . Define
a function on the adele group GA = GL2 (A) by
ϕ =
O
ϕv
(meaning ϕ({gv }) =
Q
v
ϕv (gv ), where gv ∈ Gv )
v
[4] Another tradition for refering to p-adic completions, as well as to R, is as places of Q. Also, sometimes ∞ is called
the infinite prime, while the actual primes p are finite primes.
6
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
[3.1.1] Claim: (The product formula) For x ∈ Q× ,
Y
|x|v = 1
(product over all p-adic norms as well as R)
v≤∞
Proof: Since the assertion is multiplicative, it suffices to prove the product formula for units and for primes.
All norms of ±1 are 1, and all norms but the archimedean and pth of a prime p are 1. Since |p|∞ = p and
|p|p = p1 , we have the product formula.
///
The product formula Q
shows shows that ϕ is left PQ -invariant: using the Iwasawa decomposition locally
everywhere, with k ∈ v Kv ,
A ∗
a ∗
a ∗
ϕ(
k) = |Aa/Dd|s = |A/D|s · |a/d|s = |A/D|s · ϕ(
k)
0 D
0 d
0 d
Let
a
ZA = {
0
0
a
: a, a−1 ∈ A}
[3.1.2] Theorem: For g∞ ∈ GL2 (R), acting as usual on H ∪ H,
X
Es (g∞ · i) =
ϕ(γ · g∞ )
γ∈PQ \GQ
This expression gives a left GQ -invariant, ZA -invariant function (still denoted Es ) on the adele group
GA = GL2 (A):
X
ϕ(γ · g)
γ∈PQ \GQ
Proof of this will occupy the rest of this section.
[3.1.3] Remark: The notations Esgroup and Esadelic are not standard, but have the obvious descriptive utility.
In fact, we will revert to writing Es for the Eisenstein series on GA = GL2 (A).
[3.2] Disambiguation
The immediate question arises of evaluation of ϕ(γ · g∞ ). One point is that
GQ = GL2 (Q) should not be considered as only a subgroup of G∞ = GL2 (R), but also a subgroup of every
Gv = GL2 (Qv ). Thus, GQ is best considered as imbedded on the diagonal in GA . Adding a temporary
notational burden for clarity, for each place v let jv : GL2 (Q) → GL2 (Qv ) be the natural injective map, and
let
Y
Y
j =
jv : GL2 (Q) −→
GL2 (Qv )
v
v
be the natural diagonal map to the product. Then
ϕ(γ · g∞ ) = ϕ∞ (j∞ (γ) · g∞ ) ·
Y
ϕ∞ (jv (γ))
v<∞
That is, in the definition of ϕ, the archimedean g∞ ∈ G∞ does not interact with the non-archimedean groups
Gv = GL2 (Qv ).
[3.3] Comparison to classical formulation
The familiar Eisenstein series Es (z) can be obtained from
the above by reverting to a form that does not refer to anything p-adic or adelic. That is, we claim that
s
ϕ∞ (g∞ ) = Im (g∞ · i)
(for g∞ ∈ GL2 (R))
7
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
with GL2 (R) acting on H ∪ H. The argument is about the Iwasawa decomposition, namely, that any element
of GL2 (R) can be written as a product of upper-triangular and orthogonal matrices. Indeed, given a matrix
in GL2 (R), right multiplication by an orthogonal matrix can be viewed as rotating the bottom row. This
suggests the appropriate orthogonal group element: formulaically,
a
c
b
d
√ c
c2 +d2
√ d
c2 +d2
√ d
c2 +d2
√ −c
c2 +d2
·
c2 +d2
!
=
√ac+bd
c2 +d2
2
2
c
√ +d
c2 +d2
0
Thus,
ϕ∞
a
c
b
d
= ϕ∞
√ 1
c2 +d2
0
√
∗
c2 + d 2
!
=
√ 1
c2 +d2
√
0
∗
c2 + d2
1/√c2 + d2 s
1 s
= √
= 2
c + d2
c2 + d2
On the other hand, a familiar computation gives
a
c
Im
b
d
(i) =
1
1 ai + b −ai + b = 2
= 2
−
2i ci + d −ci + d
c + d2
c + d2
Since γ ∈ GL2 (Z) maps to GL2 (Zv ) at all finite places v,
(for γ ∈ GL2 (Z) and finite place v)
ϕv (γ) = 1
Thus,
X
X
ϕ(γ · g∞ ) =
γ∈PZ \GL2 (Z)
X
ϕ∞ (γ · g∞ ) · 1 =
γ∈PZ \GL2 (Z)
s
Im γg∞ · i
γ∈PZ \GL2 (Z)
Taking the popular choice
g∞ =
1
0
x
1
√
y
0
0
(with x ∈ R and y > 0)
√1
y
produces Es (x + iy) on H, as claimed.
[3.4] Well-definedness on PQ \GQ We should show that ϕ(γg∞ ) depends only upon the coset PQ γ. Any
γ ∈ GL2 (Q) is in GL2 (Zv ) for almost all v, since the entries are in Zv for almost all v, and theQdeterminant
is a v-adic unit for almost all v, so the inverse is v-integral also. Thus, in an infinite product v<∞ ϕv (γ),
all but finitely-many factors are 1.
Let χv be the character on upper-triangular v-adic matrices Pv given by
χv
a
0
b
d
a s
= d v
(with a, d ∈ Q×
v and b ∈ Qv )
The usual maximal compact [5] subgroups Kv of the groups GL2 (Qv ) are
Kv =

 GL2 (Qv )

O(2)
(for v finite)
(for v real)
The description of ϕv can be rewritten more succinctly as
(for p ∈ Pv and k ∈ Kv )
ϕv (pk) = χv (p)
[5] We will not use the fact that these are maximal among compact subgroups, despite refering to them as such.
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Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
For g∞ ∈ GR , γ ∈ GQ , and β ∈ PQ , keeping in mind that GQ maps to all groups Gv , not just to G∞ ,
Y
ϕ(β · γ · g∞ ) = ϕ∞ (β · γ · g∞ ) ·
ϕv (β · γ)
v<∞
At the archimedean place, let γg∞ = pk be an Iwasawa decomposition in Gv , with p ∈ Pv and k ∈ Kv . We
see the left equivariance of ϕv by χv , namely,
ϕv (βγg∞ ) = ϕv (βpk) = χv (β · p) = χv (β) · χv (p) = χv (β) · ϕv (pk) = χv (β) · ϕv (γg∞ )
Similarly, but now without g∞ playing any role, at a finite place v, let γ = pk be an Iwasawa decomposition
in Gv , with p ∈ Pv and k ∈ Kv . We see the left equivariance of ϕv by χv :
ϕv (βγ) = ϕv (βpk) = χv (β · p) = χv (β) · χv (p) = χv (β) · ϕv (γ)
Putting all these local equivariances together,
Y
ϕ(β · γ · g∞ ) =
χv (β) · ϕv (γ · g∞ )
(for β ∈ PQ , γ ∈ GQ , and g∞ ∈ G∞ )
v
By the product formula,
Y
v
χv
a
0
b
d
=
Y a s
= 1
d v
v
(for a/d ∈ Q× )
That is, we have the left invariance
ϕ(β · γ · g∞ ) = ϕ(γ · g∞ )
(for β ∈ PQ , γ ∈ GQ , and g∞ ∈ G∞ )
[3.5] Bijection
of cosets Changing from the Γ and Γ∞ notation, let GZ = GL2 (Z) and
PZ = {
∗
0
∗
∗
∈ GZ }, we claim that
PQ \GQ ≈ PZ \GZ
More precisely, we claim that the obvious map PZ \GZ → PQ \GQ by PZ g → PQ g is a surjection. It is
an injection because PZ = GZ ∩ PQ . That is, we claim that that every coset PQ h with h ∈ GQ has a
representative in GZ = GL2 (Z). The argument attaches meaning to both these coset spaces, and will
thereby give the bijection.
The coset space PQ \GQ is in bijection with the set of lines in Q2 , respecting the right multiplication by GQ ,
because GQ is transitive on these lines, and PQ is the stabilizer of the line {(0 ∗)}. Next, each line in Q2
meets Z2 in a free rank-one Z-module generated by a primitive vector (x, y), meaning that gcd(x, y) = 1.
Call such a Z-module a primitive Z-line in Q2 . The collection of lines in Q2 is thus in bijection with primitive
Z-lines in Z2 , by sending a line to its intersection with Z2 . The group SL2 (Z) ⊂ GL2 (Z) is already transitive
on primitive Z-lines: for gcd(x, y) = 1, let b, d ∈ Z be such that
bx + dy = gcd(x, y) = 1
Then
(x y) ·
y
−x
b
d
= (0 1)
That is, any primitive vector can be mapped to (0 1), so the action of SL2 (Z) is transitive on primitive
vectors, hence on primitive Z-lines. Thus, certainly the slightly larger group GL2 (Z) is transitive on primitive
vectors. The stabilizer subgroup of the primitive Z-line spanned by (0, 1) in GL2 (Z) is
a b
PZ = {
: a, d ∈ Z× , b ∈ Z}
0 d
9
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
This proves the bijection of coset spaces.
[3.6] The essential conclusion From
X
Es (g∞ · i) =
ϕ∞ (γ · g∞ )
(with g∞ ∈ GL2 (R))
γ∈PZ \GZ
from the bijection of coset spaces, and from the well-definedness of ϕ left modulo PQ , we have the desired
re-expression of the Eisenstein series
X
Es (g∞ · i) =
ϕ(γ · g∞ )
(with g∞ ∈ GL2 (R))
γ∈PQ \GQ
This is the first main point, and there are further advantages to the viewpoint. Computation of the constant
term is the first illustration, after the preparation of the next section.
4. Bruhat decomposition for GL2
Expressing the coset space PZ \GZ in terms of rational matrices PQ \GQ , rather than integral matrices,
simplifies natural choices of representatives immediately, as we will see. Further, more significantly, this
makes visible the unwinding and Euler factorization of the Bruhat-cell summands of the constant term, as
we see in the next section.
[4.1] Bruhat decomposition The Bruhat decomposition for GL2 (k) for any field k is [6]
Gk = Pk t Pk wNk
(with w =
0
1
−1
0
and N =
1
0
∗
)
1
This purely algebraic fact has natural extensions to GLn and the classical groups. For G = GL2 , the Bruhat
decomposition is easy to prove: of course, the little cell PQ consists of matrices with c = 0, and we must
claim that the big cell Pk wNk is exactly
Pk wNk
Indeed, given g =
a
c
b
d
b
d
∈ GL2 (k) : c 6= 0}
g
a
= {
c
with c 6= 0,
1
0
−d/c
1
=
a
c
b
d
1
0
−d/c
1
=
∗
∗
∗
0
∈ Pk w
That proves the Bruhat decomposition in this simple case.
This expression for Gk exhibits a remarkably simple set of representatives for Pk \Gk when rational rather
than only integral entries are allowed:
Pk \Gk = Pk \Pk ∪ Pk \ Pk wNk
≈ {1} ∪ w · w−1 Pk w ∩ Nk \Nk ≈ {1} ∪ wNk
[6] The validity of this decomposition for GL (k) or GL (k), and other specific groups, was known long before F.
2
2
Bruhat’s work in the 1950s. Nevertheless, it is good practice to refer to the GL2 case in terms indicated the extension
to the general case.
10
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
where the bijection of cosets is
Pk wn = w · (w−1 Pk w · n) ←− w · (w−1 Pk w ∩ Nk ) · n
Indeed, with H, N subgroups of a larger group, H · n = H · n0 for n ∈ N , if and only if n0 n−1 ∈ H. Also,
n0 n−1 ∈ N , so
H · n = H · n0 ⇐⇒ (H ∩ N ) · n = (H ∩ N ) · n0
(for n, n0 ∈ N )
[4.2] Another comparison
Another computation verifies our rewrite of the Eisenstein series, paying
attention to the Bruhat-cell parametrization. In principle, this computation is unnecessary, but it is
informative.
Using the Bruhat decomposition, the sum defining the Eisenstein series is
X
Es (g∞ ) =
X
ϕ(γ · g∞ ) = ϕ(g∞ ) +
ϕ(γ · g∞ )
γ∈wNQ
γ∈PQ \GQ
That is, the summands can be parametrized by {1} and NQ ≈ Q. This will be important in computing the
constant term in the next section.
As ϕ∞ is right O(2)-invariant and center-invariant, by the Iwasawa decomposition G∞ = P∞ · K∞ we can
take
1 x
y 0
g∞ =
(with x ∈ R and y > 0)
0 1
0 1
With this g∞ , the term γ = 1 is
ϕ(g∞ ) = ϕ∞ (g∞ ) ·
Y
ϕv (1) = |y|s · 1 = |y|s
v<∞
For the big cell contribution to the sum, we need to compute the archimedean part
ϕ∞ w
1
0
t
1
1
0
x
1
y
0
0
1
(for t ∈ Q)
and the non-archimedean
ϕv w
1
0
t
1
(for finite v, with t ∈ Q)
In the archimedean case,
w
1
0
t
1
1
0
x
1
y
0
0
1
=
0
y
−1
x+t
Right multiplying by a suitable orthogonal matrix rotates the bottom row to put the result in PR , namely,
0
y
−1
x+t
x+t
(x+t)2 +y 2
√ −y2 2
(x+t) +y
√
y
(x+t)2 +y 2
√ x+t2 2
(x+t) +y
√
√
!
=
y
(x+t)2 +y 2
0
∗
p
(x + t)2 + y 2
Thus,
ϕ∞ w
1
0
t
1
1
0
x
1
y
0
11
0
1
= s
y
(x + t)2 + y 2 ∞
!
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
For finite v, adjust the given matrix by right multiplication by GL2 (Zv ) to make the result upper-triangular.
For t ∈ Zv , the matrix is already in GL2 (Zv ), so
ϕv
−1
0
0
1
1
0
t
1
(for finite v, with t ∈ Q ∩ Zv )
= 1
For t 6∈ Zv , necessarily t−1 ∈ Zv . Thus, the matrix
can be multiplied by
1
−t−1
0
1
−1
0
0
1
1
0
t
1
−1
t
=
0
1
in GL2 (Zv ) to obtain
−1
t
0
1
·
1
−t−1
0
1
=
t−1
0
∗
t
Thus,
ϕv
0
1
−1
0
1
0
t
1
= |t|−2s
v
(for finite v, with t 6∈ Q ∩ Zv )
That is,
ϕv
−1
0
0
1
1
0
t
1
=

 1

(for |t|v ≤ 1)
|t|−2s
v
(for |t|v ≥ 1)
Combining the archimedean and non-archimedean,
ϕ w
1
0
t
1
1
0
x
1
y
0
0
1
= 
s Y  1
y
·
(x + t)2 + y 2 ∞ v<∞  −2s
|t|v
(for |t|v ≤ 1)
(for |t|v > 1)
Since o is a principal ideal domain with units ±1, we can easily parametrize t ∈ Q in a fashion conforming
to evaluation of the displayed expression. Namely, write t = d/c with c, d relatively prime, modulo ±1. Note
that for relatively prime integers c, d



(for |d/c|v ≤ 1)
(for pv not dividing c)
(for pv not dividing c)
 1
 1
 1
=
=


 2s
(for |d/c|v > 1)
|d/c|−2s
|d/c|v−2s (for pv dividing c)
|c|v (for pv dividing c)
v
That is, by the product formula,

Y 
v<∞

1
(for |d/c|v ≤ 1)
=
|d/c|−2s
v
(for |d/c|v > 1)
Y
v<∞
|c|2s
v =
1
|c|2s
∞
Then, once again, we recover the expected:
s
s
y
ys
1
y
=
=
·
(cx + d)2 + (cy)2 ∞
|cz + d|2s
(x + dc )2 + y 2 ∞ |c|2s
∞
Again, in principle the above computation is unnecessary, but it is informative to see the details of the
reversion to a classical form.
12
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
5. Application: constant term of GL2 Eisenstein series
An immediate use of the localized rewrite of the Eisenstein series is computation of the constant term
presenting each Bruhat cell’s contribution as an Euler product, by unwinding the integral defining the constant
term.
[5.1] Transition for the constant term
In these simplest situations, the constant term of any kind of
modular/automorphic form is the zero-th Fourier component, separating variables in the x + iy coordinates:
in most elementary, though far from optimally explanatory, terms,
Z
cP f (iy) = (constant term of f )(iy) =
1
f (x + iy) dx
0
Since f (x + iy) is periodic in x, we obtain the same outcome integrating over any interval [a, a + 1] in place
of [0, 1]. In fact, the integral is over the quotient R/Z
Z
cP f (iy) =
f (x + iy) dx
R/Z
Further,
integral can be written as an integral over a quotient of a subgroup of G∞ , namely, with
the 1 ∗
N ={
},
0 1
Z
cP f (iy) =
f (n · iy) dn
NZ \NR
where the Haar measure on NR is really just the usual measure on R.
b = A.
[5.1.1] Claim: R + Q∆ + Z
Proof: An adele x fails to be in Zv only for v in a finite set S of finite places v = p. At such v, we can write
xv =
a−1
a−`
+ ... +
+ ao + a1 p1 + . . .
`
p
p
The truncated sum
yv =
(with ai ∈ Z)
a−`
a−1
+ ... +
+ ao
`
p
p
is a rational number, and is v 0 -integral at all other finite v 0 . Thus,
X
y =
yv
v∈S
b
is a rational number, and x − y is everywhere locally integral, where y ∈ Q∆ . That is, x − y ∈ R + Z.
///
[5.1.2] Remark: The point of the corollary is not to get from A/Q back to R/Z, but to get from the classical
quotient R/Z to A/Q.
[5.1.3] Remark: In fact, (additive approximation) asserts that R + Q∆ is dense in A, where Q∆ is the
diagonal copy. Equivalently, the diagonal copy of Q in the finite adeles Afin is dense.
b Two elements r, r0 ∈ R have the same image in A/Q if and only if
Thus, A/Q has representatives in R + Z.
b The latter intersection is the diagonal copy of Z, since a rational number that is in Zv
r − r0 ∈ Q ∩ (R + Z).
for all v < ∞ is in Z. Thus, R/Z injects to A/Q.
13
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
Applying this to the coordinate in the subgroup N of GA = GL2 (A),
NZ · NR · NA ∩ Kfin
= NA
and
NZ \ NR · (NA ∩ Kfin ) = NQ \NA
For right Kfin -invariant f on the adele group, for n∞ ∈ N∞ and no ∈ Nfin ,
f (n∞ · g∞ ) = f (n∞ · g∞ · no ) = f (n∞ · no · g∞ )
because G∞ and Gfin commute as subgroups of GA . Thus, evaluating the constant term on g∞ ,
Z
Z
Z
cP f (g∞ ) =
f (n∞ · g∞ ) dn∞ =
f (n∞ · g∞ · no ) dno dn∞
NZ \NR
Z
NZ \NR
NA ∩Kfin
Z
Z
f (n∞ · no · g∞ ) dno dn∞ =
=
NZ \NR
NA ∩Kfin
f (n · g∞ ) dn
NQ \NA
[5.2] Unwinding With
Es (g∞ ) =
X
ϕ(γ · g∞ )
γ∈PQ \GQ
the constant term of Es along P is the adelic integral
Z
cP Es (g) =
Es (ng) dn
NQ \NA
Parametrizing PQ \GQ via the Bruhat decomposition makes computation nearly trivial:
Z
Z
Z
X
X
X
Es (ng) dn =
ϕ(γng) dn =
NQ \NA
NQ \NA γ∈P \G
Q Q
w∈PQ \GQ /NQ
ϕ(γng) dn
NQ \NA γ∈P \P wN
Q Q
Q
By the Bruhat decomposition, PQ \GQ /NQ has exactly two representatives, 1, w, and the constant term
becomes, upon unwinding the second sum-and-integral,
Z
Z
Z
Z
X
ϕ(ng) dn +
ϕ(wγng) dn =
ϕ(ng) dn +
ϕ(wng) dn
NQ \NA
NQ \NA γ∈N
Q
NQ \NA
NA
Because ϕ is left NA -invariant, the first of the two summands is
Z
ϕ(ng) dn = ϕ(g) · vol (NQ \NA )
(the small Bruhat cell contribution)
NQ \NA
Since the integral in the second summand unwound, it factors over primes
Z
Y Z
ϕ(wng) dn =
ϕv (wngv ) dn
NA
v≤∞
Nv
[5.3] Essential features of p-adic integrals We do not need many formulaic details about integrals on
×
Qp , since we will only consider Z×
p -invariant integrands f (x), that is, f (η · x) = f (x) for all η ∈ Zp and
x ∈ Qp .
14
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
It is reasonable to normalize the additive Haar measure on Qp so that the compact, open subgroup Zp has
total measure 1.
The pn distinct compact, open subgroups a + pn Zp ⊂ Zp are translates of each other, and mutually disjoint,
so the total measure of a + pn Zp is p−n for a ∈ Zp . By translation-invariance, the total measure of a + pn Zp
is p−n for any a ∈ Qp .
For η ∈ Z×
p,
η · (a + pn Zp ) = ηa + pn Zp
Thus, multiplication by units preserves Haar measure. [7] The expression
Z×
p = Zp − pZp
1 −n
1
n ×
. Thus, with continuous
shows that the measure of Z×
p is 1 − p . Similarly, the measure of p Zp is (1 − p p
×
Zp -invariant f ,
Z
f (x) dx =
Zp
∞ Z
X
`=0
p`
f (x) dx =
Z
×
p
∞
X
Z
`
f (p ) ·
1 dx =
p`
`=0
×
p
Z
∞
X
`=0
1
f (p` ) · (1 − ) p−`
p
[5.4] Evaluation of local factors: non-archimedean case
For g ∈ G∞ , so that gv = 1, the finiteprime local factors in the Euler product for the big Bruhat cell are readily evaluated, as follows. Above, we
computed

(for |t|v ≤ 1)
 1
1 t
=
ϕv w
0 1
 −2s
(for |t|v > 1)
|t|v
With the v-adic factor corresponding to prime p, the v-adic local factor is
Z
Z
|t|−2s
dt = 1 +
v
1 dt +
|t|v ≤1
|t|v >1
= 1+ 1−
∞
X
|p−` |−2s
·
v
`=1
Z
1 dt = 1 +
p−` Z×
p
∞
X
(p` )−2s · p`−1 (p − 1)
`=1
1 p1−2s
1 − p1−2s + p1−2s − p−2s
1 − p−2s
ζv (2s − 1)
=
=
=
1−2s
1−2s
1−2s
p 1−p
1−p
1−p
ζv (2s)
where ζv (s) is the v th Euler factor of the zeta function. Thus, the finite-prime part of the big-cell summand
is ζ(2s − 1)/ζ(2s).
[5.5] Evaluation of local factors: archimedean case The archimedean factor of the big-cell summand
of the constant term is
Z s
y
dt =
2
2
R (x + t) + y ∞
= y 1−s ·
=
y 1−s
√
Z
Z
ys
1
1
s dt = y s ·
s dt = y 1+s ·
dt
2
2
2
2
2
2 s
R (x + t) + y
R t +y
R (ty) + y
Z
Z Z ∞
Z Z ∞
2
1 du
1
y 1−s
y 1−s
u(1+t2 ) s du
dt
=
·
e
u
dt
=
·
eu+t us− 2
dt
2
s
Γ(s) R 0
u
Γ(s) R 0
u
R (t + 1)
Z
1
π Γ(s − 12 )
π −(s− 2 ) Γ(s − 12 )
ζ∞ (2s − 1)
= y 1−s ·
= y 1−s ·
Γ(s)
π −s Γ(s)
ζ∞ (2s)
(with ζ∞ (s) = π −s/2 Γ(s/2))
[7] Quite generally, a compact group of automorphisms of a topological group must preserve the Haar measure on
the latter.
15
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
[5.6] Conclusion of constant-term computations
Thus, with ξ(s) the completed zeta function
ξ(s) = ζ∞ (s) · ζ(s), the constant term of Es is
cP Es (x + iy) = y s +
ξ(2s − 1) 1−s
·y
ξ(2s)
The present point is that rewriting the Eisenstein series as an automorphization of a product of local data
makes the computation of the constant term far more natural, and more genuinely representative of the
corresponding computation for larger groups.
6. Application: Hecke operators on GL2 Eisenstein series
The rewritten Eisenstein series will show that the Hecke operators are not global things, but are local, just
acting on the local components ϕv . Indeed, the local components ϕv are eigenfunctions for the local version
of Hecke operators, with eigenvalues depending on the parameter s.
The pth Hecke operator Tp on weight-0 automorphic
[6.1] Classical description of Hecke operators
forms f for Γ = GL2 (Z) is
Tp f (z) =
X
f (γ · z)
(where Θp = integer matrices with det = p)
γ∈Γ\Θp
with action [8] by linear fractional transformations
a
c
b
d
:z→
az + b
.
cz + d
[6.2] Hecke operators on rewritten Eisenstein series Directly computing, with g∞ ∈ G∞ ,
Tp Es (g∞ ) =
X
Q
v<∞ jv .
ϕ γ · j∞ (δ) · g∞
δ∈Γ\Θp γ∈PQ \GQ
δ∈Γ\Θp
Let jo =
X
X
Es (j∞ (δ) · g∞ ) =
Replace γ by γ · δ −1 in GQ , to obtain
ϕ γ · jo (δ −1 ) · g∞
X
X
δ∈Γ\Θp γ∈PQ \GQ
The finite-prime factor jo (δ −1 ) commutes with the archimedean-prime factor g∞ , so this is
X
X
ϕ γ · g∞ · jo (δ −1 ) =
δ∈Γ\Θp γ∈PQ \GQ
X
X
ϕ∞ (j∞ (γ) · g∞ ) ·
Y
ϕv (jv (γ · δ −1 ))
v<∞
δ∈Γ\Θp γ∈PQ \GQ
At all finite places v 0 but v ∼ p, δ −1 is in the local maximal compact Kv0 = GLv0 (Zv0 ), so ϕv0 (γ·δ −1 ) = ϕv0 (γ)
for v 0 6= v. Thus, suppressing jv in the notation,
P
Tp Es (g∞ ) =
X
ϕ(γ · g∞ ) ·
γ∈PQ \GQ
δ∈Γ\Θp
ϕv (γ · δ −1 )
ϕv (γ)
[8] That is, the weight-0 situation allows us to avoid worry over what to do with the determinant in GL . In the
2
classical holomorphic case, the so-called slash operator is a normalization that accommodates this, usually without
explanation or motivation.
16
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
Thus, the pth Hecke operator’s effect is local at v ∼ p. Further, this situation correctly suggests that we
should hope that ϕv is an eigenfunction for the effect of Tp , with eigenvalue depending on the complex
parameter s.
[6.3] Hecke operators as integral operators
Continue to let v correspond to prime p. The function
ϕv on Gv = GL2 (Qv ) is left Pv -equivariant by χv , and right Kv -invariant. Each of the right translates
g → ϕv (g · δ −1 ) with g ∈ Gv retains the left Pv , χv -equivariance, but cannot be expected to retain right
Kv -invariance.
Nevertheless, we claim that the sum over δ ∈ Γ\Θp recovers the right Kv -invariance. That is, apparently,
Γ\Θp , or its image projected to Gv , is stable under right multiplication by Kv . As it stands, this doesn’t
make sense, since Θp itself (projected to Gv ) is not literally stable under right multiplication by Kv .
e v be the v-adic analogue of Θp , namely,
As on other occasions, the necessary claim suggests itself: let Θ
elements of Gv = GL2 (Qv ) with entries in Zv and determinant of p-adic ord 1. Then we must claim that
e −1
the natural map Θ−1
p → Θv /Kv induces a bijection
e −1
Θ−1
p /Γ −→ Θv /Kv
e v induces a bijection
Equivalently, inverting, Θp → Kv \Θ
ev
Γ\Θp −→ Kv \Θ
e v has the same representatives as Γ\Θp , namely, [9]
Indeed, Kv \Θ
1 b
p 0
(with b ∈ Z, 0 ≤ b < p)
0 p
0 1
Thus, giving Kv total measure 1, using the right Kv -invariance of ϕv ,
Z
X
X
ϕv (g · δ −1 ) =
ϕv (g · h) =
δ∈Γ\Θp
ϕv (g · h) dh
e −1
Θ
e −1 /Kv
h∈Θ
e −1
Letting η be the characteristic function of Θ
v , this integral is an integral operator attached to the right
translation action of Gv on functions on Gv :
Z
X
ϕv (g · δ −1 ) =
η(h) ϕv (g · h) dh = (η · ϕv )(g)
Gv
δ∈Γ\Θp
The integral expression makes right Kv -invariance clear, by changing variables in the integral:
Z
Z
Z
η(h) ϕv (gk · h) dh =
η(k −1 h) ϕv (g · h) dh =
η(h) ϕv (g · h) dh
(for k ∈ Kv )
Gv
Gv
Gv
b
e v , if gcd(a, c) = 1, then either a or c is
in Θ
d
1
0
0
1
1 b0
in Z×
.
Thus,
left
multiplication
by
either
or
puts
g
into
the
form
. Necessarily
v
−c/a 1 1 −a/c
0 d0
00
1 b
ordv d0 = 1, so further left multiplication gives the form
. Since Zv /pZv ≈ Z/pZ, further left multiplication
0 p
1 ∗
by
∈ Kv gives the indicated representatives parametrized by b. When gcd(a, c) = p, a similar argument
0 1
p b0
gives the form
, and now the b0 entry can be made 0.
0 1
[9] The v-adic argument is easier than that over Z: given g =
17
a
c
Paul Garrett: Transition: Eisenstein series on adele groups (May 18, 2016)
since η is left and right Kv -invariant.
[6.4] Hecke eigenvalues Now we can prove that ϕv is an eigenvector for the localized version of Tp (with
v ∼ p), and compute its eigenvalue. The v-adic Iwasawa decomposition is Gv = Pv ·Kv . Thus, up to constant
multiples, there is a unique left Pv , χv -equivariant, right Kv -invariant function on Gv . Thus, every such is a
multiple of ϕv .
e −1 , necessarily η · ϕv = λs · ϕv for some λs ∈ C. To
In particular, with η the characteristic function of Θ
v
determine λs , it suffices to evaluate at g = 1, using ϕv (1) = 1. Thus,
Z
Z
λs =
η(h) ϕv (h) dh =
e −1
Θ
v /Kv
Gv
=
X
b
χv
1
0
b
p
−1 + χv
p
0
0
1
−1 X
ϕv (h) dh =
δ∈Γ\Θp
= p · χv
1
0
1 s p−1 s
= p · −1 + = p1−s + ps
p
1 v
v
This is the pth Hecke eigenvalue of Es :
Tp Es = (p1−s + ps ) · Es
18
ϕv (δ −1 )
∗
p−1
+ χv
p−1
0
0
1
```
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