Denominators of Eisenstein cohomology classes for GL_2 over imaginary quadratic fields

0.48 MB·English

by  Tobias Berger

DENOMINATORS OF EISENSTEIN COHOMOLOGY CLASSES
FOR GL OVER IMAGINARY QUADRATIC FIELDS
2
7
0
0 TOBIASBERGER
2

n Abstract. WestudythearithmeticofEisensteincohomologyclasses(inthe
a sense of G. Harder) for symmetric spaces associated to GL2 over imaginary
J
quadraticfields. Weproveinmanycasesalowerboundontheirdenominator

5 in terms of a special L-value of a Hecke character providing evidence for a
conjecture of Harder that the denominator is given by this L-value. We also

] prove under some additional assumptions that the restriction of the classes
T
to the boundary of the Borel-Serrecompactification of the spaces is integral.
N Such classes are interesting for their usein congruences with cuspidal classes
. to prove connections between the special L-value and the size of the Selmer
h groupoftheHeckecharacter.
t
a
m
[

1. Introduction
2
v The relationshipbetween the cohomologyof an arithmetic subgroupΓ of a con-
1
nected reductive algebraic group G and the automorphic spectrum of Γ has been
3
studied extensively. In particular, it is well-known that part of the cohomology
5
6 can be described by cuspidal automorphic forms. G. Harder initiated a program
0 to describe the entire cohomologyin terms of cusp forms and Eisensteinseries (to-
6 gether with their residues and derivatives). Using Selberg’s and Langlands’ theory
0
ofEisensteinseriesheconstructedin[26]acomplementtothecuspidalcohomology
/
h forthegroupsGL overnumberfields. TheseEisensteinclassescanbedescribedas
2
t cohomology classes with nontrivial restriction to the boundary of the Borel-Serre
a
m compactification of a symmetric space associated to G.
For arithmetic applications one would like to know if this analytically defined
:
v decomposition respects the canonical rational and integral structures on group co-
i
X homology. Harder proved for GL2 that the decomposition is, in fact, rational. By
the work of Franke and Schwermer [15] a decomposition of the cohomology of a
r
a general reductive group into cuspidal and Eisenstein parts and a rationality result
for the groups GL are now known. Harder also considered the behavior with re-
n
spect to the integral structure, in particular the case when this decomposition is
rational but not integral, which corresponds to an Eisenstein class with integral
restriction to the boundary having a denominator. For a detailed exposition of
Harder’s programwe refer to [27].
We continue this analysis of the arithmetic of Eisenstein cohomology classes in
the case of GL over an imaginary quadratic field F. In this case, the associated
2
symmetric space is a 3-dimensional real manifold, and the cohomology in degrees
1 and 2 is the most interesting. We prove a lower bound on the denominator of
degree 1 Eisenstein classes in terms of a special L-value of a Hecke character, as
2000 Mathematics Subject Classification. 11F75,11F67,22E41.
1
2 TOBIASBERGER
conjectured by Harder. As an example of the results proven, suppose m n 0,
≥ ≥
let p>max 3,m be a prime split in F, and χ:F A C a Hecke character
{ } ∗\ ∗F → ∗
withsplitconductorcoprimetopofinfinitytypezm+2z n (seeTheorem29forthe
−
complete statement of our result). We construct an Eisenstein cohomology class
Eisω (for a coefficient system depending on m and n) that is an eigenvector for
χ
the Hecke operators at almost all places such that the p-part of its denominator
is divisible by the p-part of Lalg(0,χ). Here Lalg(0,χ) is an integral normalization
of the special L-value (see Theorem 3). In Proposition 16 we analyze when the
restriction of Eisω to the boundary of the Borel-Serre compactification of the
χ
symmetric space is integral. In particular, we prove this when m = n, p > m+1,
and χc(x):=χ(x) equals χ(x) for all x A .
∈ ∗F
Such classes are interesting because of the implications for the Selmer group
of the p-adic Galois character associated to χ 1: The situation here should be
−
compared to the classical Eisenstein series of weight 2 for Γ (p) with a character ǫ
1
used by Ribet in [46]. Its q-expansion is p-integral and the constant term involves
an L-value of ǫ. Via the congruence (of q-expansions) of the Eisenstein series
withacuspidalHeckeeigenformRibet provedthe conversetoHerbrand’stheorem.
In our case the symmetric space is not hermitian but one might try to use the
integral structure coming from Betti cohomology, as carried out for GL in [29]
2/Q
and [51]. If there exists an integral cohomology class with the same restriction
to the boundary as Eisω then our result shows that there exists a congruence
χ
modulo Lalg(0,χ) between Eisω , multiplied by its denominator, and a cuspidal
χ
cohomology class. Via the Eichler-Shimura-Harder isomorphism and the Galois
representations attached to cuspidal automorphic representations by the work of
Tayloret al. (see [52])one can then constructelements in the Selmer groupofχ 1
−
and obtain a lower bound on its size in terms of Lalg(0,χ). For this application of
the results in this paper in the case of constant coefficients see [3].
Note that for this application only the case m = n is of interest since cuspidal
cohomology classes do not exist otherwise. Also, since the interior cohomology
for complex coefficients in degrees 1 and 2 are isomorphic, we restrict our study
to degree 1. For an analysis of denominators of degree 2 Eisenstein cohomology
classes associated to unramified characters see [13].
Wegiveabriefsketchofourproofofthelowerboundonthedenominatorinthe
special case of constant coefficient systems (corresponding to m = n = 0): In this
case we can treat split or inert primes p>3. Fix embeddings F ֒ Q֒ Q ֒ C
→ → p →
andletpbethecorrespondingprimeidealofF dividingp. LetG=Res (GL )
F/Q 2/F
and B the Borel subgroup of upper-triangular matrices. For any (sufficiently
small) compact open subgroup K G(A ) let S be the differentiable mani-
f ⊂ f Kf
foldG(Q) G(A)/K K ,whereK =U(2)C G(R). AnEisensteincocyclefor
f ∗
H1(S ,C\)isdescribed∞byapairo∞fHeckecharac⊂tersφ=(φ ,φ )withφ (z)=z
and φKf (z)=z 1 and a choice of a function Ψ in the ind1uc2ed repres1e,n∞tation
2,∞ − φf
VKf = Ψ:G(A ) CΨ(bg)=φ (b)Ψ(g) b B(A ),Ψ(gk)=Ψ(g) k K .
φf,C { f → | f ∀ ∈ f ∀ ∈ f}
We denote this Eisensteincocycle by Eis(Ψ ). In Section3.2 we will makepartic-
φf
ularchoicesforΨ (andcorrespondingK ),thenewvectorΨnew andthespherical
φf f φf
vectorΨ0 . We provethatΨtwist,acertainfinite twistedsumofΨ0 , is amultiple
φf φf φf
of Ψnew which will allow us to translate between the two. The cohomology class
φf
[Eis(Ψ0 )] is by construction an eigenvector for the Hecke operators at almost all
φf
DENOMINATORS OF EISENSTEIN COHOMOLOGY CLASSES 3
places (see Lemma 9) and we prove in Proposition 16 that its restriction to the
boundary is integral if LLaalglg(−(01,φ,φ11//φφ22)) is, and proceed to show this is the case if
(φ /φ )c =φ /φ .
1 2 1 2
We knowfromthe workofHarderthatthe cohomologyclass[Eis(Ψ )]is ratio-
φf
nal, i.e., it lies already in the cohomology with coefficients in a finite extension of
F. Since we are interested in the p-adic properties we study, in fact, its image in
H1(S ,F ). Thedenominatorδ([Eis(Ψ )])ofthe Eisensteincohomologyclassis
Kf p φf
the idealby whichithas to be multiplied to lie inside the imageofthe cohomology
with integral coefficients. We prove that
δ([Eis(Ψ0 )]) (Lalg(0,φ /φ )).
φf ⊆ 1 2
Bythe functoriality ofthe evaluationpairingacocyclerepresentsanintegralcoho-
mologyclassexactlywhenits pairingagainstallintegralcyclesisintegral. Explicit
generatorsfor the integral homology are not known in our case, but we can obtain
the desired lower bound on the denominator by integrating Eis(Ψ ) against one
φf
carefully chosen integral cycle. The (relative) cycle we use is motivated by the
classical modular symbol: we integrate along the path
σ :R GL (C)
>0 2
→
1 0
t ,
7→ 0 t
(cid:18) (cid:19)
or rather a sum of such paths, one for each connected component of S .
Kf
This “toroidal” integral vanishes in general for Ψ0 but we show that for Ψtwist
φf φf
the result, up to p-adic units, is
Eis(Ψtwist) L(0,φ1)L(0,φ−21).
φf ∼ L(0,φ /φ )
Zσ 1 2
We would like to conclude from this that multiplication by at least Lalg(0,φ /φ )
1 2
is necessary to make our Eisenstein cohomology class integral. For this we need
to control the p-adic properties of the numerator. To achieve this we use results
by Hida and Finis on the non-vanishing modulo p of the L-values Lalg(0,θφ±i 1) as
θ varies in an anticyclotomic Z -extension for q = p. We replace Eis(Ψtwist) by
q 6 φf
another “twisted” version Eisθ(Ψtwist) for a finite order character θ of conductor
φf
qr, defined by
1 x
Eisθ(Ψtφwfist)(g)= θ−1(x)Eis(Ψtφwfist)(g 0 −1qr ),
x∈(OXq/qr)∗ (cid:18) (cid:19)q
where is the ring of integers of the completion of F at q. The sum of paths
q
O
makingupthecycleisalsoweightedbyvaluesofθ. SeeSection4.1forthedefinition
of this cycle σ . Up to units the result of this toroidal integral is
θ
Eisθ(Ψtwist) L(0,φ1θ)L(0,φ−21θ−1).
φf ∼ L(0,φ /φ )
Zσθ 1 2
The results ofHida and Finis allow us (under certain conditions onthe conductors
oftheφ )tofindacharacterθ suchthatthenumeratorisap-adicunit. Apartfrom
i
differences in the conditions on the conductors Hida deals only with split p, whilst
Finis also treats inert p for constant coefficients. Given a character χ satisfying
certain assumptions we prove in Theorem 29 the existence of characters φ and
1
4 TOBIASBERGER
φ with χ=φ /φ for which the L-values in the numerator can be simultaneously
2 1 2
controlled. Thisinvolvestheconstructionofcharacterswithprescribedramification
and a careful analysis of Artin roots numbers.
The twisting by θ also has the effect ofmaking Eisθ(Ψtwist) vanish atthe 0- and
φf
-cusps of each connected component. By a result of Borel (see Proposition 6)
∞
it therefore represents a relative cohomology class with respect to these boundary
components. Weprovethatthisrelativecohomologyclassisagainrationalandthat
itsdenominatorboundsthatofEisθ(Ψtwist)frombelow. Wecanthereforeinterpret
φf
the toroidal integral as an evaluation pairing between relative cohomology and
homologyanddeducethattheidealgeneratedbyLalg(0,φ /φ )givesalowerbound
1 2
on the denominator of the relative cohomology class represented by Eisθ(Ψtwist).
φf
We conclude the desired bound on the denominator of [Eis(Ψ0 )] by using the
φf
divisibilities
δ([Eis(Ψ0 )]) δ([Eis(Ψtwist)]) δ([Eisθ(Ψtwist)]) δ([Eisθ(Ψtwist)] ).
φf ⊆ φf ⊆ φf ⊆ φf rel
Our results generalize and extend the work in [40] for F =Q(i) and unramified
φ /φ ,wherethe toroidalintegraliscalculatedforthe sphericalvector. K¨onigpro-
1 2
ceedstoshowinhiscasethattheL-valuegivesanupperboundonthedenominator.
Beforethis,Eisensteincohomologyfor imaginaryquadraticfields hadbeenstudied
in [24], [25], and [58]. Previous work on calculating or bounding denominators for
GL overQandtotallyrealfieldsinclude[22],[35],[43],[45],[51],and[57]. [35]and
2
[51] also use twisting techniques and a result by Washington on the non-vanishing
modulo p of Dirichlet L-values in cyclotomic towers. New about our method for
getting a lower bound is that we introduce the auxiliary cocycle Eisθ(Ψtwist) and
φf
prove that it represents a relative cohomology class, which allows us to work just
with the toroidal integral, making the calculation of additional boundary integrals
as in [22], [35] unnecessary. Our method does not allow to prove upper bounds
becauseof the transitionto the finite twistedsum, but one mightbe able to getan
upper bound by applying this idea to prove a lower bound on the denominator of
the dual cohomology class in degree 2. In principle, our method should extend to
general CM-fields, where Hida’s result is still applicable. Since the arithmetically
interestingclassesappearinthemiddledegreesthiswould,however,benotationally
more cumbersome (but see [43]).
These results generalizepart ofmy thesis [2] under C. Skinner atthe University
of Michigan, where this problem was considered in the case of constant coefficient
systems and split p. The author would like to thank Thanasis Bouganis, Vladimir
Dokchitser,Gu¨nter Harder,JoachimSchwermer,andChris Skinner for helpful dis-
cussions and an anonymous referee for improvements to the introduction and cor-
rections in the statement of Theorem 3. This article was written during visits to
the Max Planck Institute in Bonn and the Erwin Schr¨odinger Institute in Vienna.
The author would like to thank both for their hospitality and support.
2. Notation and Definitions
2.1. Basic notation. Let F be an imaginary quadratic field, σ its nontrivial au-
tomorphism, the differentofF,andd =Nm( )theabsolutediscriminant. For
F
D D
a place v of F let F be the completion of F at v. We write for the ring of
v
O
integers of F, for the closure of in F , P for the maximal ideal of , π
v v v v v
for a uniformizOer of F , and ˆ for O . Complex conjugationis denoOted by
v O vfiniteOv
Q
DENOMINATORS OF EISENSTEIN COHOMOLOGY CLASSES 5
z z. We use the notations A,A and A ,A for the adeles and finite adeles
f F F,f
7→
of Q and F, respectively, and write A and A for the group of ideles. Let p>3
∗ ∗F
be a prime of Z that does not ramify in F. Fix embeddings F ֒ Q ֒ Q ֒ C
→ → p →
and let p be the corresponding prime ideal of F over p.
2.2. The algebraic group and symmetric spaces. For any algebraic group
H/QandanyringAcontainingQwewriteH(A)forthegroupofA-valuedpoints.
We shall abbreviate H =H(R). We consider the algebraic group
∞
G:=Res (GL ).
F/Q 2/F
ThegroupG /F =GL containstheBorelsubgroupofuppertriangularmatrices
0 2/F
B , its unipotent radical U , the maximal split torus T , and the center Z . The
0 0 0 0
restriction of scalars gives corresponding subgroups B/Q,T/Q,U/Q and Z/Q of
0 1
G. We single out the element w = G(Q).
0 1 0 ∈
(cid:18)− (cid:19)
The positive simple root defines a homomorphism
α :B /F G /F
0 0 m
→
t
01 t∗2 7→t1/t2
(cid:18) (cid:19)
and we denote by α the corresponding homomorphism B/Q Res G . From
F/Q m
→
[26] we take the notation α for α :B(A) C , where :F A C is
| | ||◦ A → ∗ || ∗\ ∗F → ∗
the idelic absolute value x x = x . Here we take the usual normalized
7→ | | v| v|v
absolute values for the local absolute values, in particular, x = x x at the
Q | ∞|∞ ∞ ∞
complex place.
In G = GL (C) we choose the subgroup K = U(2) Z (C) = U(2) C
2 0 ∗
∞ ∞ · ·
containing the maximal compact subgroup of unitary matrices. The symmetric
space X = G /K can be identified with three-dimensional hyperbolic space
∞ ∞
H =R C.
3 >0
×
The Lie algebra g=Lie(G/Q) is a Q-vector space and we define g =g R.
Q
∞ ⊗
It carries a positive semidefinite K -invariant form, the Killing form
∞
1
X,Y = trace(adX adY),
h i 16 ·
and with respect to this form we have an orthogonal decomposition g =k p,
∞ ∞⊕
where k =Lie(K ) and
∞ ∞
1 0 0 1 0 i
p=RH RE RE :=R R R .
⊕ 1⊕ 2 0 1 ⊕ 1 0 ⊕ i 0
(cid:18) − (cid:19) (cid:18) (cid:19) (cid:18)− (cid:19)
Put
0 1 0 i
S :=1/2 1 i p .
± ± 1 0 ⊗R − i 0 ⊗R ∈ C
(cid:18) (cid:18) (cid:19) (cid:18)− (cid:19) (cid:19)
A maximal open compact subgroup of G(A ) is given by
f
a b
GL ( )= :a,b,c,d ,ad bd .
2 O c d ∈O − ∈O∗
(cid:26)(cid:18) (cid:19) (cid:27)
We will deal with thebfollowing congruence subgrboups: For anbideal N in and a
O
finite place v of F put N =N . We define
v v
O
a b
K1(N)= GL ( ),a 1,c 0 mod N ,
c d ∈ 2 O − ≡
(cid:26)(cid:18) (cid:19) (cid:27)
b
6 TOBIASBERGER
a b
K1(N )= GL ( ),a 1,c 0 mod N ,
v c d ∈ 2 Ov − ≡ v
(cid:26)(cid:18) (cid:19) (cid:27)
and
U1(N )= k GL ( ):det(k) 1 mod N .
v 2 v v
{ ∈ O ≡ }
For any compact open subgroup K G(A ) the adelic symmetric space
f f
⊂
S :=G(Q) G(A)/K K
Kf \ ∞ f
has several connected components. In fact, strong approximation implies that the
fibers of the determinant map
SKf ։π0(Kf):=A∗F,f/det(Kf)F∗
are connected. Any γ G(A ) gives rise to an injection j : G G(A) with
f γ
∈ ∞ →
j (g ):=(g ,γ) and, after taking quotients, to a component
γ
∞ ∞
Γ G /K S ,
γ\ ∞ ∞ → Kf
where Γ := G(Q) γK γ 1. This component is the fiber over det(γ). Choosing
γ f −
∩
a system of representatives for π (K ) we therefore have
0 f
S = Γ H .
Kf ∼ γ\ 3
[det(γ)a]∈π0(Kf)
WedenotetheBorel-SerrecompactificationsofS andΓ H byS andΓ H ,
Kf γ\ 3 Kf γ\ 3
respectively. Following [5] we write e(P) = H /A = U (R) for each rational
3 P ∼ P
BorelsubgroupP ofG. Here U denotesits unipotent radicalandA the identity
P P
component of P(R)/U (R), and the action of A on H is the geodesic action.
P P 3
The boundary of Γ H is the union of tori Γ e(P)=:e(P) with Γ =Γ
γ 3 γ,P ′ γ,P γ
\ \ ∩
P(Q) over a set of representatives for the Γ -conjugacy classes of Borel subgroups
γ
(equivalentlyofB(Q) G(Q)/Γ =P1(F)/Γ ). We recallfrom[26] 2.1and[25]p.
\ γ ∼ γ §
110 that ∂S is homotopy equivalent to
Kf
(1) ∂S˜Kf :=B(Q)\G(A)/KfK∞ ∼= Γγ,Bη\H3,
[det(γ)a]∈π0(Kf)[η]∈Pa1(F)/Γγ
where Bη(Q) = η 1B(Q)η for η G(Q) and the boundary component Γ H
− γ,Bη 3
∈ \
gets embedded in ∂S˜ via g j (g ):=η(g ,γ).
Kf ∞ 7→ η,γ ∞ ∞
2.3. Hecke characters. A Hecke character of F is a continuous grouphomomor-
phism λ:F A C . Such a character corresponds uniquely to a character on
∗\ ∗F → ∗
ideals prime to the conductor (see [31] 8.2), which we will also denote by λ. The
§
archimedean part λ : C C is of the form z zazb for t C,a,b Z. We
∞ ∗ → ∗ 7→ (zz)t ∈ ∈
will say that λ has infinity type zazb. We define the (incomplete) L-series L(s,λ)
(zz)t
for Re(s) 0 by the Euler product
≫
L(s,λ):= (1 λ(Pv)Nm(Pv)−s)−1,
−
vY∤fλ
where f is the conductor of λ. This can be continued to a meromorphic function
λ
on the whole complex plane and satisfies a functional equation (see e.g., [31] 8.6
§
or [11] 37).
Define the character λc by λc(x) = λ(σ(x)). Since σ just permutes the Euler
factors we have L(s,λ)=L(s,λc). Also let λ (x):=λ(σ(x)) 1 x.
∗ −
| |
DENOMINATORS OF EISENSTEIN COHOMOLOGY CLASSES 7
Recall from [11] p.91 and [41] XIV Theorem 14 the definition of the global root
number W(λ) appearing in the functional equation. Note that W(λ) = W(λ˜) for
λ˜ the associated unitary character λ/λ. If λ = λ then one shows using the
∗
functional equationthat W(λ)= 1. F|or|λ of infinity type zm with m Z we
± (zz)m/2 ∈
have
W(λ)=i m(Nm(f )) 1/2 τ (λ) λ( 1),
− λ − v Dv−
vY|fλ vY∤fλ
where the Gauss sum τ is given by
v
τv(λv)= (λeF)(ǫπ−ordv(fλD)).
ǫ∈Ov∗/X(1+fλ,v)
Here e is the standard additive character of F A defined by e = e Tr
F F F Q F/Q
\ ◦
in terms of the standard additive character e of Q A normalized by e (x ) =
Q Q
e2πix∞. Put τ(λ)= τ (λ). \ ∞
v|fλ v
We will use the following formula of Weil as stated in [1] Proposition 2.4:
Q
Proposition 1. Suppose that λ and λ are unitary Hecke characters of infinity
1 2
types (k ,j ) and (k ,j ) with relatively primes conductors f and f . Then
1 1 2 2 1 2
W(λ λ ) if (k j )(k j ) 0,
W(λ )W(λ )λ (f )λ (f )= 1 2 1− 1 2− 2 ≥
1 2 1 2 2 1 (( 1)νW(λ1λ2) if (k1 j1)(k2 j2)<0,
− − −
where ν =min k j , k j . (cid:3)
1 1 2 2
{| − | | − |}
For ease of reference we record the following:
Lemma 2. For λ : F A C with infinity type zazb with a,b Z we denote
∗\ ∗F → ∗ ∈
by the ring of integers in the finite extension of F obtained by adjoining the
λ p
O
values of the finite part of λ. Then for any x A
∈ ∗F,f
ord (λ(x))= a ord (x ) b ord (x ).
p p p p p
− · − ·
Proof. Let v be any finite place of F. Since λ has finite order on it suffices to
Ov∗
prove the statement for λ(π ) for any uniformizer π . If h is the class number of
v v
F, we have Ph =(α) for α and α for w =v. Now
v ∈O ∈Ow∗ 6
1=λ((α,α,...))=λ (α)λ (α) λ (α).
v w
∞
w=v
Y6
Since λ (α) we deduce that
w=v w ∈Oλ∗
6
Q h ordp(λ(πv))=ordp(λv(α))= ordp(λ (α)).
· − ∞
(cid:3)
Define Ω C to be the complex period of a N´eron differential ω of an elliptic
∈
curveE definedoversomenumber fieldsuchthatE hascomplex multiplicationby
, E has good reduction at the place above p and ω is a non-vanishing invariant
O
differential on the reduced curve E.
LetλbeaHeckecharacterofinfinitytypezazb witha,b Z. Preciselyfora>0
∈
and b 0 or a 0 and b>0 the L-value L(0,λ) is critical in the sense of Deligne.
≤ ≤
Damerellshowedinthiscasethatπmax( a, b)Ω a bL(0,λ)isanalgebraicnumber
− − −| − |
8 TOBIASBERGER
in C. We recall the following results (due to, amongst others, Shimura, Coates-
Wiles, Katz, Hida, Tilouine, de Shalit, and Rubin) about the integrality of the
special L-value at s=0:
Theorem 3. Let λ a Hecke character of infinity type zazb with conductor prime
to p. Assume a,b Z and a>0 and b 0. Put
∈ ≤
Lalg(0,λ):=Ωb a 2π −bΓ(a) L(0,λ).
−
√d ·
(cid:18) F(cid:19)
(a) If p is split then
(1 λ(p))(1 λ∗(p)) Lalg(0,λ)
− − ·
lies in the ring of integers of a finite extension of F .
p
(b) If p is inert and a > 0,b = 0 then for any ideal b coprime to 6p and the
conductor of λ
(Nm(b) λ 1(b)) Lalg(0,λ)
−
− ·
lies in the ring of integers of a finite extension of F .
p
References. If p is split then the normalization in (a) is the one appearing in the
p-adic L-function constructed by Manin-Vishik, Katz, and others. Together, [36]
Chapters4 and8, [38] Theorem5.3.0,and[34] TheoremII provethatit is a p-adic
integer in F . With our fixed embedding F ֒ F this shows that the value lies
p p
→
in a finite extension of F and is p-integral. See also [32] Theorem 1.1 and [11]
p
Theorem IIb.4.14 and II.6.7.
Part (b) uses the relation of elliptic units to special values of L-functions. For
the proof in the case when λ is the power of a Gr¨ossencharacter of a CM elliptic
curve and F has class number one see, for example [49] 7, in particular, Theorem
7.22. To extend to the general case use the arguments in§[11] Chapter II. (cid:3)
Remark. (1) If p is split then Lemma 2 shows that for a 2 the factor (1
≥ −
λ (p)) is a p-unit.
∗
(2) If F hasclassnumber one,p>a,andλis the powerofa Gr¨ossencharacter
ofaCMellipticcurvethen[12]Lemma3.4.5provesthattherealwaysexists
an ideal b such that Nm(b) λ 1(b) is prime to p.
−
−
(3) For completeness we want to mention that for inert primes p additional
divisibilities have been obtained in [37], [39], [48], [17], and [9].
2.4. ModulesandSheaves. ThegroupGL (F)actsontheF-vectorspaceMn :=
2
Symn(F2) ofhomogeneouspolynomials of degree n in two variablesX andY with
coefficients in F by right translation:
a b
c d .XiYn−i =(aX +cY)i(bX +dY)n−i.
(cid:18) (cid:19)
Applying first the field automorphism σ to the entries a,b,c and d, we get an-
n
other representationM . We also have one-dimensionalrepresentations F[k,ℓ] for
(k,ℓ) Z2,onwhichg Gactsbymultiplicationbydetk(g) σ(det(g))ℓ. Weobtain
the re∈presentations M(∈m,n,k,ℓ):=Mm Mn F[k,ℓ].·Let M(m,n,k,ℓ) :=
F F ∨
⊗ ⊗
Hom (M(m,n,k,ℓ),F). There is an isomorphism of GL (F)-modules
F 2
M(m,n,k,ℓ)∨ ∼=M(m,n,−m−k,−n−ℓ)
DENOMINATORS OF EISENSTEIN COHOMOLOGY CLASSES 9
induced by the pairing
, :M(m,n,k,ℓ) M(m,n, m k, n ℓ) F,
h i × − − − − →
1 1
XjYm−jXkYn−k×XµYm−µXνYn−ν 7→(−1)j+k mj − nk − δj,m−µδk,n−ν.
(cid:18) (cid:19) (cid:18) (cid:19)
Thisis the coordinatizedversionofthe pairinginducedby the determinantpairing
on F2 (cf. [31] p. 169).
For an -module N we denote N A by N for any -algebra A. Denote by
A
O ⊗O O
M(m,n,k,ℓ) the polynomials with -coefficients. Note that M(m,n,k,ℓ) :=
∨
Hom (M(m,On,k,ℓ), ) corresponds uOnder the duality above to O
O O
( aµ,ν mµ nν XµYm−µXνYn−ν|aµ,ν ∈O)⊂M(m,n,k,ℓ).
µ,ν (cid:18) (cid:19)(cid:18) (cid:19)
X
We now define localcoefficientsystems onthe symmetric spaces. For Γ G(Q)
⊂
an arithmetic subgroup and N an [Γ]-module we define a sheaf of -modules on
O O
Γ H by
3
\
N(U):={f :πΓ−1(U)→N locally constant :
e f(βx)=β.f(x)∀x∈πΓ−1(U) and β ∈Γ},
where π :H Γ H is the canonical projection.
Γ 3 3
→ \
Let K G(A ) be a compact open subgroup and M an F[G(Q)]-module.
f f
⊂
Assume that there exists an -lattice M in M such that M = M ˆ is
ˆ
stable under K . (For M = MO(m,n,k,ℓ)Oand K GL (ˆ) oneOcan taOke⊗MO =
f f 2
⊂ O O
M(m,n,k,ℓ) .) For each open subset U S we let
O ⊂ Kf
f(βg)=β.f(g),f(g) g M
f ˆ
MO(U):=(f :π−1(U)→M locally constant (cid:12) g π−1(U) and β ∈G(Q)O),
(cid:12)∀ ∈ ∈
f (cid:12)
whereπ :G(A)/K K S istheprojection.(cid:12)Thisdefinesasheafof -modules
∞ f → Kf (cid:12) O
on S (cf. [55] 1.4, [40] 1.5, and [13] 1.2). For any -algebra R we define M
Kf § § § O R
as M R, where R is the constant sheaf associated to R.
FoOrγ⊗ G(A )letM :=M γ.M . ThenM isalocallyfree,finitelygenerafted
f γ ˆ γ
∈ ∩
-mfodule with an action by Γ =GO(Q) γK γ 1. The two constructions of M
γ f −
O ∩ O
and M are compatible with j ; one checks that j (M )=M .
γ γ γ∗ ∼ γ
O f
2.5. Cfohomology. Forasheaf onatopologicalspfaceX,wfedenotebyHi(X, )
F F
(resp. Hi(X, )) the i-th cohomology group of (resp. with compact support),
c F F
andtheinteriorcohomology,i.e.,theimageofHi(X, )inHi(X, ),byHi(X, ).
c F F ! F
Let M be an F[G(Q)]-module with M M an -lattice as above and R an
i O ⊂ O
-algebra. Since S ֒ S is a homotopy equivalence, we have a canonical
O Kf → Kf
isomorphism
Hi(S ,M )=Hi(S ,i M )
Kf R ∼ Kf ∗ R
and in what follows we will replace i M by M and also write M for the sheaf
f R R f R
∗
j i M on ∂S , for j :∂S ֒ S .
∗ ∗ R Kf Kf → Kff f f
f
10 TOBIASBERGER
The decomposition of the adelic symmetric space into connected components
gives rise to canonical isomorphisms (see [40] 1.6 and [13] 1.2)
§ §
Hi(S ,M )= Hi(Γ H ,M R)
Kf R ∼ γ\ 3 γ ⊗
[det(γ)M]∈π0(Kf)
f g
and
Hi(∂S˜Kf,MR)∼= Hi(Γγ,Bη\H3,Mγ ⊗R).
[det(γ)M]∈π0(Kf)[η]∈PM1(F)/Γγ
f g
The above cohomology groups and isomorphisms are all functorial in R.
For an arithmetic subgroup Γ G(Q) and an [Γ]-module N we can in many
⊂ O
cases relate the sheaf cohomology Hi(Γ H ,N ) to group cohomology Hi(Γ,N )
3 R R
\
(for the proof see, e.g., [23]):
e
Proposition 4. For -algebras R in which the orders of all finite subgroups of Γ
O
are invertible there is a natural R-functorial isomorphism
Hi(Γ H ,N )=Hi(Γ,N ).
\ 3 R ∼ R
(cid:3)
e
The lemma in [13] 1.1 shows that for any -algebra R, R [1] satisfies the
§ O ⊗OO 6
conditions of the proposition for any arithmetic subgroup Γ G(Q).
⊂
For complex coefficient systems we have analytic tools available. For a C -
∞
manifold X (like S , ∂S˜ , or Γ H ) denote by Ωi(X) the space of C-valued
Kf Kf \ 3
C -differential i-forms and by Ωi(X,M ) = Ωi(X) M the space of M -
∞ C C C C
⊗
valuedsmoothi-forms. BythedeRhamTheorem(cf. [21]IV.9.1,or[31]Appendix
Theorem 2) we have
Hi(Γ\H3,MC)∼=Hi(Ω•(H3,MC)Γ).
Furthermore, the de Rham cohomology groups are canonically isomorphic to rela-
g
tive Lie algebra cohomology groups. For the definition of the latter we refer to [6]
Chapter 1. The tangent space of H at the point K G /K can be canoni-
3
∞ ∈ ∞ ∞
cally identified with g /k . For g G let L :H H be the left-translation
g 3 3
∞ ∞ ∈ ∞ →
by g and D the differential of this map. Assume that the G(Q)-action on M
Lg C
extends to a representation of G . Let ω : Z(R) C be the character de-
scribing the action on MC and w∞rite C∞(ΓM\CGL2(C))(ω→M−1C)∗for those functions in
C∞(Γ\GL2(C)) on which translation by elements in Z(R) acts via ωM−1C.
We can then identify the C-vector spaces
Ωi(H3,MC)Γ ∼=HomK∞(Λi(g∞/k∞),C∞(Γ\GL2(C))(ωM−1C)⊗MC),
by mapping an M -valued differential form ω˜ to the (g,K )-cocycle ω given by
C
ω(g)(θ ... θ ) := g 1.ω˜(gK )(D (θ ),...,D (θ )). T∞he differentials of the
1∧ ∧ i − ∞ Lg 1 Lg i
complexes corresponds and we get (cf. [6] VII Corollary 2.7)
Hi(Γ\H3,MC)∼=Hi(g∞,K∞,C∞(Γ\GL2(C))(ωM−1C)⊗MC).
Similarly, one obtains
g
Hi(SKf,MC)∼=Hi(g∞,K∞,C∞(G(Q)\G(A)/Kf)(ωM−1C)⊗MC)
and
g
Hi(∂S˜Kf,MC)∼=Hi(g∞,K∞,C∞(B(Q)\G(A)/Kf)(ωM−1C)⊗MC).
g