Eichler-Shimura isomorphism and group cohomology on arithmetic groups

0.18 MB·

by  Santiago Molina Blanco

1
EICHLER-SHIMURA ISOMORPHISM AND GROUP
COHOMOLOGY ON ARITHMETIC GROUPS
SANTIAGOMOLINA
7
1
0 Introduction
2
The Eichler-Shimura isomorphism establishes a bijection between the space of
n
modular forms and certain cohomology groups with coefficients in a space of poly-
a
J nomials. Moreprecisely,letk ≥2beanintegerandletΓ⊆SL2(Z)beacongruence
3 subgroup, then we have the following isomorphism of Hecke modules

(0.1) M (Γ,C)⊕S (Γ,C)≃H1(Γ,V(k)∨),
] k k
T
whereV(k)∨ isthedualoftheC-vectorspaceofhomogenouspolynomialsofdegree
N
k−2,M (Γ,C)isthe spaceofmodularformsofweightk andS (Γ,C)⊂M (Γ,C)
k k k
.
h is the subspace of cuspidal modular forms (see [6, Thm. 8.4] and [4, Thm. 6.3.4]).
t This isomorphism can be interpreted in geometric terms. Indeed, a modular
a
m form of weight k can be interpreted as a section of certain sheaf of differential
forms on the open modular curve attached to Γ. With this in mind, the Eichler-
[
ShimuraisomorphismcanbeobtainedcomparingdeRhamandsingularcohomology,

1 noticing that the singular cohomology of the open modular curve is given by the
v
group cohomology H•(Γ,V(k)∨). The aim of this paper is to omit this geometric
1
interpretation and to provide a new group cohomologicalinterpretation.
1
6 The restriction of the Eichler-Shimura isomorphism to the spaces of cuspidal
0 modular forms is given by the morphisms
0
1. ∂± :Sk(Γ,C)−→H1(Γ,V(k)∨),
0 where
7
γz0 γz0
1 ∂±(f)(γ)(P)= P(1,−τ)f(τ)dτ ± P(1,τ¯)f(−τ¯)d(−τ¯).
:
v Zz0 Zz0
i foranyz inHthePoincar´ehyperplane,γ ∈ΓandP ∈V(k). Infact,themorphism
X 0
defining the cuspidal part of (0.1) is given by
r
a
S (Γ,C)⊕S (Γ,C) −→ H1(Γ,V(k)∨)
k k
(f ,f¯) 7−→ (∂++∂−)(f )+(∂+−∂−)(f∗),
1 2 1 2
where f∗ ∈ S (Γ,C) is obtained from f by complex conjugating its Fourier co-
2 k 2
efficients. The image of ∂± lies in the subspaces H1(Γ,V(k)∨)± ⊂ H1(Γ,V(k)∨)
1 0
where the natural action of ω = normalizing Γ acts by ±1.
0 −1
(cid:18) (cid:19)
In a general setting, F is a totally real number field of degree d, G is the multi-
plicativegroupofaquaternionalgebraAoverF,andφisaweightk =(k ,··· ,k )
1 d
cuspidal automorphic form of G with level U ⊂ G(A∞) and central character
ψ : A×/F× → C×. We assume that ψσi(x) = sign(x)ki|x|µi, for any archimedean
place σ :F ֒→R. In this scenario,by an automorphicformwe meana function on
i
Hr×G(A∞)/U,whereHisthePoincar´eupperplaneandr isthecardinaloftheset
Σ of archimedeanplaces where A splits, with values in V (k )∨ (see §1.2
σi∈∞\Σ µi i
for a precisedefinition ofV (k )), that satisfies the usualtransformationlaws with
µi i N
1
Theauthor issupportedinpartbyDGICYTGrantMTM2015-63829-P.
1
2 SANTIAGOMOLINA
respectto the weight-k -actions ofG(F). The interestingcohomologysubgroupsto
i
consider are:
n
Hr(G(F)+,A(V (k),C)U)= Hr(Γ ,V (k)∨), Γ =G(F)+∩g Ug−1,
ψ gi ψ gi i i
i=1
M
where V (k) is the tensor product of the polynomial spaces V (k ) (j =1,··· ,d),
ψ µi j
G(F)+ ⊆G(F)isthesubgroupoftotallypositiveelements,{g } ⊂G(A∞)is
i i=1,···,n
asetofrepresentativesofthedoublecosetspaceG(F)+\G(A∞)/U,andA(V (k),C)U =
ψ
C(G(A∞)/U,V (k)∨). Similarly as in the classical case, for any character ε :
ψ
G(F)/G(F)+ −→±1 we can define a morphism
∂ε :S (U,ψ)−→Hr(G(F)+,A(V (k),C)U)(ε),
k ψ
from the set S (U,ψ) of automorphic cuspforms of weight k, level U and central
k
character ψ, to the ε-isotypical component of Hr(G(F)+,A(V (k),C)U). Such a
ψ
map is given by:
∂ǫφ= ǫ(γ)∂φγ,
γ∈G(FX)/G(F)+
where ∂φ∈Hr(G(F)+,A(V (k),C)) is the class of the cocycle
ψ
g1τ1 g1···grτr
(G(F)+)r ∋(g ,g ,··· ,g )7−→ ··· P (1,−z)hφ(z,g),PΣidz,
1 2 r Σ
Zτ1 Zg1···gr−1τr
withP ∈ r V (k ), PΣ ∈ d V (k )andz =(z ,··· ,z ),(τ ,··· ,τ )∈
Σ j=1 µj j j=r+1 µj j 1 r 1 r
Hr. Ourresultwillprovidea groupcohomologicalinterpretationto the morphisms
N N
∂ε, for any character ε.
LetF ≃Rd betheproductofthearchimedeancompletionsofF,letG bethe
∞ ∞
LiealgebraofG(F )andletK ⊆G(F )beamaximalcompactsubgroup. Then
∞ ∞ ∞
the (G ,K )-module generated by φ is isomorphic to D (k), the tensor product
∞ ∞ ψ
of discrete series of weight k at archimedean places in Σ and polynomial spaces
j
V (k )atarchimedeanplacesnotinΣ. Thisimpliesthatanyφ∈S (U,ψ)provides
µj j k
an element
φ∈H0(G(F),A(D (k),C)U); A(D (k),C)U :=Hom (D (k),AU),
ψ ψ (G∞,K∞) ψ
whereAU isthe(G ,K )-moduleofsmoothadmissiblefunctionsf :G(A)/U →C.
∞ ∞
Our main result (Theorem 2.4) can be rewritten as follows:
Theorem 0.1. There exists an exact sequence of G(F)-modules
0→A(V (k)(ε),C)U →A(Iε(k),C)U →A(Iε(k),C)U →···→A(D (k),C)U →0,
ψ 1 2 ψ
such that, up to an explicit constant, the morphism ∂ǫ is given by the corresponding
connection morphism
H0(G(F),A(D (k),C)U)−→Hr(G(F),A(V (k)(ε),C)U)≃Hr(G(F)+,A(V (k),C)U)(ε).
ψ ψ ψ
We obtain the above exact sequence from extensions of the (G ,K )-modules
σi σi
of discrete series D (k ) at every place σ ∈ Σ. The archimedean local nature of
µi i i
theseconnectionmorphisms∂εimpliesthattheG(A∞)-representationgeneratedby
∂εφ coincides with the restriction to G(A∞) of π , the automorphic representation
φ
attached to φ.
The image of a cuspidal automorphic representation π through the morphisms
φ
∂ε isusedinmanypaperstogiveagroupcohomologicalconstructionofcyclotomic
andanti-cyclotomicp-adicL-functions andStickelbergerelements attachedto qua-
dratic extensions of a totally real number field (see for instance [7], [5] and [1]).
The explicit form of ∂ε given in Theorem 2.4 provides the interpolation properties
of these objects.
Another application is the construction of Stark-Heegner points. By means of
the connection morphisms ∂ε±1, where (ε+,ε−) is a well chosen pair of characters,
EICHLER-SHIMURA ISOMORPHISM AND GROUP COHOMOLOGY ON ARITHMETIC GROUPS3
one can construct a complex torus C[L:Q]/Λ attached to a weight 2 automorphic
representation π with field of coefficients L. It is conjectured that such complex
φ
torus coincides with the abelian variety of GL -type attached to π . In [3], we
2 φ
use the cohomologicaldescription of ∂ε±1 to construct Stark-Heegner points in the
complex torus, that we conjecture to be global points in the corresponding abelian
variety. Suchpointsareconjecturallydefinedoverclassfieldsofquadraticextensions
of F and satisfy explicit reciprocity laws.
Notation. Throughout this paper, we will denote by dθ = dθ the Haar
S1 SO(2)
measure of S1 =SO(2) such that vol(S1)=π.
R R
LetF beanumberfield. Foranyplacev ofF,wedenotebyF itscompletionat
v
v. GivenafinitesetofplacesS ofF,wedenotebyF theproductofcompletionsat
S
everyplaceinS. WedenotebyF theproductofcompletionsateveryarchimedean
∞
place. Similarly, for any subset Σ of archimedeanplaces, F will be the product
∞\Σ
of completions at every archimedean place not in Σ. We denote by A the ring of
adeles of F. For any set S of places of F, we write AS for the ring adeles outside
S. Consistent with this notation, we denote by A∞ the ring of finite adeles of F.
1. Discrete series
1.1. Finitedimensionalrepresentations. LetAbeaquaternionalgebradefined
over a local field F. Let K/F be an extension where A splits. For any natural
numberk ∈N,letPK ≃Symk−2(K2)bethefiniteK-vectorspaceofhomogeneous
k−2
polynomials of degree k−2. We have a well defined action of GL (K) on PK
2 k−2
given by
a b a b
(1.2) ∗P (x,y):=P (x,y) =P (ax+cy,bx+dy).
c d c d
(cid:18)(cid:18) (cid:19) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
If we fix an embedding
ι:A−→A⊗ K ≃M (K),
F 2
then PK is equipped with an action of A×.
k−2
We denote by det:A→F the reduced norm of A, and let us consider
A+ ={a∈A: det(a)∈F2}.
We write V(k)K = PkK−2⊗det2−2k with the natural action of A+. It is clear that
the centreofA× acts triviallyonV(k) . Notice that, ifk is even,the actionofA+
K
on V(k) extends to a natural action of A.
K
1.2. Discreteseriesandexactsequences. AssumethatF =RandA=M (R).
2
Let GL (R) be the Lie algebra of GL (R). For any k ∈ Z and µ ∈ C, we define
2 2
I (k) as the (GL (R),SO(2))-module of smooth admissible vectors in
µ 2
k
(f :GL2(R)+ →C: f(cid:18)(cid:18) t1 tx2 (cid:19)g(cid:19)=sign(t1)k(t1t2)µ2 (cid:18)tt12(cid:19)2 f(g)).
µ
Write Vµ(k)=V(k)C⊗det2. If we assume that k ≥2, we have the well defined
morphism of (GL (R),SO(2))-modules
2
a b 2−k+µ
ι:Vµ(k)−→Iµ(2−k); ι(P) c d =(ad−bc) 2 P(c,d).
(cid:18) (cid:19)
Moreover,we have a (GL (R),SO(2))-invariant pairing (see [2, §2])
2
h, i :I (k)×I (2−k)−→C; (f,g)7−→ f(θ)g(θ)dθ,
I µ −µ
ZSO(2)
4 SANTIAGOMOLINA
providing the morphism of (GL (R),SO(2))-modules
2
ϕ¯:I (k)−→V (k)∨; hϕ¯(f),Pi :=hf,ι(P)i .
µ −µ V I
Composing with the natural GL (R)+-morphism
2
(1.3) V (k)∨ −→V (k); F 7−→P (x,y)=hF,(Yx−Xy)k−2i ,
−µ µ F V(X,Y)
we obtain a map
ϕ:I (k) −→ V (k);
µ µ
ϕ(f)(x,y) = hϕ¯(f),(Yx−Xy)k−2i = f(θ)(ysinθ+xcosθ)k−2dθ
V(X,Y)
ZS1
Remark 1.1. Notice that we have the symmetry
hF,P i = hF,hG,(Yx−Xy)k−2i i
G V V(X,Y) V(x,y)
= (−1)khG,hF,(Xy−Yx)k−2i i =(−1)khG,P i ,
V(x,y) V(X,Y) F V
for all F,G∈V (k)∨.
µ
The kernelof ϕ is D (k) the Discrete Series (GL (R),O(2))-module of weightk
µ 2
and central character x 7→ sign(x)k|x|µ. This definition implies that D (k) lies in
µ
the following exact sequence of (GL (R),SO(2))-modules:
2
ϕ
(1.4) 0−→D (k)−→I (k)−→V (k)−→0.
µ µ µ
Since any g ∈GL (R)+ can be written uniquely as g =u·τ(x,y)·κ(θ), where
2
y1/2 xy−1/2 cosθ sinθ
u∈R+, τ(x,y)= ∈B, κ(θ)= ∈SO(2),
y−1/2 −sinθ cosθ
(cid:18) (cid:19) (cid:18) (cid:19)
we have that
Iµ(k)= Cft; ft(u·τ(x,y)·κ(θ))=uµyk2etiθ.
t≡kM(mod2)
The (GL (R),SO(2))-module structure of I (k) can be described as follows: Let
2 s
L,R∈GL (R) be the Maass differential operators defined in [2, §2.2]
2
∂ ∂ 1 ∂ ∂ ∂ 1 ∂
L=e−2iθ −iy +y − , R=e2iθ iy +y + .
∂x ∂y 2i∂θ ∂x ∂y 2i∂θ
(cid:18) (cid:19) (cid:18) (cid:19)
Then, the (GL (R),SO(2))-module I (k) is characterizedby the relations:
2 s
k+t k−t
(1.5) Rf = f ; Lf = f ;
t t+2 t t−2
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
(1.6) κ(θ)f =etiθf ; uf =uµf ,
t t t t
for any κ(θ)∈SO(2) and u∈R+ ⊂GL (R)+.
2
Write z = x+iy and z¯= x−iy. For n ∈ {0,1,··· ,k−2}, let us consider the
elements P ∈ V (k), P (x,y) = znz¯k−2−n. It is clear that {P } is a
n µ n n n=0,···,k−2
basis for the C-vector space V (k). We compute that
µ
ϕ(f)(x,y) = f(θ)((2i−1)(z−z¯)sinθ+2−1(z+z¯)cosθ)k−2dθ
ZS1
= 22−kf(θ)(ze−iθ+z¯eiθ)k−2dθ
ZS1
k−2
k−2
= 22−k P (x,y) f(θ)e−i(2n−k+2)θdθ
n
n
n=0(cid:18) (cid:19) ZS1
X
By orthogonality, we deduce that ϕ(f )=22−kπ k−2 P (x,y).
2n−k+2 n n
(cid:0) (cid:1)
EICHLER-SHIMURA ISOMORPHISM AND GROUP COHOMOLOGY ON ARITHMETIC GROUPS5
Since κ(θ)P =e(2n−k+2)iθP , the morphism of C-vector spaces
n n
2k−2 k−2 −1
(1.7) s:V (k)−→I (k); s(P )= f ,
µ µ n 2n−k+2
π n
(cid:18) (cid:19)
defines a section of ϕ as SO(2)R+-modules.
Remark 1.2. Since (1.3) is an isomorphism, we can define a non-degenerate
GL (R)+-invariant bilinear pairing V (k)×V (k)→C
2 µ −µ
hP ,Qi=hF,Qi , F ∈V (k)∨, Q∈V (k),
F V −µ C −µ
which is symmetric or antisymmetric depending onthe parity of k, by Remark 1.1.
Moreover,by the definition of (1.3),
P(s,t)=hP,Q i, Q (x,y)=(ys−xt)k−2.
s,t s,t
In particular,
(1.8) P(−1,i)=i2−khP,P i=ik−2hP ,Pi.
0 0
Since s is a section of ϕ, we compute
(1.9) hP,Qi=hϕ¯(s(P)),Qi =hs(P),ι(Q)i ,
V I
for all P,Q∈V (k).
µ
1.3. The (GL (R),O(2))-module of discrete series. We want to give structure
2
of (GL (R),O(2))-module to I (k). Hence, we have to define the action of ω =
2 µ
1
∈O(2)\SO(2). That is to say,we have to define ω ∈End(I (k)) such
−1 µ
(cid:18) (cid:19)
that
(i) ωf ∈Cf ; (ii) ω2 =1; (iii) ωR=Lω.
t −t
If we write ωf = λ(t)f , condition (ii) implies that λ(t)λ(−t) = 1. Moreover,
t −t
condition(iii) implies thatλ(t)=λ(t+2). We obtaintwo possible (GL (R),O(2))-
2
module structures for I (k): Letting λ(t) = 1 for all t ≡ k mod 2, or letting
µ
λ(t) = −1 for all t ≡ k mod 2. Write I (k)± for the (GL (R),O(2))-module such
µ 2
that ωf =±f , respectively.
t −t
By abuse of notation, write also V (k) and V (k) (in case µ ∈ R) for the
µ µ R
GL (R)-representations
2
Vµ(k)=Vµ(k)C =PkC−2⊗|det|2−k2+µ, Vµ(k)R =PkR−2⊗|det|2−k2+µ .
Remark 1.3. With this GL (R)-module structure, the pairing h·,·i introduced in
2
Remark 1.2 is not a GL (R)-invariant in general. In fact one can show that
2
hgP,Qi=sign(detg)khP,g−1Qi.
Notethat,foranyf ∈I (k)±,wehavethatωf(θ)=±f(−θ),hencewecompute
µ
that,
ϕ(ωf)(x,y) = ωf(θ)(xcosθ+ysinθ)k−2dθ
ZS1
= ± f(−θ)(xcosθ+ysinθ)k−2dθ
ZS1
= ± f(θ)(xcosθ−ysinθ)k−2dθ =±ω(ϕ(f))(x,y)
ZS1
Thisimpliesthattheexactsequenceof(GL (R),SO(2))-modules(1.4)providesthe
2
exact sequences of (GL (R),O(2))-modules
2
(1.10) 0−→D (k)−→ι I (k)+ −→V (k)−→0,
µ µ µ
(1.11) 0−→D (k)−ι◦→I I (k)− −→V (k)(ε)−→0,
µ µ µ
6 SANTIAGOMOLINA
where ε : GL (R) → ±1 is the character given by ε(g) = signdet(g), D (k) is the
2 µ
(GL (R),O(2))-module with fixed action of ω given by ωf(θ) = f(θ), and I is the
2
automorphism of (GL (R),SO(2))-modules
2
I :D (k)−→D (k): I(f )=sign(t)f .
µ µ t t
Note that ι◦I is a monomorphism of (GL (R),O(2))-modules because I(ωf) =
2
−ω(I(f)).
1.4. Matrix coefficients. Let us consider A(C) the (GL (R),SO(2))-module of
2
admissible C∞ functions f :GL (R)+ →C. For any f ∈I (2−k), I claim that
2 0 −µ
ϕ :I (k)−→A(C), ϕ (f)(g )=hg f,f i , g ∈GL (R)+,
f0 µ f0 ∞ ∞ 0 I ∞ 2
provides a well defines morphism of (GL (R),SO(2))-modules. Indeed, for any
2
element of the Lie algebra G∈GL (R),
2
d d
Gϕ (f)(g ) = (ϕ (g exp(tG)))| = (hg exp(tG)f,f i )|
f0 ∞ dt f0 ∞ t=0 dt ∞ 0 I t=0
= ϕ (Gf)(g ).
f0 ∞
1.5. R-structures of Discrete series. Aswecanseein[2,§2.2],RandLarenot
in GL (R), they are Caley transformations in GL (C) of elements in GL (R). In
2 2 2
fact, GL (R) is generated by
2
∂ ∂ ∂ ∂
R+L = −2ysin(2θ) +2ycos(2θ) +sin(2θ) ; u ;
∂x ∂y ∂θ ∂u
∂ ∂ ∂ ∂
i(R−L) = −2ycos(2θ) −2ysin(2θ) +cos(2θ) ; and .
∂x ∂y ∂θ ∂θ
If we define h := f +f ∈ I (k)± and g := i(f −f ) ∈ I (k)±, it is easy to
t t −t µ t t −t µ
compute that
k+t k−t ∂
(R+L)h = h + h , h =−tg ,
t t+2 t−2 t t
2 2 ∂θ
(cid:18) (cid:19) (cid:18) (cid:19)
k+t k−t
i(R−L)h = g − g , ωh =±h
t t+2 t−2 t t
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
κ(θ)h =cos(tθ)h −sin(tθ)g , ωg =∓g .
t t t t t
Hence the R-vector space I (k)± ⊂ I (k)± generated by h and g defines a
µ R µ t t
(GL (R),O(2))-module over R.
2
We check that the morphisms ϕ : I (k)+→V (k) and ϕ : I (k)−→V (k)(ε)
µ µ C µ µ C
descend to morphisms of (GL (R),O(2))-modules over R
2
ϕ+ :I (k)+ −→V (k) , ϕ− :I (k)− −→V (k) (ε).
µ R µ R µ R µ R
Hence the kernel D (k) ⊂ D (k) of ϕ+ defines a (GL (R),O(2))-module over R,
µ R µ 2
generatedbyh ,suchthatD (k) ⊗ C=D (k). Nevertheless,theautomorphism
k µ R R µ
of (GL (R),SO(2))-modules I : D (k)→D (k) does not descend to an automor-
2 µ µ
phism of (GL (R),SO(2))-modules over R since I(h )=−sign(t)ig . In fact,
2 t t
I(D (k) )=iD (k) ⊂D (k).
µ R µ R µ
We obtain the exact sequences of (GL (R),O(2))-modules over R
2
(1.12) 0−→D (k) −→ι I (k)+ −→V (k) −→0,
µ R µ R µ R
(1.13) 0−→D (k) −ι◦→I iI (k)− −→iV (k) (ε)−→0.
µ R µ R µ R
EICHLER-SHIMURA ISOMORPHISM AND GROUP COHOMOLOGY ON ARITHMETIC GROUPS7
2. Connection morphisms
In this section, we assume that G is the multiplicative group of a quaternion
algebra that splits at the set of archimedean places Σ. Write r = #Σ. Let us
consider the C-vector space A(C) of functions f :G(A)−→C such that:
• There exists an open compact subgroup U ⊆ G(A∞) such that f(gU) =
f(g), for all g ∈G(A).
• UnderafixedidentificationG(F )≃GL (R)r,f | ∈C∞(GL (R)r,C).
Σ 2 G(FΣ) 2
• Fixing K , a maximal compact subgroup of G(F ) isomorphic to O(2)r,
Σ Σ
we assume that any f ∈ A(C) is K -finite, namely, its right translates by
Σ
elements of K span a finite-dimensional vector space.
Σ
• We assume that any f ∈ A(C) is Z-finite, where Z is the centre of the
universal enveloping algebra of G(F ).
Σ
Write ρ for the action of G(A) given by right translation, then (A(C),ρ) defines a
smoothG(A∞)-representationanda(G ,K )-module,whereG istheLiealgebra
∞ ∞ ∞
of G(F ) and K = K ×G(F ). Moreover, A(C) is also equipped with the
Σ ∞ Σ ∞\Σ
G(F)-action:
(h·f)(g)=f(h−1g), h∈G(F),
where g ∈ G(A), f ∈ A(C). Let us fix an isomorphism G(F ) ≃ GL (R)r that
Σ 2
maps K to O(2)r and let V be a (G ,K )-module. We define
Σ ∞ ∞
A(V,C):=Hom (V,A(C)),
(G∞,K∞)
endowed with the natural G(F)- and G(A∞)-actions.
Remark2.1. Notethatifthe(G ,K )-moduleV comesfromafinitedimensional
∞ ∞
G(F )-representation V,
∞
A(V,C)≃C(G(A∞),Hom(V,C))=C(G(A∞),V∨),
whereV∨isseenasaG(F)-modulebymeansoftheusualinjectionG(F)֒→G(F ),
∞
and the action of G(F) on C(G(A∞),V∨) is given by (h∗f)(g)=h(f(h−1g)).
Fix σ ∈ Σ, µ ∈ C and let us consider D (k), V (k) and V (k)(ε) as (G ,K )-
µ µ µ ∞ ∞
modules by means of the projection G(F )→G(F ). The exact sequences (1.10)
∞ σ
and (1.11) provide the connection morphisms
∂ǫσ :Hi(G(F),A(V ⊗D (k),C))−→Hi+1(G(F),A(V ⊗V (k)(ǫ ),C)),
σ µ µ σ
for any of the two characters ǫ :G(F )/G(F )+→±1.
σ σ σ
LetV bea(G ,K )-representationoverRsuchthatV =V ⊗C. Thisimplies
R ∞ ∞ R
thatwehaveawelldefinedcomplexconjugationonV byconjugatingonthesecond
factor. Thus, we have a complex conjugation on A(V,C), given by
A(V,C)∋φ7−→φ¯∈A(V,C); φ¯(v)=φ(v¯).
Lemma 2.2. Assume that V = V ⊗ C for some (G ,K )-module V over R,
R R ∞ ∞ R
and let µ∈R. Then, for any φ∈Hi(G(F),A(V ⊗D (k),C)), we have
µ
∂σǫσ(φ)=ǫσ(c)·∂σǫσ(φ)
for any c∈G(F )\G(F )+
σ σ
Proof. We denote by A(V ⊗D (k),C)±1 ⊂A(V ⊗D (k),C) the subspaces where
µ µ
complex conjugation acts by ±1, respectively. Since exact sequences (1.10) and
(1.11) descend to exact sequences (1.12) and (1.13), we obtain
0−→A(V⊗V (k)(ε ),C)±εσ −→A(V⊗I (k)εσ,C)±εσ −→A(V⊗D (k),C)±1 −→0.
µ σ µ µ
Hence the connection morphism satisfies
δǫσ Hi(G(F),A(V ⊗D (k),C)±1) ⊆Hi+1(G(F),A(V ⊗V (k)(ε ),C)±εσ)
σ µ µ σ
and th(cid:0)e result follows. (cid:1) (cid:3)
8 SANTIAGOMOLINA
2.1. Explicitcomputationofthe connection morphisms. Letusconsiderthe
section s:V (k) →I (k) of (1.7).
µ C µ
We compute that
hf ,ιP i = emiθP (−sinθ,cosθ)dθ
m n I n
ZS1
= i2n−k+2emiθeniθe(n−k+2)iθdθ =πi2n−k+2δ(2n−k+2+m),
ZS1
where δ(n) is the Dirac delta. Thus, g−1P = k−2α (g )P , where
∞ n=0 n ∞ n
α (g )= ik−2−2nhf ,ι(g−1P)i P= ik−2−2nhg f ,ι(P)i .
n ∞ π k−2−2n ∞ I π ∞ k−2−2n I
Since V (k) is generated by {P ,··· ,P }, we deduce that α = 0 unless n ∈
µ 0 k−2 n
{0,··· ,k−2}. Since matrix coefficient morphism are (G ,K )-module morphisms
σ σ
by §1.4, we can compute on the one side
ik−2−2n
Rα (g ) = hg (Rf ),ι(P)i
n ∞ ∞ k−2−2n I
π
ik−2−2n
= (k−n−1) hg f ,ι(P)i =(n+1−k)α (g ),
∞ k−2n I n−1 ∞
π
ik−2−2n
Lα (g ) = hg (Lf ),ι(P)i
n ∞ ∞ k−2−2n I
π
ik−2−2n
= (n+1) hg f ,ι(P)i =−(n+1)α (g ).
∞ k−2n−4 I n+1 ∞
π
On the other side, we have that s(P )= 2k−2 k−2 −1f . Hence,
n π n 2n−k+2
2k−2 k−2 −1 (cid:0) (cid:1)
(n<k−2)Rs(P ) = (n+1)f =(k−2−n)s(P ),
n 2n−k+4 n+1
π n
(cid:18) (cid:19)
2k−2 k−2 −1
(n>0) Ls(P ) = (k−n−1)f =ns(P ).
n 2n−k n−1
π n
(cid:18) (cid:19)
Assume that φ˜∈A(V ⊗I (k),C) andthe actionof(G ,K )on V is trivial. For
µ σ σ
any f ∈I (k) and v ∈V, we will usually denote by φ˜ (f) the expression φ˜(v⊗f).
µ v
We aim to compute
h (g ):=φ˜ (s(g−1P))(g ,g), g ∈G(F )+ ≃GL (R)+,
P ∞ v ∞ ∞ ∞ σ 2
forallg ∈G(Aσ),P ∈V (k),andv ∈V. Sinceh (g )= k−2α (g )φ˜ (s(P ))(g ,g),
µ P ∞ n=0 n ∞ v n ∞
we compute
P
k−2
Rh = (Rα )φ˜ (s(P ))+α φ˜ (Rs(P )) =
P n v n n v n
nX=0(cid:16) (cid:17)
k−2 k−3
k−1
= (n+1−k)α φ˜ (s(P ))+ α φ˜ (f )+ (k−2−n)α φ˜ (s(P ))
n−1 v n 22−kπ k−2 v k n v n+1
n=0 n=0
X X
k−1 k−1
= 2k−2 α φ˜ (f )−α φ˜ (f ) = 2k−2α φ˜ (f ),
k−2 v k −1 v 2−k k−2 v k
π π
k−2 (cid:16) (cid:17)
Lh = (Lα )φ˜ (s(P ))+α φ˜ (Ls(P )) =
P n v n n v n
nX=0(cid:16) (cid:17)
k−2 k−2
k−1
= (−n−1)α φ˜ (s(P ))+ α φ˜ (f )+ nα φ˜ (s(P ))
n+1 v n 22−kπ 0 v −k n v n−1
n=0 n=1
X X
k−1 k−1
= 2k−2 α φ˜ (f )−α φ˜ (f ) = 2k−2α φ˜ (f ),
0 v −k k−1 v k−2 0 v −k
π π
(cid:16) (cid:17)
EICHLER-SHIMURA ISOMORPHISM AND GROUP COHOMOLOGY ON ARITHMETIC GROUPS9
Notice that R = e2iθ2iy( ∂ − 1 ∂ ) and L = −e−2iθ2iy( ∂ − 1 ∂ ) with
∂τ 4y∂θ ∂τ¯ 4y∂θ
τ = x + iy. Since s is a morphism of SO(2)R+-modules, h is a function of
P
GL (R)+/SO(2)R+ ≃ H, thus h is a function on τ and τ¯. Let us compute ∂hP
2 P ∂τ
and ∂hP: By (1.9) and (1.8),
∂τ¯
∂h y−1e−4πiθ (k−1)
P(τ,τ¯) = R(h )= i2−khg P ,Piφ˜ (f )
∂τ 2i P 2πiye2iθ ∞ 0 v k
(k−1) (k−1) φ˜ (f )
= (g−1P)(1,−i)φ˜ (f )= P(1,−τ) v k (τ,τ¯,g),
2πiye2iθ ∞ v k 2πi f
k
∂h −y−1e2iθ (1−k)
P(τ,τ¯) = L(h )= ik−2hg ωP ,Piφ˜ (f )
∂τ¯ 2i P 2πiye−2iθ ∞ 0 v −k
(1−k) (1−k) φ˜ (f )
= (g−1P)(1,i)φ˜ (f )= P(1,−τ¯) v −k (τ,τ¯,g),
2πiye−2iθ ∞ v −k 2πi f
−k
byRemark1.3,whereφ˜ (f )f−1andφ˜ (f )f−1areseenasfunctionsofGL (R)+/SO(2)R+ ≃
v k k v −k −k 2
H. A similar (and classical) calculation shows that φ˜ (f )f−1 and φ˜ (f )f−1 are
v k k v −k −k
holomorphic and anti-holomorphic,respectively.
For any P ∈V (k), g ∈G(Aσ), v ∈V and φ∈A(V ⊗D (k),C) the expressions
µ µ
φ(v⊗f )
k
(2.14) ω (P,v,g)(τ) := P(1,−τ) (τ,g)dτ,
φ
2πif
k
φ(v⊗f )
−k
(2.15) ω¯ (P,v,g)(τ¯) := P(1,−τ¯) (τ¯,g)dτ¯,
φ
2πif
−k
define holomorphic and anti-holomorphic forms in H, respectively. Moreover, it is
easy to check that
ω (P,v,g)(γ−1τ)=ω (γP,v,γg)(τ), ω¯ (P,v,g)(γ−1τ¯)=ω¯ (γP,v,γg)(τ¯),
φ γφ φ γφ
for any γ ∈ G(F)∩G(F )+. Assume that c ∈ Hn(G(F),A(V ⊗D (k),C)) is
σ φ µ
represented by the n-cocycle φ : G(F)n → A(V ⊗ Dµ(k),C). Then ∂σǫσ(cφ) is
represented by the (n + 1)-cocycle dnφ˜, where φ˜(γ) ∈ A(V ⊗ I (k) ,C) is any
µ C
preimage of φ(γ) for all γ ∈ G(F)n. We consider the (n+1)-cocycle ∂εσ(φ) =
σ
dnφ˜− dnb, where b(γ)(g)(v ⊗ P) = φ˜(γ)(v ⊗ s(P))(1,g). We compute, for all
P ∈V (k), v ∈V, γ =(γ ,··· ,γ )∈G(F)n, α∈G(F) and g ∈G(Aσ),
µ 1 n
∂ǫσφ(α,γ)(g)(v⊗P) = α (φ˜−b)(γ) (v⊗s(P))(1,g)+
σ
(cid:16)n+1 (cid:17)
+ (−1)i(φ˜−b)(αγ )(v⊗s(P))(1,g),
i
i=1
X
whereαγ =(α,γ ,··· ,γ γ ,··· ,γ )fori=1,··· ,n,andαγ =(α,γ ,··· ,γ ).
i 1 i−1 i n n+1 1 n−1
Since (φ˜−b)(γ)(v⊗s(P))=0 by construction, we obtain
∂ǫσφ(α,γ)(g)(v⊗P) = α φ˜(γ) (v⊗s(P))(1,g)−α b(γ) (v⊗P)(g)
σ
= φ˜((cid:16)γ)(v⊗(cid:17)s(P))(α−1,α−1g)−φ(cid:0)˜(γ)((cid:1)v⊗s(α−1P))(1,α−1g)
Since φ˜(γ)(v ⊗f ) = φ(γ)(v ⊗f ) and φ˜(γ)(v ⊗f ) = ǫ (c)φ(γ)(v ⊗f ), we
k k −k σ −k
deduce from the above computations that, for any α∈G(F)∩G(F )+,
σ
α−1i
∂ǫσφ(α,γ)(g)(v⊗P) = (k−1) ω (α−1P,v,α−1g)−ǫ (c)ω¯ (α−1P,v,α−1g)
σ φ(γ) σ φ(γ)
Zi
αi
= (1−k) ω (P,v,g)−ǫ (c)ω¯ (P,v,g).
αφ(γ) σ αφ(γ)
Zi
10 SANTIAGOMOLINA
Remark 2.3. We have a well defined action of G(F)/G(F)+ on Hr(G(F)+,M),
for any G(F)-module M, given by
cγ(α ,··· ,α )=γ c(γ−1α γ,··· ,γ−1α γ) ,
1 r 1 r
for γ ∈G(F), and α ∈G(F)+. The i(cid:0)mage of the restriction m(cid:1)ap
i
Hr(G(F),M)−→Hr(G(F)+,M)
lies in H0(G(F)/G(F)+,Hr(G(F)+,M)).
Letψ :A×/F× →C× beaHeckecharactersuchthat,foranyarchimedeanplace
σi :F ֒→R, ψσi(x)=sign(x)ki|x|µi. Let Dψ(k) be the (G∞,K∞)-module obtained
by making the tensor product of D (k ) at the place σ , if σ ∈Σ, and V (k ) at
µi i i i µj j
the place σ , if σ 6∈ Σ. An element of D (k) is f ⊗PΣ, where f = f ,
j j ψ k k σi∈Σ ki
f ∈D (k ) are the elements defined above, and PΣ ∈ V (k ). Let V (k)
ki µi i σi6∈Σ µi i N ψ
bethe(G ,K )-moduleobtainedbymakingthetensorproductofV (k )atallthe
∞ ∞ N µi i
places σ . For any character ǫ:G(F )/G(F )+ ≃G(F)/G(F)+ →±1, we denote
i ∞ ∞
by Hr(G(F)+,A(V (k),C))(ǫ) the ε-isotypical component, namely, the subspace
ψ
of Hr(G(F)+,A(V (k),C)) such that the action of G(F)/G(F)+ is given by the
ψ
character. By the above remark, the restriction map provides an isomorphism
Hr(G(F),A(V (k)(ε),C))≃Hr(G(F)+,A(V (k),C))(ǫ).
ψ ψ
Usingtheabovecomputations,weaimtogiveanexplicitformulafortheconnection
morphism:
Theorem2.4. LetφbeaweightkautomorphicformofG(A)withcentralcharacter
ψ. Then φdefines an element of φ∈H0(G(F),A(D (k),C)). For achoice of signs
ψ
at the places at infinity
ǫ:G(F )/G(F )+ ≃G(F)/G(F)+ →±1,
∞ ∞
the composition of the connection morphisms δ , for σ ∈Σ,
ǫσ
∂ :H0(G(F),A(D (k),C))−→Hr(G(F),A(V (k)(ǫ),C))≃Hr(G(F)+,A(V (k),C))(ǫ),
ǫ ψ ψ ψ
can be computed as follows:
r
∂ φ= (1−k ) ǫ(γ)∂φγ,
ǫ j
jY+1 γ∈G(FX)/G(F)+
where ∂φ∈Hr(G(F)+,A(V (k),C)) is the class of the cocycle
ψ
g1τ1 g1···grτr φ(f ⊗PΣ)
(G(F)+)r ∋(g ,g ,··· ,g )7−→ ··· P (1,−z) k (z,1,g)dz,
1 2 r Σ (2πi)rf
Zτ1 Zg1···gr−1τr k
for any P =PΣ⊗P ∈V (k), z =(z ,··· ,z ),(τ ,··· ,τ )∈Hr.
Σ µ 1 r 1 r
Proof. Let S ⊂ Σ be a subset of archimedean places such that #S = s < r.
Assume that σ =σ ∈Σ\S and let k be its correspondingweight and µ=µ . Let
j j
V = D (k )⊗ V (k )(ε ), where S′ =Σ\(S∪{σ}). Thus,
σi∈S′ µi i σi∈∞\(Σ\S) µi i σi
the composition of the connection morphisms corresponding to σ ∈ S, provides a
i
N N
morphism
δ :H0(G(F),A(D (k),C))−→Hs(G(F),A(V ⊗D (k),C)).
S ψ µ
By the previous computations, if P ∈V (k), g ∈G(Aσ), v ∈V,
µ
αi
∂ǫσ∂ φ(α,γ)(g)(v⊗P)=(1−k) ω (P,v,g)−ǫ (c)ω¯ (P,v,g).
σ S α∂Sφ(γ) σ α∂Sφ(γ)
Zi