An Eisenstein ideal for imaginary quadratic fields by Tobias Theodor Berger A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2005 Doctoral Committee: Professor Christopher M. Skinner, Chair Associate Professor Brian D. Conrad Associate Professor Fred M. Feinberg Associate Professor Lizhen Ji Associate Professor Kannan Soundarajan
141
Embed
An Eisenstein ideal for imaginary quadratic fleldsdept.math.lsa.umich.edu/research/number_theory/theses/tobias... · An Eisenstein ideal for imaginary quadratic flelds by Tobias
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
An Eisenstein ideal for imaginary quadratic fields
by
Tobias Theodor Berger
A dissertation submitted in partial fulfillmentof the requirements for the degree of
Doctor of Philosophy(Mathematics)
in The University of Michigan2005
Doctoral Committee:
Professor Christopher M. Skinner, ChairAssociate Professor Brian D. ConradAssociate Professor Fred M. FeinbergAssociate Professor Lizhen JiAssociate Professor Kannan Soundarajan
ACKNOWLEDGEMENTS
The support and encouragement of many people over the years has inspired me
to pursue mathematics and has sustained me whilst working on my Ph.D. It gives
me great pleasure to be able to thank these people here.
My advisor on this thesis was Chris Skinner, and I would particularly like to thank
him for his insight, guidance, and encouragement that helped me to navigate my way
through tricky technical issues and past seemingly dead ends. I always immensely
valued the time that he was able to give me and the patience he showed me as I
took my first tentative steps in this field. Secondly, I would like to thank my wife,
Hannah Melia, who when necessary helped to distract me and at other times kept
me on target, and was constantly supportive and encouraging throughout.
I am very grateful to Brian Conrad who generously organized the extremely useful
VIGRE seminars and helped me to learn the finer points of mathematical exposition.
In addition I would like to thank Trevor Arnold, Gunther Harder, Lizhen Ji, Christian
Kaiser, Kris Klosin, Mihran Papikian, James Parson, Dinakar Ramakrishnan, Karl
Rubin, Eric Urban, and Uwe Weselmann for helpful and enlightening discussions.
Last but not least, I am forever indebted to my parents and grandparents who
encourage me in all my pursuits and supported me in many ways throughout my
The cohomology group H i(G, A) is then isomorphic to ker(d′i)/im(d′i−1).
If A has the additional structure of an S-module for a commutative ring S and
the G-action is S-linear, the above discussion carries over to the category of S[G]-
modules and the group cohomology groups H i(G,A) are S-modules.
(c) For an arithmetic subgroup Γ ⊂ G(Q) and an O[Γ]-module N we can in many
cases relate the sheaf cohomology groups H i(Γ\H3, NR) to the cohomology groups
H i(Γ, NR):
Proposition 2.2 ([HaCAG] Satz 2.9.1). For O-algebras R in which the orders
of all finite subgroups of Γ are invertible there is a natural R-functorial isomorphism
H i(Γ\H3, NR) ∼= H i(Γ, NR).
25
Sketch of proof. We recall that the sheaf cohomology groups H i(Γ\H3, ·) are defined
as the right derived functors of the global section functor. We note that the functor
NR 7→ NΓR used for the definition of group cohomology is the composite of NR 7→ NR
and NR 7→ H0(Γ\H3, NR). Under the assumption in the proposition the functor
NR 7→ NR is exact. In addition one can show that this functor takes injective
R[Γ]-modules to acyclic R-module sheaves. It therefore maps an injective resolution
of NR to a resolution of NR by acyclic sheaves. Taking global sections one gets a
complex whose cohomology by definition gives the groups H1(Γ, NR) but which is
also naturally isomorphic to the cohomology groups H i(Γ\H3, NR).
Remark 2.3. A lemma in [F] shows that for any O-algebra R, R ⊗O O[16] satisfies
the conditions of the proposition for any arithmetic subgroup Γ ⊂ G(Q).
2.9.3 Complex coefficient systems
From now on we further assume that M and N are finite-dimensional C-vector
spaces. We then have analytic tools to handle these cohomology groups. For a
C∞-manifold X denote by Ωi(X) the space of C-valued C∞ differential i-forms with
exterior derivative di, and by Ωi(X,M) = Ωi(X) ⊗C M the space of M -valued
smooth i-forms. Note that Ω0(X,M) = C∞(X, M). We write ΩiX for the sheaf of
C∞ differential i-forms.
Proposition 2.4 (de Rham Theorem, [Hi93] Appendix Theorem 2). For a
locally constant sheaf F on X having values in the category of finite dimensional
C-vector spaces there is a natural isomorphism
H i((Ω•X ⊗C F)(X); d• ⊗ idF) ∼= H i(X,F).
Sketch of proof. By the Lemma of Poincare (which states that the higher de Rham
cohomology groups of the open unit disc in Cn all vanish) the complex formed by
the sheaves Ω•X ⊗C F provides a resolution of F . Furthermore, one shows that
the sheaves ΩiX ⊗C F are acyclic. The sheaf cohomology H i(X,F) is therefore
naturally isomorphic to the cohomology of the complex obtained by taking the global
sections.
26
For X = SKf, ∂SKf
, and Γ\H3 we defined locally constant sheaves M and N ,
respectively. In these cases we let Ωi(X, M) := (ΩiX⊗CM)(X). De Rham’s Theorem
implies that
H i(SKf, M) ∼= H i(Ω•(SKf
, M)),
H i(∂SKf, M) ∼= H i(Ω•(∂SKf
, M)),
and
H i(Γ\H3, N) ∼= H i(Ω•(Γ\H3, N)).
Note that
Ωi(H3, N)Γ ∼= Ω•(Γ\H3, N)
via ω 7→ ω π for the canonical projection π : H3 → Γ\H3 (cf. [BW] VII §1).
Similarly,
Ωi(SKf, M) ∼= (Ωi(G(A)/KfK∞)⊗C M)G(Q)
and
Ωi(∂SKf, M) ∼= (Ωi(G(A)/KfK∞)⊗C M)B(Q).
For X = Γ\H3 the natural isomorphisms of Proposition 2.2 and 2.4 compose
to give an isomorphism between de Rham cohomology and group cohomology. For
future reference we want to state this isomorphism explicitly for degree 1:
Proposition 2.5. The natural isomorphism
H1(Ω•(H3, N)Γ) ∼= H1(Γ\H3, N) ∼= H1(Γ, N)
is induced by any of the following maps on closed 1-forms: For a choice of basepoint
x0 ∈ H3 assign to a closed 1-form ω with values in N the (inhomogeneous) 1-cocycle
Gx0(ω) : γ 7→∫ γ.x0
x0
ω.
Proof. This is well-known but since we cannot find a reference we give the argument
here (see, however, [Co] Proof of Lemma 3.3.5.1 for a more general version). First
one checks that Gx0 is well-defined. It is independent of the choice of path because
27
dω = 0. Also it is easy to check that the class of the cocycle is independent of the
choice of x0.
Since the functor N → NΓ is left-exact and by Propositions 2.2 and 2.4 the
functors N 7→ H i(Ω•(H3, N)Γ) and N 7→ H i(Γ, N) are erasable functors on C[Γ]-
modules (see [Hart] III.1 for the definition of erasable additive functors). This implies
that both
(H i(Ω•(H3, ·)Γ))i≥0 and (H i(Γ, ·))i≥0
are universal δ-functors. (We again refer to [Hart] III.1 for the definition and proper-
ties of δ-functors.) By the universality of both δ-functors there is a unique sequence
of isomorphisms H i(Ω•(H3, ·)Γ) → H i(Γ, ·) for each i ≥ 0, starting with the canoni-
cal isomorphism in degree 0, which commute with the connecting homomorphism δi
for each short exact sequence of C[Γ]-modules. It suffices therefore to show that the
map on closed 1-forms given above defines a morphism H1(Ω•(H3, ·)Γ) → H1(Γ, ·)extending the one in degree 0.
As we recalled above, H1(Γ, N) is calculated by taking Γ-invariants of the acyclic
resolution N → A•(Γ, N) and computing the cohomology of the resulting complex.
The de Rham cohomology group is calculated as the cohomology of the complex
NΓ → Ω•(Γ\H3, N) = Ω•(H3, N)Γ. This is the complex of Γ-invariants of the
complex N → Ω•(H3, N). Since H3 is contractible the Poincare Lemma mentioned
above in Proposition 2.4 implies that this latter complex is exact and therefore a
resolution of N . Note also that both resolutions are functorial in N .
For any x0 ∈ H3 the morphism f 0 : C∞(H3, N) → A0(Γ, N) given by φ 7→ (g 7→φ(g.x0)) commutes with the maps from N (in each case taking an element m ∈ N
to the constant map equal to m):
Nε−−−→ C∞(H3, N)
d0−−−→ Ω1(H3, N)d1=0 −−−→ 0∥∥∥yf0
yf1
Nε−−−→ A0(Γ, N)
d0−−−→ A1(Γ, N)d1=0 −−−→ 0
To make the above diagram commute f 1 must take a closed 1-form ω to the homo-
28
geneous 1-cocycle
(γ1, γ2) 7→ F (γ2.x0)− F (γ1.x0) =
∫ γ2.x0
γ1.x0
ω,
for F ∈ C∞(H3, N) with d0(F ) = ω. Applying Stokes’s Theorem gives the expression
as an integral. With the correspondence between homogeneous and inhomogeneous
cocycles recalled above this is the map given in the statement of Proposition 2.5.
After taking Γ-invariants of the resolutions f 0 and f 1 induce δ-functorial maps on the
cohomology groups in degree 0 and 1 and so must be the canonical isomorphisms.
The de Rham cohomology groups are also canonically isomorphic to relative Lie
algebra cohomology groups. For the definition of the latter we refer to [BW] Chapter
1. The tangent space of H3 at the point x0 := 1K∞ ∈ G∞/K∞ can be canonically
identified with g∞/k∞. For g ∈ G∞ let Lg : H3 → H3 be the left-translation by g
and DLg the differential of this map. Assume that the G(Q)-action on M extends
to a representation of G∞. Let ωM : Z(R) → C∗ be the character describing the
action on M and write C∞(Γ\GL2(C))(ω−1M ) for those functions in C∞(Γ\GL2(C))
on which translation by elements in Z(R) acts via ω−1M .
conductor of µi. If only one of µ1,v, µ2,v is ramified then dv(φ) = µ2,v(−1)
Nm(Prv)
.
Proof. Using the Bruhat decomposition we have
res(Eis(g, φ · |α|z/2, Ψ)) =
∫
U(Q)\U(A)
Eis(ug, φ · |α|z/2, Ψ)du
= ωz(g, φ, Ψ) +
∫
U(A)
ωz(w0ug, φ, Ψ)du
and we obtain the first part of c(φ) from the calculation at the infinite place, which
is done in [Ha79] pp.71-72 and [HaGL2] pp. 71-73.
At the finite places where µ1/µ2 is unramified, it is a standard calculation (cf.
[B92] pp. 478, 497) that the integral∫
U0(Fv)Ψ0
φv |α|z/2v
(w0uvgv)duv gives a multiple of
the corresponding spherical function Ψ0
w0.φv |α|−z/2v
(which equals Tφv |α|z/2
v(Ψ0
φv |α|z/2v
) by
definition), the factor being given by
∫
U0(Fv)
Ψ0v(w0uv)duv =
Lv(µ1/µ2, z − 1)
Lv(µ1/µ2, z).
We now give the calculation for µ1,v/µ2,v ramified. It is easy to check that
Tφv |α|z/2
v: V
K1(Psv)
φv |α|z/2v
→ VK1(Ps
v)
w0.(φv |α|z/2v )
,
so Tφv |α|z/2
v(Ψv)(g) must be a multiple of
Ψw0.(φv |α|z/2
v )(g) =
µ2,v(a)µ1,v(d)|ad|1−z/2 if g =
a b
0 d
1 0
πs−rv 1
k, k ∈ K1(Ps
v)
0 otherwise,
43
the newvector spanning VK1(Ps
v)
w0.(φv |α|z/2v )
. The multiple is given by
dv(φ) = Tφv |α|z/2
v(Ψv)(
1 0
πs−rv 1
).
Next note that
Tφv |α|z/2
v(Ψv)(gv) =
∫
U0(Fv)
Ψv(w0uvgv)duv
=
∫
U0(Ov)
Ψv(w0uvgv)duv +∞∑
n=1
∫
π−nv O∗v
Ψv(w0
1 x
0 1
gv)dx.
The important cases for us are:
(I) r = 0, i.e., µ1,v is unramified, but s > 0
(II) s− r = 0, i.e., µ2,v is unramified, but r > 0
The situation for other values of r and s is messy; in certain cases dv(φ) can be zero.
We will always be able to put ourselves in the situation of Case I or II.
In Case I we have that [
1 0
πs−rv 1
] = [
1 0
0 1
] (‘[ ]’ indicating the double coset
in B(Fv)\G(Fv)/K1(Ps
v)), so we can determine dv(φ) by evaluating at the identity
matrix. The terms in the infinite sum all turn out to be zero since for n ≥ 1 and
x ∈ O∗v
w0 ·1 π−n
v x
0 1
=
−πn
v x−1 1
0 −π−nv x
1 0
πnv x−1 1
does not lie in the same double coset as
1 0
1 1
and so gets mapped to zero by Ψ.
The constant dv(φ) is therefore given by
∫
Ov
Ψ(
0 1
−1 0
1 x
0 1
)dx =
∫
Ov
Ψ(
0 1
−1 0
)dx =
vol(Ov)µ1,v(−1)µ2,v(−1) = µ2,v(−1),
where we have used that 0 1
−1 0
=
−1 1 + πs
0 −1
1 0
1 1
1 + πs −1
−πs 1
.
44
In Case II we can evaluate at
1 0
1 1
. For ease of calculation we calculate
Tφv |α|z/2
v(Ψv)(w0) which gives us dv(φ) up to a factor of µ1,vµ2,v(−1), which equals
µ1,v(−1) by assumption. Again the infinite sum does not contribute anything since
w0
1 π−n
v x
0 1
w0 =
−πn
v x−1 1
0 −π−nv x
0 1
−1 πnv x−1
lies in the double coset [
1 0
1 1
], which is different from [
1 0
πrv 1
] in this case.
We are left with
∫
Ov
Ψv(w0
1 x
0 1
w0)dx =
∫
Ov
Ψv(
−1 0
0 −1
1 0
−x 1
)dx =
=
∫
Prv
µ1,v(−1)µ2,v(−1)dx =µ1,v(−1)
Nm(Prv)
.
Both cases can therefore be summarized by
Tφv |α|z/2
v(Ψ
φv |α|z/2v
) =µ2,v(−1)
Nm(Prv)·Ψ
(w0.φv)|α|−z/2v
.
3.3.3 Restriction to particular boundary components
Recall from (2.5) that the boundary of the Borel-Serre compactification of the
adelic symmetric space is given by
∂SKf=
∐
[det(ξ)]∈HK
∐
[η]∈P1(F )/Γξ
Γξ,Bη\e(Bη),
where HK = F ∗\A∗F /det(Kf )C
∗, Γξ = G(Q) ∩ ξKfξ−1 for ξ ∈ G(Af ), Γξ,P =
Γξ ∩ P (Q) for parabolic subgroups P/Q, and Bη(Q) = η−1B(Q)η for η ∈ G(Q).
This is homotopy equivalent to
∂SKf= B(Q)\G(Q)/KfK∞ ∼=
∐
[det(ξ)]∈HK
∐
[η]∈P1(F )/Γξ
Γξ,Bη\H3,
45
where the boundary component Γξ,Bη\H3 gets embedded in ∂SKfvia g∞ 7→ jη,ξ(g∞) :=
η(g∞, ξ).
We will be interested in the restriction of cohomology classes to the boundary com-
ponents Γξ,P\e(P ) ∼ Γξ,P\H3. So in the next lemma we clarify the relation between
the various descriptions of the boundary restrictions of a class in H1(SKf, Mm,n
C ).
Lemma 3.6 (Definition/Lemma). (a) For [ω] ∈ H1(∂SKf, Mm,n
C ), ω a Lie algebra
cocycle, the restriction to H1(Γξ,P\H3, Mm,nC ) is given by the class of ωξ
P := j∗η,ξ(ω)
if P = Bη.
(b) For [ω] ∈ H1(SKf, Mm,n
C ), ω a Lie algebra cocycle, the restriction to
H1(Γξ,P\H3, Mm,nC )
is given by the class of
resξP (ω)(g) =
∫
UP (Q)\UP (A)
ω(ugξ)du
for g ∈ G∞, where UP is the nilpotent part of the parabolic P . We have resξP (ω) =
(res(ω))ξP , where res(ω) is the constant term defined in section 2.10.
Remark 3.7. The “constant term of ω with respect to P” is given by
∫
UP (Q)\UP (A)
ω(ug)du.
Since we will not be using this general notion, “constant term” will always refer to
the one with respect to B, as defined in Section 2.10.
Proof. If one takes for granted the statements about the constant term recalled in
Section 2.10 and the comments at the start of this section on the embedding of
the boundary component, then the restriction to Γξ,P for P = Bη is given by
res(ω)(jη,ξ(g)) =∫
U(Q)\U(A)ω(uηgξ)du. The latter equals
∫UP (Q)\UP (A)
ω(ugξ)du since
ω is invariant under multiplication by η−1 on the left.
Alternatively, one can refer to Proposition 2.2.3 of [Z] which proves that the
restriction of [ω]|jξ(Γξ\H3) to the boundary component Γξ,P\H3 is given by the class
46
of ∫
Γξ,UP\UP (R)
ω(u∞gξ)du∞.
Strong approximation for UP (A) shows that this agrees with the expression given
above.
3.3.4 Translation to group cohomology
By Proposition 2.5 the de Rham cohomology group H1(Γξ,P\H3, N) is naturally
isomorphic to the cohomology group H1(Γξ,P , N) for any C[Γξ,P ]-module N . After
a choice of basepoint x0 ∈ H3 this isomorphism is given by mapping a closed 1-form
ω with values in N to the following 1-cocyle:
Gx0(ω) : γ 7→∫ γ.x0
x0
ω.
For later considerations it will be convenient to translate the restrictions ωξP of
adelic boundary cohomology classes ω to group cohomology for the arithmetic sub-
groups Γξ,P , i.e. to make the isomorphism
(3.4) H1(∂SKf, MC) ∼=
⊕
[det(ξ)]∈HK
⊕
[η]∈P1(F )/Γξ
H1(Γξ,Bη , Mξ ⊗C)
(cf. Section 2.9.2) explicit for M an O[16, G(Q)]-module. Note that Mξ ⊗ C = MC.
To combine this with the results of the previous section on the restriction of Lie
algebra cohomology classes to a boundary component, recall from Section 2.9.3 how
to translate a Lie algebra cocycle ω ∈ HomK∞(g∞/k∞, C∞(Γ\GL2(C))(ω−1∞ )⊗MC)
to a closed 1-form ω: Reversing the isomorphism given there, for x∞ = g∞K∞ ∈ H3
and T ∈ Tx∞H3 let
ω(x∞)(T ) := g∞.ω(g∞)(DL−1g∞
T ).
For a particular choice of basepoint we then get a fairly nice expression for the
image of a boundary cohomology class in H1(∂SKf, Mm,n
C ) (represented by a relative
Lie algebra 1-cocycle ω) in the group cohomology of Γξ,P :
47
Lemma 3.8. For P = Bη let x0 = η−1∞ K∞. Then Gx0ω
ξP is given on Uη by
η−1∞
1 x
0 1
η∞ 7→
∫ 1
0
(
1 tx
0 1
).ω(
1 tx
0 1
, ηfξ)(
0 x
0 0
dt.
Here we denote by ηf and η∞ the images of η ∈ G(Q) in G(Af ) and G∞ respectively.
Proof. By definition,
Gη−1∞ K∞(ωξP )(η−1
∞
1 x
0 1
η∞) =
∫ η−1∞ ( 1 x0 1 )K∞
η−1∞ K∞ωξ
P .
To calculate the path integral we apply the following lemma, adapted from [Wes]:
Lemma 3.9 ([Wes] Lemma 5.1). Given h : R → G∞ a differentiable homomor-
phism and g ∈ G∞ , define c : R → H3 by c(t) := h(t) · g · x0. For a0, a1 ∈ R let
yi := c(ai) and denote h := (Dh)0T0 ∈ g∞. Then one has the following equality:
∫ y1
y0
ω =
∫ a1
a0
(h(t)g).ω(h(t)g, gf )(g−1hg)dt.
We take y0 = η−1∞ , g = η−1
∞ , h(t) = η−1∞
1 xt
0 1
η∞ ∈ G∞, a0 = 0, a1 = 1, and
obtain:
Gη−1∞ K∞(ωξP )(η−1
∞
1 x
0 1
η∞) =
∫ 1
0
(η−1∞
1 tx
0 1
).ωξ
P (η−1∞
1 tx
0 1
)(
0 x
0 0
)dt.
With ωξP (g∞) = ω(ηg∞ξ) we get the expression given in the lemma.
48
We record for later:
Lemma 3.10. For φ = (φ1, φ2) : T (Q)\T (A) → C∗, Ψ ∈ VKf
φf, and ω = ω0(φ, Ψ)
(see (3.1)) we get that the image of [ω] in H1(Γξ,Bη ,Mm,nC )) is represented by the
1-cocycle
Gη−1∞ K∞(ωξBη)(η−1
∞
a x
0 d
η∞) = Ψ(ηfξ)
x∫ 1
0
1 tx
0 1
.Y mX
ndt if φ ∈ S1,
x∫ 1
0
1 tx
0 1
.XmY
ndt if φ ∈ S1.
Proof. Recall the definition of S+ and S− from Section 2.4. One checks that for
x ∈ C one has
xS+ − xS− =
0 x/2
x/2 0
≡
0 x
0 0
mod k∞.
In a similar manner as the preceding lemma one can show that Gx0ωξP is always
zero on η−1∞
a 0
0 d
η∞. This follows from h being a multiple of H in this case, along
which ω0 vanishes.
3.4 Hecke eigenvalues of Eisenstein cohomology class
Next we want to calculate the effect of the Hecke operators on our Eisenstein class
and its image under the restriction map. We recall briefly the definition of the Hecke
operators from Section 2.9:
For a place v of F , a non-negative integer s and g ∈ GL2(Fv) we define an action
of the double coset K1(Psv)gK1(Ps
v) on f ∈ C∞(G(Q)\G(A)/K1(Psv)) by
([K1(Psv)gK1(Ps
v)]f)(h) =∑
i
f(hx−1i ),
where K1(Psv)gK1(Ps
v) =⊔
i K1(Ps
v)xi.
49
We are especially interested in the action of
Tv,s = T (Pv) = K1(Psv)
1 0
0 πv
K1(Ps
v).
By the definition of the Eisenstein cohomology class Eis(φ, Ψ), it suffices to check
the effect of the Hecke operator Tv,s on Ψv ∈ VK1(Ps
v)
φv |α|z/2v
: Since our Ψv ∈ VK1(Ps
v)
φv |α|z/2v
are
newvectors we get that
Tv,s(Ψv) = av(φ|α|z/2)Ψv for av(φ|α|z/2) ∈ C.
For our application in Chapter VII we only need to consider places where either
both µi are unramified or only one of them is ramified.
We first consider the places where both µi are unramified. We can therefore obtain
av(φ|α|z/2) by evaluating at the identity. We get by Lemma 3.3
av(φ|α|z/2) = Tv,0(Ψv)(1) =∑
i
Ψv(x−1i ) = Ψv(
π−1
v 0
0 1
) +
∑
a∈Ov/Pv
Ψv(
1 − a
πv
0 π−1v
) =
= µ1,v(π−1v ) · |π−1
v |z/2 + Nm(Pv)µ2,v(π−1v ) · |πv|z/2
= Nm(Pv)z/2µ1,v(π
−1v ) + Nm(Pv)
1−z/2µ2,v(π−1v ).
At the places where µ1 is unramified, i.e. r = 0, but µ2 is ramified (s > 0) we get
av(φ|α|z/2) as value of Tv,s(Ψv)(
1 0
1 1
). However, like in the proof of Prop. 3.5 the
computation is easier if we evaluate at w0; obtaining the value at
1 0
1 1
by then
multiplying by µ2,v(−1). Using the second case of Lemma 3.3, we obtain
av(φ|α|z/2)µ2,v(−1) = Tv,s(Ψv)(w0) =∑
i
Ψv(w0x−1i ) =
∑
a∈Ov/Pv
Ψv(w0
1 − a
πv
0 π−1v
) =
= Ψv(
π−1
v 0
0 1
w0) +
∑
a∈Ov/Pv ,a6=0
Ψv(
a−1 π−1
v
0 aπ−1v
1 0
−π/a 1
)
50
= µ1,v(π−1v )µ2,v(−1)Nm(Pv)
z/2 + 0.
The case r > 0, s−r = 0 provides a useful check for our calculations. We can eval-
uate at the identity, and we get the eigenvalue av(φ|α|z/2) = Nm(Pv)1−z/2µ2,v(π
−1).
We observe that all these eigenvalues are invariant if we replace φ|α|z/2 with
w0.(φ|α|z/2). This is explained by the fact shown in Prop. 3.5 that for all finite
places there is some constant dv(φ) so that
Ψw0.ηv = dv(φ)
∫
U0(Fv)
Ψηv(w0 · uvgv)duv.
Furthermore, this tells us that the image of the Eisenstein series under the restriction
map has again the same eigenvalues for the action of the Tv,s’s on the boundary
cohomology. To conclude we summarize our calculations:
Lemma 3.11. For Ψv ∈ VK1(Ps
v)
φv |α|z/2v
we have Tv,s(Ψv) = av(φ|α|z/2)Ψv, where
av(φ|α|z/2) =
Nm(Pv)z/2µ1,v(π
−1v ) + Nm(Pv)
1−z/2µ2,v(π−1v ) if r = s = 0,
Nm(Pv)z/2µ1,v(π
−1v ) if r = 0, s > 0,
Nm(Pv)1−z/2µ2,v(π
−1v ) if r = s > 0.
3.5 Examples and properties of algebraic Hecke characters
In this section we want to give a list of examples of Hecke characters and their
properties that will be used in subsequent proofs.
Definition 3.12. We will call a Hecke character λ : F ∗\A∗F → C∗ anticyclotomic if
λc = λ, where λc(x) := λ(x).
Remark 3.13. 1. For finite order characters (i.e. λ∞ = 1) this agrees with the usual
definition (e.g. [Ti] Definition 3.4). In [dS] §II.6 a different notion (λ = λ∗) is used,
one based on an involution of Hecke characters of type (A0) preserving criticality:
λ 7→ λ∗, where λ∗(x) = λ(x)−1|x|AF. This arises in Katz’s work [K76], in particular,
in the p-adic functional equation. For infinity types z and z the two definitions
agree (e.g. for the (inverse of) the Großencharakter arising from an elliptic curve
51
with complex multiplication). This follows from λλ = | · |m+nAF
if λ∞ = zmzn (see
below). Hida suggests in [Hi03] the definition λc = λ−1, which, of course, agrees
with our notion for unitary characters, relates, however, characters with different
infinity types in the general case.
2. Note that our definition implies that the conductor of an anticyclotomic char-
acter λ is stable under complex conjugation. Furthermore, for infinity types with
λ∞(−1) = −1 the conductor of λ must contain a non-split prime.
Lemma 3.14. Let λ be a Hecke character of F with infinity type λ∞ = zmzn. Then
λλ = | · |m+nAF
.
Proof. Denote by (x) the fractional F -ideal generated by the (finite part) of an idele
x ∈ A∗F and by Oλ the ring of integers in the finite extension of F containing the
values of the finite parts of λ and λ.
We will show for each place v that λvλv(xv) = |xv|m+nv . For the infinite place this
is obvious. For finite places v we first note that λ has finite order on O∗v, so λλ|O∗v
is trivial. Take now a uniformizer πv. If h is the class number of F , we can write
(πv)h = (α) for α ∈ O → A∗
F . One checks that λv(α)Oλ = α−mα−nOλ. Also λ(πhv )
differs from λv(α) only by roots of unity in O∗λ. This shows that (λλ)h
v(πv)Oλ =
(πvπv)−h(m+n)Oλ. We deduce (λλ)v(πv) = Nm(πv)
−(m+n).
Lemma 3.15. For F 6= Q(√−1),Q(
√−3) and m + n even let χ0 be the unramified
Hecke character of infinity type zmzn that descends to a trivial character on the class
group of F . Then χ0 is anticyclotomic.
Proof. We first show existence and uniqueness: Since F 6= Q(√−1),Q(
√−3) we
have F ∗ ∩ ∏v O∗
v = O∗ = ±1. Since (−1)m+n = 1, χ0 therefore is well-defined
on F ∗ · C∗ ∏v O∗
v. The additional condition of χ0 being trivial on F ∗\A∗F,f/
∏v O∗
v
determines uniquely the character on A∗F .
Interpreting χ0 as a character on ideals we need to show that χ0(p) = χ0(p) for
all prime ideals p of F . This follows from the preceding lemma and χ0(p)χ0(p) =
χ0((Nm(p))) = Nm(p)−(m+n).
52
Lemma 3.16. Any unramified Hecke character λ of an imaginary quadratic field F
is anticyclotomic.
Proof. We first note that finite order unramified characters, i.e., characters with
trivial infinity type, descend to characters on the ideal class group
Cl(F ) ∼= F ∗\A∗F /C∗ ∏
v
O∗v,
and are therefore anticyclotomic since [a] = [a]−1 in Cl(F ).
For F 6= Q(√−1),Q(
√−3) if the unramified character λ has infinity type λ∞(z) =
zmzn then m + n is even and we let χ0 be as in the preceding lemma. We consider
now λ/χ0. Since this is a finite order unramified character we deduce that λ is
anticyclotomic.
For F = Q(√−1) or Q(
√−3) we use that their class number is one: This implies
that the finite order unramified character λ(x)λ(x)
is trivial.
Example 3.17. Our main theorem will demand unramified characters χ with infinity
type z2. The proof of Lemma 3.16 shows that all such characters are given by
composition of characters on the ideal class group with χ0.
We will have some freedom in how to factor χ as µ1/µ2 (where (µ1, µ2) : T (Q)\T (A) →C∗ is then used in the definition of the Eisenstein cohomology class). At some point
it will be necessary to find such µi that are anticyclotomic. For this the following
result by Greenberg in [G85] will be useful:
Lemma 3.18. Let F be an arbitrary imaginary quadratic field. Then there exists an
anticyclotomic Grossencharacter µG of infinity type z−1 whose conductor is divisible
precisely by the ramified primes of F . If F 6= Q(√−1),Q(
√−3) its restriction to the
units at the ramified places can be taken to have order 2.
Remark 3.19. We will frequently use the inverse of this character (which is also
anticyclotomic). An important property of these characters is that, for F different
53
from Q(√−1),Q(
√−3), their conductors contain a prime with respect to which −1
and 1 are non-congruent (i.e., some prime with residue characteristic p ≥ 3).
Proof. For Q(√−1) and Q(
√−3) Greenberg takes the Großencharaktere associated
to specific elliptic curves. In the other cases he uses the same method of construction
as in Lemma 3.15 (i.e., extension to A∗F from F ∗ ∏
v O∗vC
∗). At the ramified places
v the character is defined to be trivial on the local norm of the units (viewed as a
subgroup of O∗v). We refer to [G85] for the proof, especially that this character is
anticyclotomic.
The following character will be useful because of its minimal ramification (see also
[Ti] Lemme 2.5):
Lemma 3.20. Let ` ≥ 5 be a rational prime and l a prime of F dividing `. Then
there exists a Hecke character µl with conductor l of infinity type z.
Proof. Since ` ≥ 5, l separates the roots of unity and so the character is well-defined
on F ∗ · C∗U(l). Since the ray class group F ∗\A∗F,f/U(l) is finite we can trivially
extend to a continuous character on A∗F .
We will be using the following properties of the values of algebraic Hecke charac-
ters: Suppose p is a prime in the imaginary quadratic field F . Denote the underlying
rational prime by p.
Lemma 3.21. For µ : F ∗\A∗F → C∗ with infinity type zazb for a, b ∈ Z we let Oµ
denote the ring of integers in the finite extension Fµ of Fp obtained by adjoining the
values of the finite part of µ. Denote by v0 the place corresponding to p.
(1) For x ∈ A∗F with xv0 ∈ O∗
v0and xv0 ∈ O∗
v0we have µ(x) ∈ O∗
µ.
(2) If
a ≥ 0
a ≤ 0
then
µ−1(πv0) ∈ Oµ
µ(πv0) ∈ Oµ
.
Proof. Since µ has finite order on O∗v it suffices for (1) to show that µ(πw) ∈ O∗
µ
for w 6= v0, v0. Denote the prime ideal corresponding to πw by Pw. If h is the class
54
number of F , we have Phw = (α) for α ∈ O and α ∈ O∗
v for all v 6= w. Now
1 = µ((α, α, . . .)) = µ∞(α)µw(α)∏
v 6=w
µv(α).
Since∏
v 6=w µv(α) ∈ O∗µ and µ∞(α) = αaαb ∈ O∗
v0we deduce that µw(α) ∈ O∗
µ, i.e.,
valp(µw(α)) = 0, which implies µ(πw) ∈ O∗µ.
For (2) the same argument for w = v0 shows that valp(µw(α)) = −valp(µ∞(α)).
3.6 Integrality and rationality results
Definition 3.22. Let p be a prime of Z split in the imaginary quadratic field F
and let p be one of the primes above it. Let φ = (µ1, µ2) : T (Q)\T (A) → C∗ be
a character with φ ∈ S1(m,n, 0, 0) or φ ∈ S1(m,n, 0, 0). We put χ := µ1/µ2. Let
Oχ denote the ring of integers in the finite extension Fχ of Fp obtained by adjoining
the values of the finite part of both µi and Lalg(0, χ) ∼ L(0,χ)Ω2 , where Ω is a complex
period depending only on F . (For the algebraicity and p-adic integrality of the special
L-function value see Theorem 2.1.)
Let
H1(X, MOχ) := im(H1(X, MO ⊗O Oχ) → H1(X, M ⊗F Fχ))
for X ⊂ SKf. We will need the following results:
Proposition 3.23. Assume that the conductors of µ1 and µ2 are coprime to (p).
For constant coefficient systems (i.e., the infinity type of µ1 equals z and of µ2 equals
z−1) we have
[ω0(φ, Ψφ)], [ω0(w0.φ, Ψw0.φ)] ∈ H1(∂SKsf,Oχ).
(Here Ψφ and Ψw0.φ are the functions defined in Definition 3.4.)
Proof. In Section 3.3 we made explicit the isomorphism
H1(∂SKsf, R) ∼=
⊕
[det(ξ)]∈HKs
⊕
[η]∈P1(F )/Γξ
H1(Γξ,Bη , R),
55
which is functorial for O[16]-algebras R. Here HKs := A∗
F /det(Ks)F ∗ for Ks :=
KsfK∞, Γξ = G(Q) ∩ ξKs
fξ−1, and Γξ,Bη = Γξ ∩ η−1B(Q)η. Using this description
we associated to ω0(φ, Ψφ) a group cohomology class for R = C in Lemma 3.10. We
claim that it lies in the image of the natural map from H1(Γξ,Bη ,Oχ). Analyzing
the expression in Lemma 3.10 we need to show for all η and ξ and for all matricesa x
0 d
∈ Γξ,B that xΨφ(ηfξ) and xΨw0.φ(ηfξ) lie in Oχ. One checks that it is
sufficient to prove this for a specific choice for the set of representatives ξ and η. If
we can find a set of representatives ξ and η whose p- and p-components are units
(i.e., they are elements of GL2(Op) :=∏
v|p GL2(Ov)) then the definition of Ψφ at
places away from the conductors of the µi together with Lemma 3.21 shows that
each Ψφ(ηfξ) and Ψw0.φ(ηfξ) lies in O∗χ. For ξ this can be achieved because HKs is
a generalized ideal class group. The Chebotarev density theorem implies that each
class is represented by a prime different from p and p so we can choose ξ such that its
p- and p-components equal 1. This implies, in particular, that Γξ∩G(Qp) ⊂ GL2(Op),
so x and x from above also lie in Oχ.
For finding [η] ∈ P1(F )/Γξ = B(Q)\G(Q)/Γξ with η ∈ G(Q)∩GL2(Op) we claim
that GL2(Fp) :=∏
v|p GL2(Fv) satisfies
GL2(Fp) = B0(F )GL2(Op).
The Iwasawa decomposition implies GL2(Fp) = B0(Fp)GL2(Op). By the Chinese
Remainder Theorem we get F ·∏v|pOv =∏
v|p Fv and F ∗ ·∏v|pO∗v =
∏v|p F ∗
v and so
B0(Fp) = B0(F )B0(Op).
Together these prove our claim. Applying the claim to η′ ∈ GL2(F ) one gets a
decomposition η′ = bk with b ∈ B0(F ) and k ∈ GL2(Op). Then [η′] = [k] with
k ∈ G(Q) ∩GL2(Op).
Lemma 3.24. Assume m = n = 0, that the conductors of µ1 and µ2 are coprime
to (p), and that the conductor of χ = µ1/µ2 is coprime to the discriminant of F .
56
Assume further that at each place occuring in the conductor only one of the µi is
ramified. Then we have either
[res(Eis(ω0(φ, Ψφ))] ∈ H1(∂SKsf,Oχ)
or
[res(Eis(ω0(w0.φ, Ψw0.φ))] ∈ H1(∂SKsf,Oχ).
If χc = χ (where χc(x) := χ(x)), then the two constant terms differ only by a p-adic
unit and are both integral.
Remark 3.25. Given χ with conductor coprime to (p) and the discriminant of F one
can always factor χ = µ1/µ2 with µ1 := µl for a prime l not dividing the conductor
of χ and coprime to (p), where µl is the character defined in Lemma 3.20. Then µ1
and µ2 = µl/χ satisfy the conditions in the Lemma above.
Proof. By Proposition 3.5 the first constant term in the lemma is
ω0(φ, Ψφ)− 2π√dF
· L(−1, χ)
L(0, χ)· ω0(w0.φ, Ψw0.φ)
and we put C(χ) := 2π√dF· L(−1,χ)
L(0,χ). The second constant term is
ω0(w0.φ, Ψw0.φ)− 2π√dF
· L(−1, χ−1| · |2)L(0, χ−1| · |2) · ω0(φ, Ψφ).
Applying the functional equation (see Section 2.6) and using χχ = | · |2AF(see Lemma
3.14) one checks that the second constant term equals
ω0(w0.φ, Ψw0.φ)− C(χ)−1 · ω0(φ, Ψφ).
We note that
C(χ) =2π√dF
· L(0, χ| · |−1)
L(0, χ)∼ Lalg(0, χ| · |−1)
Lalg(0, χ),
where ‘∼’ indicates equality up to the p-adic units given by the Euler factors (1 −λ(p))(1−λ∗(p)) for λ = χ and χ| · |−1. Hence C(χ) lies in Fχ by Theorem 2.1. Since
either C(χ) ∈ Oχ or C(χ)−1 ∈ Oχ, the first statement of the Lemma follows from
Proposition 3.23.
57
For the statement for anticyclotomic characters we first observe that by the func-
tional equation
C(χ) = u(χ) · L(0, χ)
L(0, χ)
for
u(χ) :=(Nm(fχ))−1/2
W (χ)
with fχ the conductor of χ and χ = χ| · |−1. If χc = χ, then L(0, χ) = L(0, χc) =
L(0, χ), so L(0,χ)L(0,χ)
= 1. It remains to check that u(χ) ∈ O∗χ. Since the conductor of χ
is coprime to p we only need to analyze the root number W (χ) (see Section 2.6 for
its definition). The Gauss sums are p-integral because fχ is coprime to p. (This uses
Lemma 3.21.) Since fχ is coprime to the discriminant (which equals Nm(D) for the
different D) we also get that the factors χ(D−1v ) of the root number lie in O∗
χ.
We want to record here that for unramified characters χ one has u(χ) = 1. This
follows from χ(D−1) = −1, which is a consequence of F being imaginary quadratic
Checking the action of K∞ on the Lie algebra (see Section 1.4), we get
ω∞(
1 0
1 u
)(H) = 2
(uu)z/2+1
(1 + uu)z+2.
This gives rise to Beta-Function integrals, which converge for Re(z) > −1. The
archimedean integral therefore contributes
i
4π
∫
C∗
(uu)z/2+1
(1 + uu)z+2
du ∧ du
uu=
1
2
Γ(z/2 + 1)Γ(z/2 + 1)
Γ(z + 2).
The preceding analysis also shows that all the local integrals converge absolutely
for Re(z) > −1 and that their product exists so the integral over A∗F converges
absolutely.
To conclude the proof of the proposition by analytic continuation it suffices to
prove that for any ξ ∈ G(Af )
∫
σξ
Eis(Ψnew) =
∫ ∞
0
Eis(
1 0
0 ξt
, Ψnew
φ|α|z/2)(H
2)dt
t
converges to a holomorphic function in z for Re(z) ≥ 0. The following argument is
adapted from [S02a] Proposition 3.5 and [Wes] Proposition 2.4 and shows conver-
gence of the integral for Eis(Ψnew) ∈ HomK∞(g∞/k∞, C∞(G(Q)\G(A)/Knewf )(ω−1)⊗
Mm,nC ) for all m,n.
For c > 0 let
Ic(z) :=
∫ c
1/c
Eis(
1 0
0 ξt
, Ψnew
φ|α|z/2)(H
2)dt
t.
For any c > 0 this is a holomorphic function for all z with Re(z) ≥ 0 since the
Eisenstein cohomology class is holomorphic for z in this region (cf. [Ha82] p. 123).
72
It suffices therefore to show that Ic(z) converges locally uniformly for all z with
Re(z) ≥ 0 as c →∞. Recall that Eis(Ψnewφ|α|z/2) = Eis(ωz(φ, Ψnew
φ|α|z/2)). If we write
ωz(g, φ, Ψnewφ|α|z/2) = (α1(g∞)S+ + α2(g∞)
H
2+ α3(g∞)S−)Ψnew
φ|α|z/2(gf )
and let
(αi, Ψnewφ|α|z/2) ∈ V
Knewf
φ|α|z/2 ⊗Mm,nC : (g∞, gf ) 7→ αi(g∞)Ψnew
φ|α|z/2(gf ), i = 1, 2, 3
then
Ic(z) =
∫ c
1/c
Eis(α2, Ψnewφ|α|z/2)(
1 0
0 ξt
)
dt
t.
Put Ez(g) = Eis(α2, Ψnewφ|α|z/2)(g). Note that the constant term res(Ez)(g) vanishes
for g =
1 0
0 ξt
and g =
ξ 0
0 t
w0 since
res(Eis(ωz(φ, Ψnewφ|α|z/2)))(g) = ωz(g, φ, Ψnew
φ|α|z/2)+d(φ, Ψnewφ|α|z/2)ω−z(g, w0.φ, Ψnew
w0.(φ|α|z/2)),
which vanishes for these g on multiples of H. It follows that
Ic(z) = I1c (z) + I2
c (z),
where
I1c (z) =
∫ 1
1/c
Ez(
1 0
0 ξt
)− res(Ez)(
1 0
0 ξt
)
dt
t,
I2c (z) =
∫ 1
1/c
t−(m+n)
Ez(
1 0
0 t
ξ 0
0 1
w0)− res(Ez)(
1 0
0 t
ξ 0
0 1
w0)
dt
t.
The expression for I2c (z) follows from the identity
1 0
0 t
= w−1
0
1 0
0 t−1
w0
t 0
0 t
and a change of variables.
We note that Ez ∈ A(G(Q)\G(A)/Kf )⊗Mm,nC , where A(G(Q)\G(A)/Kf ) is the
space of automorphic forms, a certain subspace of C∞(G(Q)\G(A)/Kf ) of functions
73
of moderate growth (see [B92] §3, [HC] IV Theorem 7). Therefore standard growth
estimates for automorphic forms on Siegel sets (see [Langl] Lemma 3.4, [Schw] §1.10,
[HC] I Lemma 10) imply that for any g ∈ G(A) and r ∈ R there exists a constant
C(g, r, z) > 0, locally uniform in z, such that
‖Ez(
1 0
0 t
g)− res(Ez)(
1 0
0 t
g)‖ ≤ C(g, r, z)tr, 0 < t ≤ 1
for ‖ · ‖ the norm on Mm,nC given by the scalar product defined in Section 2.5. From
this it follows that I1c (z) and I2
c (z) converge absolutely and locally uniformly for all
z with Re(z) ≥ 0 as c →∞. The limits therefore define holomorphic functions in z,
as claimed above.
4.2.3 Calculation of the toroidal integral for a twisted version of Ψφ
As recalled at the start of this chapter we defined a particular Ψφ|α|z/2 ∈ Vφ|α|z/2
for φ = (µ1, µ2), given by∏
v/∈S Ψnewv
∏v∈S Ψ0
v. Denote from now on by Ψ′′φ|α|z/2 the
multiple twisted sum
Ψ′′φf |α|z/2
f
(g) =∑v∈S
∑
x∈(Ov/Prv)∗
µ−11 (x)Ψ
φf |α|z/2f
(g
1 x
πrv
0 1
v
),
where Prv ‖ cond(µ1) = M1.
Lemma 4.4 shows that Ψ′′φf |α|z/2
f
equals Ψnew up to L-factors. We conclude that
the toroidal integral for the Eisenstein series Eis(Ψ′′φf |α|z/2
f
) has the following value:
Lemma 4.7.∑
[ξ]∈HKnew
∫
σξ
Eis(Ψ′′φf |α|z/2
f
) =
=
∫
F ∗\A∗F
Eis(
1 0
0 x
, Ψ′′
φf |α|z/2f
)(H
2)d∗x =
L(µ1, z/2)L(µ−12 , z/2)
L(µ1/µ2, z)·
· (µ−12 (M1)Nm(M1)
−z/2) · 1
2
Γ(z/2 + 1)Γ(z/2 + 1)
Γ(z + 2)·∏v∈S
µ−12,v(−1)(µ2/µ1)(P
rv)Nm(Pr
v)z
74
4.3 Twisting by a finite character
In order to determine a bound for the denominator of the Eisenstein cohomology
class, we will also need the toroidal integral for the following twisted sum: Let
θ : F ∗\A∗F → C∗ be a finite order character of prime-power conductor qr, with q an
odd prime of Z such that (q, M1M2) = 1, where Mi is the conductor of µi.
Throughout this section we assume q is inert in F . The modification necessary
for q split is notationally cumbersome, but all one has to do is to repeat the twisting
process twice, once for each place above q.
Let η := φ|α|z/2 : T (Q)\T (A) → C∗ and η =: (η1, η2). For Ψ ∈ Vη put
Eisθ(g, Ψ) :=∑
x∈(Oq/Prq)∗
θq(x)Eis(g
1 −x/q
0 1
q
, Ψ).
Note that
Eisθ(g, Ψ) = Eis(g, Ψθ),
where Ψθ = Ψθq(gq)
∏v 6=q Ψv(gv) and Ψθ
q(gq) =∑
x mod q θ(x)Ψq(gq
1 −x/q
0 1
q
).
We can apply our analysis in section 4.1 to relate Ψnew,θη,q to some newvector:
Firstly, by Lemma 4.3, Ψ′(g) := Ψnewη,q (g)θq(det(g)) is the spherical function for Vηqθq
(we use here that the conductors of ηi and θ are relatively prime). Lemma 4.4 tells us
that Ψ′′(g) =∑
x mod q θ(x)Ψ′(g
1 −x/q
0 1
q
) is the newvector in Vηqθq , multiplied
by θ−1q (−1)(η2/η1)(q
r) · L−1q (η1/η2, 0). Untwisting by θq(det(g)) we deduce that
Lemma 4.8.
Ψnew,θη,q (g) = Ψnew
ηθ,q(g)θq(−det(g)) · (η2/η1)(qr) · L−1
q (η1/η2, 0).
This implies the following:
Corollary 4.9. Ψnew,θηf
∈ VKθ
f
ηf θ for Kθf :=
∏v 6=q K1(M1,vM2,v) · (K1((qr)) ∩ U1((qr)))
(see Section 3.2 for the definition of U1((qr))).
75
The translation of our toroidal integral over copies of C∗ to an integral over F ∗\A∗F
is now slightly more complicated, since Ψθη is not right-invariant under
1 0
0 xq
for
xq ∈ O∗q . We have instead Ψθ
η(g
1 0
0 xq
) = Ψθ
η(g)θq(xq) by Lemma 4.8. This leads
to
Lemma 4.10.
∑
[ξ]∈HKθ
∫
C∗Eis(
1 0
0 ξx∞
, Ψnew,θ
(µ1,µ2)|α|z/2)(H
2)d∗x∞ =
=
∫
F ∗\A∗F
Eis(
1 0
0 x
, Ψnew,θ
(µ1,µ2)|α|z/2)θ(x)(H
2)d∗x.
Proof. We again want to replace the argument by
1 0
0 ξx∞xf
for xf ∈ O∗. As we
just showed, Eis(
1 0
0 ξx∞xf
, Ψnew,θ
(µ1,µ2)|α|z/2) = Eis(
1 0
0 ξx∞
, Ψnew,θ
(µ1,µ2)|α|z/2)θq(xq).
Since by assumption θ is unramified away from q, θ∞ = 1, and by choosing our
representatives ξ to be unramified at q we can replace θq(xq) by θ(ξx∞xf ) and hence
obtain the right hand side after a change of variables.
For Re(z) ≥ 0 the value of the integral in Lemma 4.10 is now given by
Proposition 4.11.
∫
F ∗\A∗F
Eis(
1 0
0 x
, Ψnew,θ
(µ1,µ2)|α|z/2)θ(x)(H
2)d∗x =
=L(µ1θ, z/2)L(µ−1
2 θ−1, z/2)
LS(µ1/µ2, z)· Γ(z/2 + 1)Γ(z/2 + 1)
Γ(z + 2)·
· 1
2
((θµ2)
−1(M1qr)Nm(M1q
r)−z/2) · θ(−1)(µ2/µ1)(q
r)Nm(qr)z,
where M1 is the conductor of µ1 and S is the finite set of places where both µi are
ramified, but µ1/µ2 is unramified.
76
Proof. Away from the place q one can quickly repeat the calculations from Prop. 4.5
to see the effect of the twist by θ. At q we are looking at the integral
∫
F ∗q
Ψnew,θ
(µ1,µ2)|α|z/2,q(
1 0
1 xq
)θ(xq)d
∗xq.
By Lemma 4.8 this equals (use θ|F ∗ = 1)
θ(−1)(µ2/µ1)(qr)Nm(qr)zL−1(µ1/µ2, z) ·
∫
F ∗q
Ψnew(µ1,µ2)|α|z/2θ,q(
1 0
1 xq
)d∗xq.
Now we are in the situation of case (4) of Proposition 4.5, and we obtain
θ(−1)(µ2/µ1)(qr)Nm(qr)zL−1(µ1/µ2, z) · (µ2θ)
−1q (qr)q−rz/2.
4.4 Relative cohomology and homology
4.4.1 Definitions
Let Γ ⊂ G(Q) be an arithmetic subgroup and M an O[Γ]-module. We define the
homology Hi(Γ\H3, M) as the homology of the complex of Γ-coinvariants of singular
chains (C•(H3)⊗M)Γ.
We recall the definition of relative singular homology and cohomology (for con-
stant coefficients R; see [B67] for the general case): For a subspace A of a manifold
X define the singular i-chains Ci(A, R) to be the free R-module generated by all sin-
gular i-simplices ∆i → A and let C•(X,A, R) := C•(X, R)/C•(A,R). The relative
homology Hi(X, A, R) is then defined as homology of this chain complex. One checks
that classes in Hi(X,A, R) are represented by relative cycles, i-chains α ∈ Ci(X, R)
such that ∂α ∈ Ci−1(A,R). Define now relative (simplicial) cohomology as homology
of the chain complex C• = Hom(C•(X, A, R), R). This implies that relative cochains
are absolute cochains (for X) vanishing on chains in A. For the corresponding def-
initions for sheaf cohomology and Borel-Moore homology and isomorphisms with
the singular theories we refer to [B67]. We revert here to singular homology and
77
cohomology because we want to make use of the explicit evaluation pairings between
them.
Proposition 4.12 ([F], Satz 3, [G67] §23). 1. For the ring R = O[16] and a
subspace A ⊂ Γ\H3 the evaluation pairing
Hi(Γ\H3, A, M∨ ⊗R)/torsion×H i(Γ\H3, A, M ⊗R)/torsion → R
is perfect and functorial in the ring R.
2. For R = C, the pairing between a de Rham (or relative Lie algebra) cocycle ω
and a differentiable singular cycle σ is given by the integral
∫
σ
ω.
4.4.2 Interpretation of the toroidal integral as evaluation pairing
We have [Eisθ(Ψ′′φ)] ∈ H1(SKθ
f,C). Here φ = (µ1, µ2) : T (Q)\T (A) → C∗. Let
SKθf
∼=⊕
[det(ξ)]∈HKθ
Γθξ\H3.
The paths σξ we described in 4.2.1 are not 1-cycles in SKθf. They are only relative
cycles (cf. [Ko] §5.2) giving rise to classes in H1(Γθξ\H3, ∂(Γθ
ξ\H3),Z). Since the
endpoints lie in the ∞- and 0-cusps (use
1 0
0 1s
K∞ = w0
1 0
0 s
K∞) they are,
in fact, relative cycles for H1(Γθξ\H3, Γ
θξ,B\e(B) ∪ Γθ
ξ,Bw\e(Bw),Z).
If we can show that Eisθ(Ψ′′φ) is a relative cocycle with respect to the ∞- and
0-cusps of each connected component, we can apply the evaluation pairing
H1(Γθξ\H3, e
′(B) ∪ e′(Bw), R)×H1(Γθξ\H3, e
′(B) ∪ e′(Bw),Z) → R,
given by ([ω], [σξ]) 7→∫
σξω (here we follow [BS] in writing e′(P ) for Γθ
ξ,P\e(P )). By
the functoriality in the O-algebra R, a multiple aω is integral only if a([ω], [σξ]) is.
Our toroidal integral now corresponds to the sum of these evaluation pairings for
each connected component. This allows us in the next section to deduce a lower
78
bound on the denominator of the Eisenstein cohomology class in terms of a special
L-value. We will show that the result of the toroidal integral is the inverse of the
special L-value up to p-adic units.
We prove now:
Lemma 4.13. Let Fθ be the finite extension of Fχ (see Definition 3.22) containing
the values of the finite order character θ. Then we have
[Eisθ(Ψ′′φ)] ∈
⊕
[det(ξ)]∈HKθ
H1(Γθξ\H3, e
′(B) ∪ e′(Bw), Fθ).
Proof. It is clear that [Eisθ(Ψ′′φ)] ∈ H1(SKθ
f, Fθ). From the form of the constant term
for res(Eis(ω0(φ, Ψφ))) (see Proposition 3.5) we deduce, by interchanging the finite
sums of the twists with the integral, that
res(Eisθ(Ψ′′φ)) = res(Eisθ(ω0(φ, Ψ′′
φ))) = ω0(φ, (Ψ′′φ)
θ) + d(φ)ω0(w0.φ, (Ψ′′w0.φ)
θ).
Here
(Ψ′′∗)
θ(g) =∑
x
θ(x)Ψ′′∗(g
1 −x/q
0 1
q
).
To check that the Eisenstein cocycle vanishes on 1-cycles of e′(B)∪e′(Bw) we now
translate to group cohomology and homology. We showed in Lemma 3.10 that the
restriction of [ω0(∗ · θ, (Ψ′′∗)
θ)] to H1(Γθξ,Bη\e(Bη), Fθ) ∼= H1(Γθ
ξ,Bη , Fθ) is represented
by the cocycle
η−1
a x
0 d
η 7→ (Ψ′′
∗)θ(ηfξ) ·
x
x
,
the two cases depending on the infinity type of ∗.We need to show therefore that (Ψ′′
∗)θ vanishes on ηfξ for η the identity matrix and
w0. We note that ξ ∈ G(Af ) can be chosen to be a diagonal matrix. Then vanishing
for η equal to the identity matrix follows immediately from∑
x∈(Oq/Prq)∗ θ(x) = 0 for
the finite order character θ. For η = w0 the vanishing follows from our definition of
the newvectors Ψnew∗ , of which Ψ′′
∗ is a multiple, and from our choice of q distinct
from the conductors of the characters µ1 and µ2.
79
4.4.3 Comparison with other methods
We briefly comment on other approaches to interpreting the toroidal integral. The
following method is used by Harder in [Ha02] for SL2(Z), in [Ko] §5.4 for Q(i), and
in [Ka] §5 for Γ1(p) ⊂ SL2(Z): Complete the relative cycles σξ (or rather their images
under powers of Tv for v the place corresponding to p) by a chain in H1(∂SKf,Z)
that is only supported on the infinity components of cusps “above 0”. Then the
evaluation pairing between an Eisenstein cohomology class supported only at cusps
“above ∞” and the completed cycle is given by the toroidal integral. However, when
these cusps coincide the calculation of the additional “boundary integral” and the
bounding of its denominator is non-trivial.
Kaiser [Ka] uses a twisting argument similar to the one in the next section (but for
the cycles, not the cohomology class) to deduce lower bounds for the denominator.
Our approach seems to explain why he can choose cycles such that the boundary
integrals vanish and provides a shorter alternative argument.
Skinner [S02a] gives a different interpretation of the toroidal integral. In [S02a]
§4 he introduces a theory of “partial Borel-Serre compactifications”. He reinterprets
the twisted Eisenstein cocycle Eisθ(Ψ′′) as a cocycle in the cohomology of the space
SKθf
⋃ξ(e
′(B) ∪ e′(Bw0)), which contains the closure of the σξ in SKθf. Since the
restriction of the twisted Eisenstein cohomology class to these cusps is trivial Eisθ(Ψ′′)
corresponds to a class cθ in H1c (SKθ
f
⋃ξ(e
′(B)∪e′(Bw0)),C). He shows that cθ is again
rational (in some finite extension of Fχ), and that if a ·Eisθ(Ψ′′) is integral, then a ·cθ
is, too. He considers for each connected component of SKfthe map of manifolds
σξ|R>0 : R>0 → jξ(Γξ\H3) ⊂ SKf.
This is a proper embedding giving rise to a map
σ∗ξ : H1c (SKθ
f
⋃
ξ
(e′(B) ∪ e′(Bw0)), R) → H1c (R>0, R) = H1
c (R>0,Z)⊗R ∼= R,
which is functorial in R and maps cθ to∫
σξEisθ(Ψ′′).
80
4.5 Bounding the denominator
Recall the definition of the denominator of a cohomology class from Definition
4.1. We are interested in bounding δ(Eis(Ψφ)) for the Ψφ defined in Section 3.2.
From now on, we consider an odd prime p of Z split in F and let p ⊂ O be one
of the primes dividing it. Let φ = (µ1, µ2) : T (Q)\T (A) → C∗ for µ1 and µ2 Hecke
characters with infinity type z and z−1, respectively, such that the conductors Mi of
µi are coprime to (p). Denote by Oφ the ring of integers in the finite extension of Fp
containing the values of µ1,f , µ2,f and Lalg(0, µ1/µ2). Here we use Theorem 2.1 on
the algebraicity and integrality of the special L-value.
For a finite order character θ of prime power conductor qr, with q a prime in
Z distinct from p and coprime to the conductors of the µi, let Oθ be the ring of
integers in the finite extension of Oφ containing the values of θ, Lalg(0, µ1θ), and
Lalg(0, (µ2θ)−1) (again we use Theorem 2.1).
Observe that
δ([Eis(Ψφ)]) ⊆ δ([Eis(Ψ′′φ)]) ⊆ Oφ,
and
δ([Eis(Ψ′′φ)])Oθ ⊆ δ([Eisθ(Ψ′′
φ)]).
In Section 4.4 we showed that the toroidal integral∑
[ξ]∈HKθ
∫σξ
Eisθ(Ψ′′φ) gives the
value of sums of evaluation pairings between relative cohomology and homology.
Their functoriality in the coefficient system implies that the denominator δ([Eisθ(Ψ′′φ)])
is bounded below by the denominator of the integral. Proposition 4.11 (for z=0) and
Lemma 4.4 imply that the integral equals L(0,µ1θ)L(0,(µ2θ)−1)L(0,µ1/µ2)
, up to units in Oθ (using
Lemma 3.21 one checks that
(θµ2)−1(M1q
r)θ(−1)(µ2/µ1)(qr)
1
2
∏v∈S
µ−12,v(−1)(µ2/µ1)(P
rv) ∈ O∗
θ .)
This means that δ([Eisθ(Ψ′′φ)]) is contained in the (possibly fractional) ideal(
L(0, µ1/µ2)
L(0, µ1θ)L(0, (µ2θ)−1)
)Oθ.
81
We would like to find finite order anticyclotomic characters θ such that the (al-
gebraic) L-factors in the denominator are p-adic units. We have at our disposal two
results on the non-vanishing modulo p of the L-values Lalg(0, θµ±1i ) as θ varies in an
anticyclotomic Zq-extension:
Theorem 4.14 (Finis [Fi2] Thm. 1.1). Let q - 2#Cl(F ) be a prime split in F ,
distinct from p. Consider Hecke characters λ of infinity type λ∞(z) = zkz1−k for a
fixed positive integer k with λ∗ = λ (where λ∗(x) = λ(x)−1|x|AF), conductor dividing
ddF q∞ for some fixed d, global root number W (λ) = 1, and such that no inert primes
congruent to -1 mod p divide the conductor of λ with multiplicity one. Then for all
but finitely many such Hecke characters
L(0, λ)Wp(λ)Ω1−2k(k − 1)!(2π√dF
)k−1 is a p− adic unit.
The p-adic root number Wp(λ) is defined as p−ordv0 (fλ) · τv0(λv0) for v0 the place
corresponding to p. For the definition of the Gauss sum τv0 see Section 2.6.
Hida has announced a similar result (for general CM fields). The pre-print [Hi04b]
includes the theorem:
Theorem 4.15 ([Hi04b] Theorem 4.3). Fix a character λ of split conductor
(i.e., such that the conductor is a product of primes split in F/Q) with infinity type
λ∞(z) = zk(
zz
)lfor k > 0 and l ≥ 0. Then
Lalg(λθ, 0) is a p− adic unit
for all but finitely many finite-order anticyclotomic characters θ of q-power conductor
for a split prime q distinct from p and coprime to the conductor of λ.
From these two results we deduce the following:
Proposition 4.16. Let µ1, µ2 as above. Assume in addition that either
(a) the characters satisfy µci = µi (for this infinity type this coincides with µ∗i = µi)
and that no inert primes congruent to −1 mod p divide either of the conductors
of µi with multiplicity one
82
(b) or the characters µi have split conductor.
Then there exists a prime q and a finite order anticyclotomic character θ of q-power
conductor such that Lalg(0, µ1θ) and Lalg(0, (µ2θ)−1) lie in O∗
θ .
Proof. All that remains to show is that for (a) Wp(λ) is a p-adic unit (we take k = 1
in Finis’ Theorem). This follows from the definition of the Gauss sum and (the proof
of) Lemma 3.21.
With this we have also proven:
Theorem 4.17. Under the same assumptions as the previous proposition we have
δ([Eis(Ψφ)]) ⊆ Lalg(0, µ1/µ2)Oφ.
Theorem 1.2 in the introduction follows from Theorem 4.17:
Corollary 4.18 (Theorem 1.2). Let χ be a Hecke character of infinity type z2 such
that χc = χ. Assume also that no inert primes congruent to −1 mod p or factors
of p divide the conductor of χ. Then there exist characters µi such that χ = µ1/µ2
and δ([Eis(Ψ(µ1,µ2))]) ⊂ (Lalg(0, χ)).
Proof. It suffices to show that χ can be factored as µ1/µ2 with characters satisfying
the conditions of the previous theorem. Put µ1 = µGχ and µ2 = µG, where µG is
the character from Lemma 3.18. Since µG is ramified only at the ramified places
in F and satisfies µcG = µG this choice φ = (µ1, µ2) satisfies the condition (a) of
Proposition 4.16.
CHAPTER V
The torsion problem
Recall from Lemma 3.24 that [res(Eis(Ψ(µ1,µ2)))] ∈ H1(∂SKsf,Oχ) if we assume
χ = µ1/µ2 to be anticyclotomic (see Definition 3.22 for Oχ and Fχ). In Chapter VI
we will show that if we can find c ∈ H1(SKsf,Oχ) with the same restriction to the
boundary as our Eisenstein cohomology class [Eis(Ψ(µ1,µ2))] ∈ H1(SKsf, Fχ) then this
implies a congruence between a certain integral multiple of the Eisenstein cohomology
class and a cohomology class in H1! (SKf
,Oχ).
The aim of this chapter is to isolate cases where we can find such a class c, which
means ruling out congruences of the Hecke eigenvalues of our Eisenstein class with
those of torsion classes in H2c . The existence of such torsion classes was shown,
for example, in R. Taylor’s thesis [T], and in calculations by Feldhusen ([F] p.26).
We manage to avoid this “torsion problem” after restricting to constant coefficient
systems and unramified χ, and excluding the two fields Q(√−1) and Q(
√−3).
Our strategy is to find an involution on the boundary cohomology such that (for
each connected component of SKsf)
H1(Γ\H3,Oχ)res³ H1(∂(Γ\H3),Oχ)−,
where the superscript ‘-’ indicates the -1-eigenspace of this involution. We prove
the existence of such an involution for all maximal arithmetic subgroups of SL2(F ),
extending a result of Serre for SL2(O). After checking that [res(Eis(Ψ(µ1,µ2)))] lies in
this −1-eigenspace, we deduce the existence of the integral lift c.
83
84
5.1 Involutions and the image of the restriction map
In this section we work with a general arithmetic subgroup Γ. Assuming that we
have an orientation-reversing involution on Γ\H3 such that
H1(Γ\H3,Oχ)res→ H1(∂(Γ\H3),Oχ)−
we show that the map is, in fact, surjective. The existence of such an involution will
be shown for maximal arithmetic subgroups in the following sections.
We first recall:
Theorem 5.1 (Poincare and Lefschetz duality). Suppose Γ ⊂ G(Q) is an
arithmetic subgroup. Let R be a Dedekind domain in which both the lowest common
multiple of the orders of stabilizers |Γx| as well as the greatest common divisor of the
indices of torsion-free subgroups of finite index in Γ are invertible. Then there are
perfect pairings
Hrc (Γ\H3, R)×H3−r(Γ\H3, R) → R for 0 ≤ r ≤ 3
and
Hr(∂(Γ\H3), R)×H2−r(∂(Γ\H3), R) → R for 0 ≤ r ≤ 2.
Furthermore, the maps in the exact sequence
H1(Γ\H3, R)res−→ H1(∂(Γ\H3), R)
∂−→ H2c (Γ\H3, R)
are adjoint, i.e.,
〈res(x), y〉 = 〈x, ∂(y)〉.
Proof. Serre states this in the proof of Lemma 11 in [Se70] for field coefficients, [AS]
Lemma 1.4.3 proves the perfectness for fields R and [U95] Theorem 1.6 for Dedekind
domains as above. Other references for this Lefschetz or “relative” Poincare duality
for oriented manifolds with boundary are [Ma99] Chapter 21, §4 and [G67] (28.18).
We use here that H3 is an oriented manifold with boundary and that Γ acts on it
properly discontinuously and without reversing orientation.
85
For the following, we just want to recall the definition of the pairings: Write
M = Γ\H3. As explained in [Ma99] there is an unique element (called the R-
fundamental class) zΓ ∈ H3(M, ∂M,R) such that ∂zΓ is the fundamental class of
∂M ∈ H2(∂M,R) induced by the R-orientation of M . The pairings are then given
by the cup product and evaluation on the respective fundamental classes.
In particular, one deduces the following lemma:
Lemma 5.2 (Poincare duality and orientation-reversing involutions). Sup-
pose Γ and R are as in the theorem and that 2 is invertible in R. Let ι be an
orientation-reversing involution on Γ\H3. Denoting by a superscript + (resp. −)
the +1-(resp. −1-) eigenspaces for the induced involutions on cohomology groups,
we have perfect pairings
Hrc (Γ\H3, R)± ×H3−r(Γ\H3, R)∓ → R for 0 ≤ r ≤ 3
and
Hr(∂(Γ\H3), R)± ×H2−r(∂(Γ\H3), R)∓ → R for 0 ≤ r ≤ 2.
Proof. That ι reverses the orientation on Γ\H3 means exactly that ι(zΓ) = −zΓ for zΓ
as in the proof of the theorem. This implies that +1- and −1-eigenspaces are “self-
orthogonal” under the duality pairing, or maximal isotropic subspaces. Since the
perfect pairing for the boundary uses the fundamental class ∂zΓ the same argument
applies to the boundary after checking that the connecting homomorphism ∂ is ι-
equivariant.
Lemma 5.3. Suppose in addition to the conditions of the previous theorem and
lemma that R is a complete discrete valuation ring with finite residue field of char-
acteristic p > 2. Suppose that we have an involution ι as in the lemma such that
H1(Γ\H3, R)res→ H1(∂(Γ\H3), R)ε,
where ε = +1 or −1. Then, in fact, the restriction map is surjective.
86
Proof. Let m denote the maximal ideal of R. Since the cohomology modules are
finitely generated (so the Mittag-Leffler condition is satisfied for lim←−H1(·, R/mr)), it
suffices to prove the surjectivity for each r ∈ N of
H1(Γ\H3, R/mr) ³ H1(∂(Γ\H3), R/mr)ε.
For these coefficient systems we are dealing with finite groups and can count the
number of elements in the image and the eigenspace of the involution; they turn
out to be the same. We observe that H1(∂(Γ\H3), R/mr) = H1(∂(Γ\H3), R/mr)+⊕H1(∂(Γ\H3), R/mr)− and that, by the last lemma,
#H1(∂(Γ\H3), R/mr)+ = #H1(∂(Γ\H3), R/mr)−.
Similarly we deduce from the adjointness of res and ∂ and the perfectness of the
pairings that im(res)⊥ = im(res) and so
#im(res) =1
2#H1(∂(Γ\H3), R/mr).
5.2 The involution for SL2(O)
We first make the following observation that will simplify the treatment of the
cohomology of the boundary components:
Lemma 5.4. For imaginary quadratic fields F other than Q(√−1) or Q(
√−3),
Γ ⊂ SL2(F ) an arithmetic subgroup, P a parabolic subgroup of ResF/Q(SL2/F ) with
unipotent radical UP , and R a ring in which 2 is invertible we have
H1(ΓP , R) ∼= H1(ΓUP, R),
where ΓP = Γ ∩ P (Q) and ΓUP= Γ ∩ UP (Q).
Proof. Serre shows in [Se70] Lemme 7 that ΓUP/ ΓP and that the quotient WP =
ΓP /ΓUPcan be identified with a subgroup of the roots of unity of F, i.e., of ±1
since F 6= Q(√−1),Q(
√−3). We recall his argument here: Suppose P = Bη for
87
η ∈ G(Q) (where Bη(Q) = η−1B(Q)η). The parabolic Bη is the stabilizer of a cusp
Dη ∈ P1(F ), the latter determined by the isomorphism B(Q)\G(Q) ∼= P1(F ) given
by [η] = [
a b
c d
] 7→ Dη := [c : d]. If g ∈ Bη(Q) denote by ω(g) the element in F ∗
such that g.x = ω(g)x for all x ∈ Dη. One gets a short exact sequence
1 → Uη(Q) → Bη(Q)ω→ F ∗ → 1.
The eigenvalues of any element of an arithmetic subgroup are integral. In particular,
if g ∈ ΓBη then ω(g) ∈ O∗, and we have the short exact sequence
1 → ΓUη → ΓBηω→ O∗ → 1
which proves the claim made at the start.
By the Inflation-Restriction sequence we deduce now that
H1(ΓP , R)∼−→ H1(ΓUP
, R)WP
since #WB = 2 ∈ R∗. Now WB ⊂ ±1 acts trivially on ΓU , and therefore
H1(ΓP , R) ' H1(ΓUP, R).
For a general arithmetic subgroup Γ ⊂ G(Q), the set Bη : [η] ∈ B(Q)\G(Q)/Γis a set of representatives for the Γ-conjugacy classes of Borel subgroups. The group
Uη is the unipotent radical of Bη. For D ∈ P1(F ) let ΓD = Γ ∩ UD, where UD is
the unipotent subgroup of SL2(F ) fixing D. Note that if Dη ∈ P1(F ) corresponds
to [η] ∈ B(Q)\G(Q) under the isomorphism of B(Q)\G(Q) ∼= P1(F ) given by right
action on [0 : 1] ∈ P1(F ) (see also Lemma 5.4) we have that UDη = Uη(Q) and
ΓDη = Γ ∩ Uη(Q) = ΓUη .
Let U(Γ) be the direct sum ⊕[D]∈P1(F )/ΓΓD. Up to canonical isomorphism this is
independent of the choice of representatives [D] ∈ P1(F )/Γ. The inclusion ΓD → Γ
defines a homomorphism α : U(Γ) → Γab.
Serre studies in [Se70] the kernel of U(Γ) → Γab. For Γ = SL2(O) he shows
(by choosing an appropriate set of representatives of P1(F )/SL2(O ∼= Cl(F )) that
88
there is a well-defined action of complex conjugation on U(SL2(O)) induced by the
complex conjugation action on the matrix entries of G∞ = GL2(C). Denoting by
U+ the set of elements of U(SL2(O)) invariant under the involution and by U ′ the
set of elements u + u for u ∈ U(SL2(O)), his result is as follows:
Theorem 5.5 (Serre [Se70] Theoreme 9). For imaginary quadratic fields F
other than Q(√−1) or Q(
√−3) the kernel of the homomorphism α : U(SL2(O)) →SL2(O)ab satisfies the inclusions
6U ′ ⊆ ker(α) ⊆ U+.
For our purposes we reinterpret this as follows:
Corollary 5.6. For imaginary quadratic fields F other than Q(√−1) or Q(
√−3),
Γ = SL2(O), and R a ring in which 2 and 3 is invertible, the image of the restriction
map
H1(Γ\H3, R) → H1(∂(Γ\H3), R)
is contained in the −1-eigenspace of the involution induced by
ι : H3 = C×R>0 → H3 : (z, t) 7→ (z, t).
Proof. We first note that since SL2(O) (in fact, even GL2(F )) has only 2- and 3-
torsion (see Proof of Lemma 1.1 in [F]) we have an isomorphism H1(Γ\H3, R) ∼=H1(Γ, R) by Proposition 2.5. From Section 2.8 we know that ∂(Γ\H3) is homotopy
equivalent to∐
[η]∈P1(F )/Γ
ΓBη\H3,
where ΓBη = Γ ∩Bη. That we have, in fact,
H1(∂(Γ\H3), R) ∼=∐
[η]∈P1(F )/Γ
H1(ΓUη , R) = H1(U(Γ), R)
follows from Lemma 5.4.
The involution ι on H3 extends canonically to H3. One checks that for γ ∈Γ we have ι(γ.(z, t)) = γ.ι(z, t). Since Γ = Γ this implies that ι operates on
89
Γ\H3 and Γ\H3, and hence on ∂(Γ\H3). We note that the involution induced
on∐
[η]∈P1(F )/Γ ΓBη\H3 (for a choice of representatives η fixed under complex conju-
gation) is given by
[(z, t)] ∈ ΓBη\H3 7→ [(z, t)] ∈ ΓBη\H3.
We define involutions on the singular cohomology groups
H1(Γ\H3, R), H1(∂(Γ\H3), R), and∐
[η]
H1(ΓBη\H3, R)
via the involution given on singular cocycles by pullback of ι on the corresponding
space. The involution on H1(U(Γ), R) = Hom(U(Γ), R) induced by the complex
conjugation action on U(Γ) in Serre’s theorem is given by ϕ 7→ ϕ, where ϕ(u) :=
ϕ(u).
We claim now that under the isomorphism
H1(∂(Γ\H3), R) ∼= H1(U(Γ), R)
the involutions on both sides correspond. As in Proposition 2.5 we get that the
isomorphism
H1(ΓBη\H3, R) ∼= H1(ΓUη , R)
is given on the level of cocycles by mapping a singular 1-cocycle f to
Gx0(f) : γ 7→ f([x0, γ.x0])
for some x0 ∈ H3, where [x0, γ.x0] denotes a 1-cycle with endpoints given by x0
and γ.x0 ∈ H3. The map on cohomology classes is independent of the choice of the
basepoint x0 and [x0, γ.x0]. We have now
Gx0(ι∗(f)) = Gι(x0)(f) ∈ Hom(ΓUη , R),
so the two involutions do indeed correspond.
We can therefore check that the image of the restriction maps is contained in the
−1-eigenspace on the level of group cohomology: The restriction map is given by
Hom(Γab, R) → Hom(U(Γ), R) : ϕ 7→ ϕ α.
90
By Serre’s theorem 0 = ϕ(α(uu)) = ϕ((α(u)) + ϕ(α(u)), so ϕ α(u) = ϕ(α(u)) =
−ϕ(α(u)) for any u ∈ U(Γ).
In order to apply Lemma 5.3 we still have to check that complex conjugation
reverses the orientation of H3, and therefore Γ\H3, since Γ (and more generally
SL2(C)) acts without reversing orientation, as we will show below. The orientation
of H3 uniquely determines the orientation of H3. By definition, H3 being orientable
means that one can find a consistent choice of generators of H3(H3,H3 − x, R) for
x ∈ H3 (for the definition of relative homology see Section 3.4). By choosing a
coordinate neighborhood of x, i.e., an open neighborhood U ⊂ H3 containing x
homeomorphic to the open unit ball in R3, one has isomorphisms (see [Ma99] p.
where the ‘ ’ denotes reduced cohomology groups and S2 is the standard 2-sphere.
This provides the connection to the “geometric” notion of orientation reversing:
Rotations preserve the generator of H2(S2, R), reflections in planes act by −1.
The planes in hyperbolic 3-space H3 = C×R>0 are either Euclidean hemispheres
or half-planes which are perpendicular to the boundary C of H3 (see [EGM] §I.1.1).
Complex conjugation on H3 is exactly a reflection in one of these half-planes, so is
orientation-reversing.
Earlier we also claimed that SL2(C) acts on H3 without reversing orientation.
This can easily be seen from the geometric definition of the action of SL2(C) via the
Poincare extension of the action on P1(C) (see Section 2.3): Since the determinant
equals one, the action on P1(C) is given by an even number of reflections in lines and
circles in C. The action on H3 is therefore given by an even number of reflections in
the corresponding hyperbolic planes.
91
Applying Lemma 5.3 we have therefore proven:
Corollary 5.7. For imaginary quadratic fields F other than Q(√−1) or Q(
√−3),
Γ = SL2(O), and R a complete discrete valuation ring in which 2 and 3 are invertible
and with finite residue field of characteristic p > 2, the restriction map
H1(Γ\H3, R) → H1(∂(Γ\H3), R)−
surjects onto the −1-eigenspace of the involution induced by
ι : H3 = C×R>0 → H3 : (z, t) 7→ (z, t).
5.3 The involution for other maximal arithmetic subgroups
Any maximal arithmetic subgroup of SL2(F ) is conjugate to one of the following
groups (see [EGM] Prop. 7.4.5): Let b be a fractional ideal and
H(b) := a b
c d
∈ SL2(F )|a, d ∈ O, b ∈ b, c ∈ b−1.
In this section we extend Theoreme 9 of [Se70] (Theorem 5.5) to these groups. After
we had discovered this generalization we found out that it had already been suggested
in [BN], but for our application we need more detail than is provided there.
Remark 5.8. Our choice of embeddings of ΓBη\H3 into the adelic boundary ∂SKf=
B(Q)\G(A)/KfK∞ in Section 3.3.3 means that the arithmetic subgroups Γ act from
the right on the set of cusps P1(F ) ∼= B(Q)\G(Q). All the actions of H(b) in this
section will therefore be written as right actions.
Note that since H(b) is the stabilizer of any lattice m⊕ n with m and n fractional
ideals of F such that m−1n = b, one can deduce
Lemma 5.9. Let a, b be two fractional ideals of F . If [a] = [b] in Cl(F )/Cl(F )2,
then H(a) = H(b)γ with γ ∈ GL2(F ). If the fractional ideals differ by the square of
an O-ideal, then γ can be taken to be in SL2(F ).
92
We first generalize a Theorem of Bianchi for SL2(O) (see [EGM] Theorem VII
2.4) to H(b). For this we need the following lemma.
Lemma 5.10. Let (x1, x2), (y1, y2) ∈ F × F . The following are equivalent:
(1) x1b + x2O = y1b + y2O.
(2) There exists σ ∈ H(b) such that (x1, x2) = (y1, y2)σ.
Proof. We follow exactly the proof for SL2(O) in [EGM].
(2) ⇒ (1) is clear.
(1) ⇒ (2): Put a = x1b + x2O = y1b + y2O. If a = (0), then there is nothing
to prove. Otherwise choose an n ∈ N and θ ∈ F ∗ such that an = (θ). Note that
x1, y1 ∈ ab−1. The equations (x1b + x2O)an−1 = (θ) and (y1b + y2O)an−1 = (θ)
show that there are α1, β1 ∈ an−1b and α2, β2 ∈ an−1 with θ = α1x1 + α2x2 and
θ = β1y1 + β2y2.
Put
σ =
y1α1+x2β2
θy2α1−x2β1
θ
y1α2−x1β2
θy2α2+x1β1
θ
.
It is easy to check that σ lies in H(b) and satisfies (2).
Definition 5.11. Define j : P1(F ) → Cl(F ) to be the map
j([z1 : z2]) = [z1b + z2O].
Clearly, j is well-defined. The preceding lemma now implies
Theorem 5.12. For Γ = H(b), the induced map
j : P1(F )/Γ → Cl(F )
is a bijection.
Proof. In light of lemma 5.10 the only thing left to show is the surjectivity of j. Given
a class in Cl(F ) take a ⊂ O representing it. By the Chinese Remainder Theorem
one can choose z2 ∈ O such that
• ord℘(z2) = ord℘(a) if ℘|a.
93
• ord℘(z2) = 0 if ℘ - a, ord℘(b) 6= 0.
Then one chooses z1 such that
• ord℘(z1b) > ord℘(z2) if ℘|a or ord℘(b) 6= 0.
• ord℘(z1b) = 0 if ℘|z2, ℘ - a, and ord℘(b) = 0.
These choices ensure that ord℘(z1b + z2O) = ord℘(a) for all prime ideals ℘.
Recall that for D ∈ P1(F ) we put ΓD = Γ ∩ UD, where UD is the unipotent
subgroup of SL2(F ) fixing D. The Theorem implies
Corollary 5.13. For Γ = H(b), if j([x1 : x2]) = j([y1 : y2]) then Γ[x1:x2] is conjugate
in Γ to Γ[y1:y2].
5.3.1 Representing elements of Γ[z1:z2]
Lemma 5.14. For any fractional ideal a (or projective O-module of rank 1),
Λ2(a) = 0.
Proof. One has a ⊕ a−1 ∼= O ⊕O (see, for example [Bour] VII, §4, Prop. 24). This
implies Λ2(a⊕ a−1) ∼= Λ2(O2) = O. Since Λ2(a⊕ a−1) = a⊗ a−1 ⊕ Λ2(a)⊕ Λ2(a−1)
we deduce Λ2(a) = 0.
Alternatively, observe that the localization of a at any prime ideal ℘ of O is a free
O℘-module of rank 1. Since the exterior product commutes with localization this
also proves the Lemma.
Recall the definition of Γ[z1:z2] from the start of Section 5.2. The following Lemma
will be useful for studying the kernel of α : U(Γ) = ⊕[D]∈P1(F )/ΓΓD → Γab:
Lemma 5.15. For Γ = H(b), Γ[z1:z2] is conjugate in H(b) to
θ1 t
0 1
θ−1 : t ∈ a−2b,
94
where a = z1b + z2O and θ is an isomorphism O ⊕ b∼→ a⊕ a−1b of determinant 1,
i.e., such that its second exterior power
Λ2θ : Λ2(O ⊕ b) = b → Λ2(a⊕ a−1b) = a⊗ a−1b = b
is the identity.
Proof. The main change to Serre’s method in [Se70] §3.6 is that we consider the lattice
L := O ⊕ b instead of O2. We claim there exists a projective rank 1 submodule E
of L containing a multiple of (z1, z2). Let E be the kernel of the O-homomorphism
L = O ⊕ b → F given by (x, y) 7→ yz1 − xz2. Since the image is a = z1b + z2O, we
get L/E ∼= a, so L/E is projective of rank 1 and L decomposes as E ⊕ L/E.
By definition Γ[z1:z2] fixes L∩λ(z1, z2), λ ∈ F, but this is exactly E. Since Γ[z1:z2]
is unipotent it can therefore be identified with HomO(L/E, E). Using the exterior
product b = Λ2(L) = Λ2(E ⊕ L/E) = E ⊗O L/E, we get that E is isomorphic
to (L/E)−1 ⊗ b (here we use the preceding lemma). This implies an isomorphism
HomO(L/E, E) = (L/E)−1 ⊗ E ∼= (L/E)−1 ⊗ (L/E)−1 ⊗ b ∼= a−2b. Choosing an
isomorphism θ : L → L/E ⊕E ∼= a⊕ a−1b of determinant 1 we can represent Γ[z1:z2]
as stated above.
Remark 5.16. 1. Alternatively, Γ[z1:z2] is conjugate to θ′ 1 0
−t 1
θ′−1 : t ∈ a−2b
for an isomorphism θ′ : O ⊕ b → a−1b ⊕ a of determinant 1. Up to conjugation by
an element in H(b), θ′ is given by
0 −1
1 0
θ.
2. For Γ = SL2(O) our Γ[z1:z2] equals Γ[a−1b] in Serre’s notation in [Se70] §3.6, not
Γ[a]. This follows from our different choice of the isomorphism j : P1(F )/Γ → Cl(F ).
5.3.2 The involution on U(Γ)
If the class of b in Cl(F ) is a square, H(b) is isomorphic to SL2(O) by Lemma
5.9, and the involution on U(SL2(O)) induced by complex conjugation and Serre’s
Theoreme 9 can easily be transferred to U(H(b)). We therefore turn our attention
95
to the case when [b] is not a square in Cl(F ). Note that this implies that [b] has
even order, since any odd order class can be written as a square.
Definition 5.17. Define an involution on H(b) to be the composition of complex
conjugation with an Atkin-Lehner involution, i.e. by
H =
a b
c d
7→ AHA−1 =
d −Nm(b)c
−bNm(b)−1 a
,
where A =
0 1
−Nm(b)−1 0
.
Like Serre, we will choose a set of representatives for the cusps P1(F )/H(b)
on which this involution acts. For this we observe that if Γ[z1:z2] fixes [z1 : z2]
then AΓ[z1:z2]A−1 fixes [z1 : z2]A
−1 = [z2 : −Nm(b)z1]. We use the isomorphism
j : P1(F )/H(b) → Cl(F ) to show that this action on the cusps is fixpoint-free. We
observe that if j([z1 : z2]) = a then j([z1 : z2]A−1) = [z2b + Nm(b)z1O] = [ab]. Note
that [a] 6= [ab] in Cl(F ) since otherwise [a2] = [Nm(a)b] = [b], i.e., [b] a square,
contradicting our hypothesis. So Cl(F ) can be partitioned into pairs (ai, aib).
Choosing [zi1 : zi
2] ∈ P1(F ) such that ai = zi1b + zi
2O we obtain
U(H(b)) =⊕
(ai,aib)
(Γ[zi1:zi
2] ⊕ AΓ[zi1:zi
2]A−1).
Our choice of representatives of P1(F )/H(b) shows that the involution operates on
U(H(b)) and, in fact, by identifying Γ[zi1:zi
2] with θ1 s
0 1
θ−1 : s ∈ a−2
i b for
θ : O ⊕ b → ai ⊕ a−1i b and AΓ[zi
1:zi2]A
−1 with θ′ 1 0
−t 1
θ′−1 : t ∈ ai
−2b−1 for
θ′ = AθA−1 : O⊕b → ai−1⊕ aib, we can describe the involution on each of the pairs
as
(s, t) ∈ a−2i b⊕ ai
−2b−1 7→ (tNm(b), sNm(b)−1).
96
5.3.3 Generalization of Serre’s Theoreme 9
Denote by U+ the set of elements of U(H(b)) invariant under the involution
H 7→ AHA−1, and by U ′ the set of elements u + AuA−1 for u ∈ U(H(b)).
Theorem 5.18. For Γ = H(b) with [b] a non-square in Cl(F ) , the kernel N of the
homomorphism
α : U(Γ) → Γab
coming from the inclusion ΓD → Γ for D ∈ P1(F ) satisfies 6U ′ ⊂ N ⊂ U+.
Remark 5.19. Given Lemma 5.9, this provides the extension of Serre’s Theorem to
all maximal arithmetic subgroups of SL2(F ).
Proof. With small modifications, we follow Serre’s proof of his Theoreme 9. As in
Serre’s case, it suffices to prove the inclusion 6U ′ ⊂ N , i.e. that 6(u + AuA−1) maps
to something in the commutator [H(b), H(b)]:
Suppose that we have 6U ′ ⊂ N , but that there exists an element u ∈ N not con-
tained in U+. Then the subgroup of N generated by 6U ′ and u has rank #Cl(F )+1.
This contradicts the fact that the kernel of α has rank #Cl(F ) (see [Se70] Theoreme
7). (The latter is proven by showing dually that the rank of the image of the restric-
tion map H1(H(b)\H3, R) → H1(∂(H(b)\H3), R) has half the rank of that of the
boundary cohomology. This we showed in the proof of Lemma 5.3).
To prove 6U ′ ⊂ N now we make use of Serre’s Proposition 6:
Proposition 5.20 ([Se70] Proposition 6). Let q be a fractional ideal of F and
let t ∈ q and t′ = t/Nm(q) so that t′ ∈ q−1. Put xt =
1 t
0 1
and yt =
1 0
−t′ 1
.
Then (xtyt)6 lies in the commutator subgroup of H(q).
Put a := z1b+z2O. If u ∈ Γ[z1:z2], identify it with θ−1
1 t
0 1
θ for some t ∈ a−2b
and θ : O ⊕ b → a ⊕ a−1b of determinant 1. One easily checks that AuA−1 then
corresponds to (AθA−1)
1 0
−tNm(b)−1 1
(AθA−1). Like Serre, we use now that
97
since [a] = [a−1], AuA−1 is also given by Corollary 5.13 by B−1θ−1
1 0
−t′ 1
θB for
t′ = tNm(b)−1Nm(a)2 and B ∈ H(b) taking
Nm(b)z2
z1
to Nm(a)−1
Nm(b)z2
z1
.
Since θ−1xtytθ is a representative of u + BAuA−1B−1, we deduce from the above
Proposition with q = a−2b that 6(u + BAuA−1B−1) and therefore 6(u + AuA−1) lie
in [H(b), H(b)].
Remark 5.21. Since w0 =
0 1
−1 0
∈ SL2(O) it is possible to unify the treatment
of all maximal arithmetic subgroups H(b) independent of b being a square in the
class group or not. However, since it would not significantly simplify notation or
exposition we did not pursue this here.
We again want to reformulate our result in the following form:
Corollary 5.22. For imaginary quadratic fields F other than Q(√−1) or Q(
√−3),
Γ = H(b) with [b] a non-square in Cl(F ), and R a ring in which 2 and 3 is invertible,
the image of the restriction map
H1(Γ\H3, R) → H1(∂(Γ\H3), R)
is contained in the −1-eigenspace of the involution induced by
ι : H3 = C×R>0 → H3 : (z, t) 7→ A.(z, t)
for A =
0 1
−Nm(b)−1 0
.
Proof. It is easy to check that AΓA−1 = Γ and that
ι(γ.(z, t)) = (AγA−1).ι(z, t).
Now we proceed exactly as in the proof of Corollary 5.6.
To be able to apply Lemma 5.3 we again show that this involution is orientation-
reversing on H3. Since A ∈ GL2(C) acts on H3 via A′ = (det(A)−12 )A ∈ SL2(C), its
98
action on H3 preserves the orientation (see end of Section 5.2, before Corollary 5.7).
Recalling further that the action of complex conjugation is orientation-reversing,
our involution is therefore orientation-reversing. For future reference we record the
application of Lemma 5.3 to our involution:
Corollary 5.23. For imaginary quadratic fields F other than Q(√−1) or Q(
√−3),
Γ = H(b) with [b] a non-square in Cl(F ), and R a complete discrete valuation ring
in which 2 and 3 are invertible and with finite residue field of characteristic p > 2,
the restriction map
H1(Γ\H3, R) → H1(∂(Γ\H3), R)−
surjects onto the −1-eigenspace of the involution induced by
ι : H3 = C×R>0 → H3 : (z, t) 7→ A.(z, t)
for A =
0 1
−Nm(b)−1 0
.
5.4 Unramified characters χ
In Chapter III we defined an Eisenstein cohomology class associated to (µ1, µ2) on
an adelic symmetric space SKsf
for a specific choice of Ksf . In this section we will show
that for unramified characters χ = µ1/µ2 we can always write the corresponding SKsf
as a disjoint union of Γ\H3 with Γ = H(b) for fractional ideals b. This allows us
to apply our results for maximal arithmetic subgroups from the previous sections
by considering the restriction maps to the boundary separately for each connected
component.
We recall now from Section 2.3: The space SKfhas several connected components,
in fact, strong approximation implies that the fibers of the determinant map
SKf= G(Q)\G(A)/(KfK∞) ³ HK := A∗
F /det(K)F ∗
(where K = KfK∞) are connected. Any ξ ∈ G(Af ) gives rise to an injection
jξ : G∞ → G(A) with jξ(g∞) = (g∞, ξ) and, after taking quotients, to a component
Γξ\G∞/K∞ → G(Q)\G(A)/K,
99
where Γξ := G(Q) ∩ ξKfξ−1. This component is the fiber over det(ξ).
Let F now be any imaginary quadratic field different from Q(√−1) and Q(
√−3).
Let χ : F ∗\A∗F /
∏v O∗
v → C∗ be an unramified Hecke character of infinity type
χ∞(z) = z2. We consider (µ1, µ2) : T (Q)\T (A) → C∗ with µ1, µ2 Hecke characters
of infinity type z and z−1, respectively, such that χ = µ1/µ2. Let
Kf = Ksf =
∏v∈S
U1(M1,v)∏
v/∈S
GL2(Ov),
for S the set of places where either of the µi are ramified (they have to be ramified
because their infinity types are z or z−1). Here U1(M1,v) = k ∈ GL2(Ov) : det(k) ≡1 mod M1,v and M1,v is the conductor of µ1,v. Put Ks = Ks
fK∞.
Assumption 5.24. The only unit in O∗ congruent to 1 modulo M1,v for some v ∈ S
is 1.
Under this assumption we have Ksf ∩GL2(F ) = SL2(O).
We want to find a set ti ∈ G(Af ), i = 1 . . . hKs with det(ti) providing a system
of representatives for HKs such that the Γti equal H(bi) for appropriate fractional
ideals. For a finite idele a, denote by (a) the corresponding fractional ideal. Since
in our case det(Ksf ) has finite index in
∏v O∗
v (HKs is a generalized ray class group
modulo the conductor of µ1), (det(ti)) has to run through a number of copies of a
set of representatives for the ideal class group Cl(F ).
We first consider the special case where #HKs is odd (this requires the class
number of F to be odd, but also imposes restrictions on Ksf and the ramification
of our factorization χ = µ1/µ2): If HKs is a group of odd order hKs then any
element of HKs is a square. Rather than the usual choice of ti as
ai 0
0 1
such that
ai ∈ A∗F,f , i = 1 . . . hKs is a set of representatives for HKs , we can therefore take
ti =
bi 0
0 bi
∈ G(Af ), where [bi]
2 = [ai] ∈ HKs . This means that in this case we
can ensure that all Γti equal Γ := G(Q) ∩Ksf = SL2(O).
Next we consider the case where #HKs is even. First, we choose a system of
100
representatives γj of
ker(HKs → Cl(F )) ∼= O∗\∏
v
O∗v/det(Ks
f ).
Then take a set of representatives ak of Cl(F )/(Cl(F ))2 in A∗F,f (represent the
principal ideals by (1)). Lastly, we choose a set b2m representing Cl(F )2.
Now we can define ti as follows: The set is given by the elements
γjakbm 0
0 bm
∈ G(Af ),
as j, k and m run through their indexing sets.
We obtain a decomposition
SKsf
∼=hKs∐i=1
Γti\G∞/K∞ ∼=hKs∐i=1
Γti\H3,
where Γti = G(Q) ∩ tiKsf t−1i , with the i-th connected component Γti\H3 being em-
bedded via jti .
Note that if ti =
γjakbm 0
0 bm
, the associated Γti = H((ak)) under Assumption
5.24. Also, by construction, either ak = 1 or [(ak)] is not a square in Cl(F ).
5.5 Integral lift of constant term
In this section we show that for χ = µ1/µ2 an unramified character we can lift the
constant term of the Eisenstein cohomology class to an integral class, i.e., that there
exists c ∈ H1(SKsf,Oχ) with the same restriction to the boundary as the Eisenstein
cohomology class.
Our setup is as follows: Let F be an imaginary quadratic field, distinct from
Q(√−1) and Q(
√−3). Suppose p is a prime of F such that the underlying rational
prime p is greater than 3 and splits in F .
Let µ1, µ2 : F ∗\A∗F → C∗ be Hecke characters of infinity type z and z−1, re-
spectively, such that χ = µ1/µ2 is unramified. Denote by S the set of places where
101
the µi are ramified. Assume in addition that 1 is the only unit in O∗ congru-
ent to 1 modulo M1,v for some v ∈ S (i.e., that Assumption 5.24 holds). Put
φ = (µ1, µ2) : T (Q)\T (A) → C∗. Let Oχ denote the ring of integers in the finite
extension Fχ of Fp obtained by adjoining the values of both µi,f and Lalg(0, χ).
In the last section we showed that under these conditions there exist ti ∈ G(Af )such that det(ti) is a system of representatives for HKs and such that the Γti are all
either SL2(O) or H(b) for fractional ideals b whose classes are non-square in Cl(F ).
Since unramified characters are anticyclotomic (see Lemma 3.16) Lemma 3.24 applies
so that [res(Eis(ω0(φ, Ψφ)))] is integral. We can further show:
Proposition 5.25.
[res(Eis(Ψφ))] ∈ H1(∂SKsf,Oχ)−,
where the latter is defined via the isomorphism to
⊕
[det(ti)]∈HKs
H1(∂(Γti\H3),Oχ)−
for the choice of ti’s from the last section and where the involutions on each of the
connected components are defined as in Corollaries 5.7 (if Γti = SL2(O)) and 5.23
(if Γti = H(b) for b a non-square in Cl(F )).
Remark 5.26. Together with Corollaries 5.7 and 5.23 this shows the existence of
an integral lift of the constant term. Note that Oχ is the ring of integers in a finite
extension of Fp so that the conditions in these corollaries are satisfied.
Proof. We shorten the notation to
ω0(Ψφ)) := ω0(φ, Ψφ)) and ω0(Ψw0.φ) := ω0(w0.φ, Ψw0.φ).
Under the assumptions in this section we have that
We need to show therefore that the cohomology class determined by the constant
term lies in the −1-eigenspace of the involution induced by u 7→ u for Γti = SL2(O)
and by u 7→ AuA−1 for Γti = H(b).
The restriction to Γti,P\H3 was denoted by [restiP (Eis(ω0(Ψφ)))] and equals, by
Lemma 3.6 and the proof of Lemma 3.24, the class of
ω0(Ψφ)tiP − ω0(Ψw0.φ)ti
P .
For P = Bη the image of the latter under Gη−1∞ K∞ (we will drop the subscript from
now on) was calculated in Lemma 3.10 and is given in our case (when m = n = 0)
by:
G(ω0(Ψφ)tiBη)(η−1
∞
1 x
0 1
η∞) = xΨφ(ηf ti)
G(ω0(Ψw0.φ)tiBη)(η−1
∞
1 x
0 1
η∞) = xΨw0.φ(ηf ti).
Note that by Lemma 5.4 we can restrict to Uη.
Case (1). We again first treat the notationally simpler case where all Γti equal
SL2(O) and the involution is complex conjugation on the matrix entries.
We claim that
G(ω0(Ψφ)tiBη)(g) = G(ω0(Ψw0.φ)
tiBη)(g) for all g ∈ Γti,Uη .
Given the form of the constant term this immediately implies that it lies in the −1
eigenspace for the involution induced by complex conjugation.
Recall that in this case ti =
γibi 0
0 bi
for some γi ∈ O∗ and bi ∈ A∗
F,f . We will
also use that the Ψφ (which were defined in Section 3.2 as a product of local factors)
103
satisfy
Ψφ(
a b
0 d
k) = µ1(a)µ2(d) for
a b
0 d
∈ B(A), k ∈
∏v
SL2(Ov) ⊂ Ksf .
Note that, in particular, Ψφ(bg) = φ−1∞ (b)Ψφ(g) for b ∈ B(F ) ⊂ G(Af ).
We are therefore left to show that Ψφ(ηf ti) = Ψw0.φ(ηf ti). For this we use the
Bruhat decomposition of matrices in GL2(F ) given by:
(5.1)
a b
c d
=
1 b/d
0 1
a 0
0 d
if c = 0,
1 a/c
0 1
ad−bcc
0
0 −c
0 1
−1 0
1 d/c
0 1
otherwise.
Since Ψφ(
a b
0 d
g) = Ψφ(
a b
0 d
)Ψφ(g) we can consider separately the cases
(a) η =
a b
0 d
for a, b, d ∈ F and
(b) η =
0 1
−1 0
1 e
0 1
for e ∈ F .
We check that for (a)
Ψφ(ηf
γibi 0
0 bi
) = µ1(γibi)µ2(bi)Ψφ(ηf )
and
Ψw0.φ(ηf
γibi 0
0 bi
) = µ2(γi)|γi|µ1(bi)µ2(bi)Ψw0.φ(ηf ).
Since γi ∈ O∗ and χ = µ1/µ2 is unramified it suffices to show that Ψφ(ηf ) =
Ψw0.φ(ηf ). In case (b) we similarly reduce to this assertion.
In (a) we get Ψφ(ηf ) = µ−11,∞(a)µ−1
2,∞(d) = da. Since w0.φ has infinity type (z, z−1)
this equals Ψw0.φ(ηf ). In (b) we need to calculate the Iwasawa decomposition of η in
104
GL2(Fv) if e /∈ Ov (at all other places Ψφ(ηv) = Ψw0.φ(ηv) = 1). It is given by 0 1
−1 0
1 e
0 1
=
e−1 0
0 e
−1 0
−e−1 −1
.
So, if e /∈ Ov then Ψφ(ηv) = (µ2/µ1)v(e) = χ−1v (e), which we claim matches
Ψw0.φ(ηv) = (µ1/µ2)v(e)|e|−2v . This follows from χc = χ and χχ = | · |2 (for the
latter see Lemma 3.14).
Case (2). We now treat the case of Γti = H(b), where the involution is induced
by H 7→ AHA−1 for A =
0 1
−N−1 0
with N = Nm(b). Considering the effect of
the involution on the cusp corresponding to Bη, we claim that
G(ω0(Ψφ)tiBη)(g) = G(ω0(Ψw0.φ)
tiBηA−1 )(AgA−1) for all g ∈ Γti,Uη .
Recall that ti =
xibi 0
0 bi
for some xi, bi ∈ A∗
F,f . We have to show that
(5.2) Ψφ(ηf ti) = Ψw0.φ(ηfA−1ti).
Again making use of the Bruhat decomposition, we need to only consider η as in
cases (a) and (b) above. Following the arguments used for Case (1), Case(a) re-
duces immediately to showing that Ψφ(ti) = Ψw0.φ(A−1ti). The left hand side equals
µ1,f (xibi)µ2,f (bi), the right hand side is
Ψw0.φ(
N 0
0 1
0 1
−1 0
xibi 0
0 bi
) = N−1Ψw0.φ(
bi 0
0 xibi
)
= N−1µ1,f (xibi)µ2,f (bi)|xi|−1f .
Equality follows from |xi|−1f = Nm(b).
For (b), one quickly checks that for η =
0 1
−1 0
the two sides in (5.2) agree.
For the general η =
0 1
−1 0
1 e
0 1
one shows that, on the one hand,
ηf
xibi 0
0 bi
=
bi 0
0 xibi
0 1
−1 0
1 exi
0 1
,
105
and on the other hand,
ηfA−1
xibi 0
0 bi
=
xibi 0
0 biN
0 1
−1 0
1 exi/N
0 1
.
Since (xixi) = (N) the valuations of exi/N agrees with that of exi. Repeating
the calculation for η = w0 and then applying the argument from Case 1(b) (since
χ is unramified we are only concerned about the valuation of the upper right hand
entry) we also obtain equality.
CHAPTER VI
Bounding the Eisenstein ideal
After defining an Eisenstein ideal in a Hecke algebra acting on cohomological
cuspidal automorphic forms, we put the results of Chapters III-V together to prove
a bound for its index in terms of a special L-value.
Recall our setup: Let F be an imaginary quadratic field, distinct from Q(√−1)
and Q(√−3). Suppose p is a prime of F such that the underlying rational prime p
is greater than 3 and splits in F .
We consider unramified Hecke characters χ : F ∗\A∗F → C∗ of infinity type z2
(i.e. χ∞(z) = z2). We gave examples of such characters in Section 3.5 and showed
that they are anticyclotomic, meaning that they satisfy χc = χ. In Corollary 4.18
we obtained characters µ1, µ2 : F ∗\A∗F → C∗ of infinity type z and z−1, respectively,
such that χ = µ1/µ2, for which we could bound from below the denominator of the
Eisenstein cohomology class
[Eis(ω0(φ, Ψφ))] ∈ H1(SKsf, Fχ),
where φ = (µ1, µ2) : T (Q)\T (A) → C∗. For the definition of Eis(ω0(φ, Ψφ)) and Ksf
see Sections 3.2 and 3.3. We recall that Fχ is the finite extension of Fp obtained by
adjoining the values of the finite part of both µi and Lalg(0, χ) (cf. Theorem 2.1) and
that we call its ring of integers Oχ. The choice of characters in the proof of Corollary
4.18 is µ1 = χ·µG and µ2 = µG, where µG is Greenberg’s character from Lemma 3.18.
The character µG has conductor (from now on denoted by M1) divisible precisely by
106
107
the primes ramified in F , and at these primes its restriction to the local units has
order 2. This also ensures that 1 is the only unit in O∗ congruent to 1 modulo the
conductors of µi, so that Ksf ∩ G(Q) = SL2(O) (cf. Assumption 5.24). We denote
by S the set of places where µG is ramified.
6.1 Diamond operators
We assume now in addition that p does not divide #Cl(F ). Let H be the p-Sylow
subgroup of the ray class group ClM1(F ) ∼= F ∗\A∗F /C∗U(M1). Since p does not
divide the class number of F we have
(6.1) H ∼=∏v∈S
Hv ⊂∏v∈S
O∗v/(1 + M1,v) ∼= (O/M1)
∗
for Hv the p-Sylow subgroups of O∗v/(1+M1,v). We define a compact open subgroup
KH,sf containing Ks
f by
KH,sf :=
∏
v/∈S
GL2(Ov)∏v∈S
UHv(M1,v),
where
UHv(M1,v) = k ∈ GL2(Ov) : det(k) ∈ Hv mod M1,v.
Since the spherical vector Ψ0φv
defined in (3.3) is right-invariant under UHv(M1,v)
due to µ1,v|O∗v having order 2 we see that ω0(φ, Ψφ) also defines a nontrivial class in
H1(∂SKH,sf
,Oχ) for the slightly larger group KH,sf (cf. the corresponding statement
for Ksf in Proposition 3.23) and Eis(ω0(φ, Ψφ)) defines a class in H1(SKH,s
f, Fχ) which
we will denote by cχ. Our arguments in chapters III-V apply to KH,sf and this
particular character φ = (χ · µG, µG) without change. Note, in particular, that
KH,sf ∩G(Q) = SL2(O) also, since −1 /∈ H (cf. Section 5.4).
By [U98] §1.2 and §1.4.5 we have for any O-algebra R an R-linear action of the
ray class group ClM1(F ) on H1(SKH,sf
, R) via the diamond operators (see Section
2.9.2). For a prime ideal q not dividing M1 the action of [q] ∈ ClM1(F ) is given by
the Hecke operator
Sv = [KH,sf
πv 0
0 πv
KH,s
f ]
108
for v the place corresponding to q and πv a uniformizer of Ov. This uses that
KH,sf ⊃ K(M1) :=
a b
c d
∈ K0
f :
a b
c d
≡
1 0
0 1
mod M1
.
We claim that the action of the diamond operators Sv, v /∈ S, is trivial if [(πv)] ∈H ⊂ ClM1(F ) (recall that we use the notation (g) for the fractional ideal generated
by a finite idele g). By (6.1) we have [(πv)] = [αO] for some α ∈ O with α ∈ H
mod M1. Therefore Sv has the same action as [KH,sf
α′ 0
0 α′
KH,s
f ] for α′ ∈ A∗F an
idele with (α′f ) = αO and α′v = 1 for each v ∈ S and v = ∞. But
α′ 0
0 α′
∈ G(A)
can be written as the product of
α 0
0 α
∈ Z(Q) → G(A) and an element of
Z∞ ·KH,sf and so its Hecke action is trivial.
We conclude that we have an action of Oχ[ClM1(F )/H] on H1(SKH,sf
, R) for any
Oχ-algebra R. Since ClM1(F )/H has order prime to p, Oχ[ClM1(F )/H] is semisimple.
For ω := µ1µ2, which can be viewed as a character of ClM1(F )/H, let eω be the
idempotent associated to ω−1, so that Sveω = ω(πv)−1eω. For R = C the idempotent
eω projects to cuspforms with central character ω via the Eichler-Shimura-Harder
isomorphism.
6.2 Main result
Recall that H i! (SKH,s
f, R) := im(H1
c (SKH,sf
, R) → H1(SKH,sf
, R)) for any Oχ-algebra
R and that H1! (SKH,s
f,Oχ) := im(H1
! (SKH,sf
,Oχ) → H1! (SKH,s
f, Fχ)).
Definition 6.1. Denote by Tχ the Oχ-subalgebra of EndOχ(eωH1! (SKH,s
f,Oχ)) gen-
erated by the Hecke operators Tv = [KH,sf
1 0
0 πv
KH,s
f ] for all primes v /∈ S.
109
Definition 6.2 (Eisenstein ideal). We call the ideal Iµ1,µ2 ⊆ Tχ generated by
Tv − µ−1
1,v(Pv)− µ−12,v(Pv)Nm(Pv)|v /∈ S
the Eisenstein ideal associated to (µ1, µ2). Here Pv is the maximal ideal in Ov.
Our main result can now be stated as:
Theorem 6.3. There is an Oχ-algebra surjection
Tχ/Iµ1,µ2 ³ Oχ/(Lalg(0, χ)
).
Proof. Recall the long exact sequence
. . . → H1c (SKH,s
f, R) → H1(SKH,s
f, R)
res−→ H1(∂SKH,sf
, R) → H2c (SKH,s
f, R) → . . .
for any Oχ-algebra R. The class cχ ∈ H1(SKH,sf
, Fχ) is annihilated by Tv−µ−11,v(Pv)−
µ−12,v(Pv)Nm(Pv), v /∈ S (cf. Lemma 3.11). Since χc = χ, its (non-trivial) restriction
to the boundary res(cχ) lies in H1(∂SKH,sf
,Oχ) (cf. Lemma 3.24). In Corollary
4.18 we showed that the denominator of cχ is bounded below by Lalg(0, χ) (i.e.,
δ(cχ) ⊂ (Lalg(0, χ)). In Chapter V, we proved that there exists c ∈ H1(SKH,sf
,Oχ)
with the same restriction to the boundary as the Eisenstein cohomology class cχ.
Note that
res(eωc) = eωres(c) = eωres(cχ) = res(cχ)
since
Sv(cχ) = ω−1(πv)cχ.
We can now prove the theorem following the proof of Proposition 6.2 in [S02a]:
Without loss of generality, we can assume that δ(cχ) ( Oχ; there is nothing to prove
otherwise. Let δ be a generator of δ(cχ). Then δcχ is an element of an Oχ-basis of
eωH1(SKH,sf
,Oχ). By construction, c0 := δ ·(eωc−cχ) ∈ H1! (SKH,s
f,Oχ) is a nontrivial
element of an Oχ-basis of eωH1! (SKH,s
f,Oχ). Extend c0 to an Oχ-basis c0, c1, . . . cd of
H1! (SKH,s
f,Oχ). For each t ∈ Tχ write
t(c0) =d∑
i=0
ai(t)ci, ai(t) ∈ Oχ.
110
Then
(6.2) Tχ → Oχ/(δ), t 7→ a0(t) mod δ
is an Oχ-module surjection. We claim that it is independent of the Oχ-basis chosen
and that it is a homomorphism of Oχ-algebras with the Eisenstein ideal Iµ1,µ2 con-
tained in its kernel. To prove this it suffices to check that each a0(Tv − µ−11,v(Pv) −
Nm(Pv)µ−12,v(Pv)), v /∈ S is contained in δOχ. This is an easy calculation. Let
tv = Tv − µ−11,v(Pv) − Nm(Pv)µ
−12,v(Pv). Then by Lemma 3.11, tvcχ = 0 and hence
tvc0 = tvδeωc ∈ δH1! (SKH,s
f,Oχ). Thus a0(tv) ∈ δOχ and (6.2) is a well-defined Oχ-
algebra surjection, coinciding with Tv 7→ µ−11,v(Pv) + Nm(Pv)µ
−12,v(Pv) mod δ. Since
Oχ/(δ) ³ Oχ/(Lalg(0, χ)) by Corollary 4.18, this proves our theorem.
Remark 6.4. 1. Note that the right-hand side in the statement of the theorem
is nontrivial if p divides Lalg(0, χ). If the left-hand side Tχ/Iµ1,µ2 is non-trivial
then a minimal prime contained in Iµ1,µ2 gives rise (via the Eichler-Shimura-Harder
isomorphism) to a cuspidal eigenform with eigenvalues congruent to those of the
Eisenstein cohomology class. Using the same notation as in the proof, the class
c ∈ H1(SKH,sf
,Oχ) exhibits a non-trivial element of the cohomological congruence
module M/(N ⊕N ′) for M = H1(SKH,sf
,Oχ), N = H1! (SKH,s
f,Oχ), and N ′ = δcχOχ.
See [P] and [Gh] for more details on congruence modules and their relationships.
2. L(0, χ) 6= 0 since 0 is the abscissa of convergence for this (non-unitary) infinity
type (see [La] XV §4).
3. Observe that for Q(√−1) and Q(
√−3) (the fields excluded above) the bound-
ary cohomology for constant coefficients and maximal compact open subgroups Kf
is trivial.
CHAPTER VII
Application to bounding Selmer groups
In this chapter we will demonstrate how to apply Theorem 6.3 to bound the size
of certain Selmer groups from below by Lalg(0, χ).
7.1 Background
We keep the following notation from the previous Chapter: p, F , p, χ, Fχ, Oχ.
Let R be the integer ring in a finite extension L of Fp. Denote its maximal ideal
by mR. Let GF := Gal(F/F ), and for Σ a finite set of places of F containing the
places above p let GΣ be the Galois group of the maximal extension of F unramified
at all places not in Σ. For any place v of F , Gv and Iv denote, respectively, the
decomposition group and the inertia subgroup for v determined by F → F v. Denote
by χp the infinite order p-adic Galois character associated to χ (see end of Section
2.6) and by ε : GF → Z∗p the p-adic cyclotomic character defined by the action of GF
on the p-power roots of unity: g.ξ = ξε(g) for ξ with ξpm= 1 for some m.
7.1.1 Fitting ideals
We recall the definition and basic properties of Fitting ideals. For details we refer
to the Appendix of [MW]. Let A be a ring and let M be a finitely generated A-module
with generators m1, . . . , mn. Let f : An ³ M be the surjective A-homomorphism
defined by f(ei) = mi for i = 1, . . . , n. Here ei denotes the ith standard basis vector
of An. The Fitting ideal FittA(M) of M is the ideal generated by the determinants
111
112
det(v1, . . . , vn) for which the column vectors v1, . . . , vn lie in ker(f). One checks that
this does not depend on the choice of the generators mi.
The following proposition contains the properties of the Fitting ideal that we will
use:
Proposition 7.1. (i) FittA(M) ⊂ AnnA(M).
(ii) For any A-algebra B we have FittB(M ⊗A B) = FittA(M) ·B.
(iii) For any ideal a ⊂ A we have FittA(A/a) = a.
(iv) If A is a complete local Noetherian ring with maximal ideal mA and M an
A-module of finite length, then
mlengthA(M)A ⊂ FittA(M).
Remark 7.2. For rings A that are complete discrete valuation rings of residue char-
acteristic p (like Oχ or R) we will write valp(#M) instead of lengthA(M).
7.1.2 Galois cohomology
For any profinite group G (e.g. GF or GΣ) call N a topological G-module if it is a
commutative topological group with a continuous action of G. Given a topological G-
module N , define the continuous cohomology groups H i(G,N) to be the cohomology
of the complex defined by continuous cochains (for details see [Ru] Appendix B.2):
Let Ci(G,N) = f : Gi → N continuous . For every i ≥ 0 there is a coboundary
map di : Ci(G,N) → Ci+1(G,N) and we set H1(G,N) := ker(di)/image(di−1). The
group H0(G,N) can be identified with NG, and for the trivial G-action on N we have
H1(G,N) = Homcts(G,N). Note also that for discrete modules any homomorphism
is continuous.
When N has the additional structure of an S-module for some ring S and the
G-action is S-linear (call this an S[G]-module), the continuous cohomology groups
are S-modules.
If 0 → T ′ → T → T ′′ → 0 is an exact sequence of topological S[G]-modules and
if there exists a continuous section (a map of sets, not necessarily a homomorphism)
113
from T ′′ → T , then we call it a topological short exact sequence (cf. [Ru] Appendix
B.2, [BW] p. 258).
Proposition 7.3. If N is a topological S[G]-module, then H1(G,N) classifies (iso-
morphism classes of) topological short exact sequences (of S[G]-modules)
0 → N → E → 1 → 0,
i.e., continuous extension classes of 1 by N , where 1 is the trivial linear representa-
tion of G on S (or S/I for some ideal I).
Proof. Given an extension
0 → Nf→ E
g→ 1 → 0
let e ∈ E project onto an S-generator of 1. Define a 1-cocycle c : G → N by
c(σ) = f−1(e− σ(e)). Since E is a continuous extension, c is a continuous 1-cocycle.
Two isomorphic extensions give rise to the same cohomology class. Note that, in
particular, if E is a split extension, i.e., if there exists a G-invariant continuous section
h such that g h = id1, then this construction yields a 1-coboundary. Furthermore,
if two extensions give rise to the same cohomology class they are isomorphic (see
argument in [Wa] Proposition 4).
Conversely, given a continuous 1-cocycle c : G → N let E be the S-module N⊕S,
where the action of G on N is extended by σ.r = rc(σ) ⊕ r for r ∈ S. Since c is
continuous this gives a topological S[G]-module. We note that if c is a 1-coboundary,
i.e., of the form σ 7→ σ(n)−n for some n ∈ N then r ∈ S 7→ (−rn, r) ∈ N ⊕S splits
the extension E = N ⊕ S.
7.1.3 Selmer groups
Selmer groups are generalizations of the ideal class groups of number fields. To
motivate the general definition of a Selmer group we recall the connection of class
groups with Galois groups via class field theory:
Example 7.4. By class field theory we have a canonical identification of the ideal
class group Cl(F ) with the Galois group of the maximal everywhere unramified
114
abelian extension of F , the Hilbert class field HF . The latter is the quotient of the
Galois group of the maximal abelian extension F ab over F by the images of all the
inertia groups.
If we are just interested in the p-Sylow subgroup Cl(F )[p∞], we can recover this
from Hom(Gal(HF /F ),Qp/Zp) by taking the Pontryagin dual Hom(·,Qp/Zp). But
Hom(Gal(HF /F ),Qp/Zp) can be identified with
ker(H1(GF ,Qp/Zp) →∏
v
H1(Iv,Qp/Zp)).
This will turn out to be the Selmer group for GF of the trivial Galois representation
Zp.
Definition 7.5 (Pontryagin dual). For a topological R-module N put N∨ =
Homcts(N,Qp/Zp). The group N∨ is an R-module via r.f(n) = f(rn), r ∈ R, f ∈N∨, n ∈ N .
Lemma 7.6. If N is a finite R-module, then
HomR(N,R∨) ∼= N∨, f 7→ (n 7→ f(n)(1)),
is an isomorphism of R-modules.
Proof. (This is Lemma 6.1.1(ii)) from [S04].) Observe that R∨ = ∪R∨[mnR]. Since
N is a finite R-module it follows that given f ∈ HomR(N, R∨) there is an n such
that im(f) ∈ R∨[mnR]. Thus the map in the lemma takes values in N∨ (and not just
Hom(N,Qp/Zp)). It is then easy to check that the map
N∨ → HomR(N, R∨), f 7→ (n 7→ (r 7→ f(rn))),
and the map in the lemma are inverses of each other.
By an R-lattice in a finite-dimensional L-space V we mean a finite R-submodule
M ⊂ V that spans V over L.
We define the Selmer group for a “p-ordinary Galois representation” following
Greenberg (cf. [G89]): Suppose given an n-dimensional L-space V and a continuous
115
representation ρ : GΣ → AutL(V ), i.e., such that there exists a GΣ-stable R-lattice
M ⊂ V on which GΣ acts continuously with respect to the mR-adic topology. Suppose
also given a Gw-stable subspace V +w for each place w of F lying over p.
Let M ⊂ V be any GΣ-stable R-lattice. For each w|p let M+w = M ∩V +
w . This is a
Gw-stable R-lattice in V +w . The module M∗ = M ⊗R R∨ is a discrete R[GΣ]-module
and for w|p the module M+,∗w := M+
w ⊗R R∨ is a discrete R[Gw]-module. Moreover,
there are canonical maps M+,∗w → M∗ coming from the inclusions M+
w → M . For
each finite set Σ′ ⊂ Σ\w|p we define a Selmer group associated to M by
Definition 7.7.
Sel(Σ′,M) = ker(H1(GΣ,M∗) → ⊕w|pH1(Iw,M∗/M+,∗
w )⊕w∈Σ′ H1(Iw,M∗)).
We write Sel(M) for Sel(∅,M).
Lemma 7.8. The Pontryagin dual of Sel(T,M) is a finitely generated R-module.
Definition 7.9. Suppose ρ : GΣ → R∗ is a continuous Galois character for some Σ
as above. Denote by R(ρ) the free rank one R-module on which GΣ acts via ρ.
We will apply the Selmer group definition in the case of 1-dimensional Galois
characters arising from Hecke characters of type (A0):
Example 7.10. For λ : F ∗\AF → C∗ a Hecke character of type (A0), i.e., with
infinity type zazb with a, b ∈ Z, we associated at the end of Section 2.6 a Galois
character
λp : GabΣ → O∗
λ,
where Σ consists of the places dividing the conductor fλ and the places dividing p,
and Oλ is the ring of integers in the finite extension Fλ of Fp containing the values of
λf . Extending λp trivially to GΣ we put M = Oλ(λp) and V = Fλ(λp) := M ⊗OλFλ.
By definition λp is “locally algebraic”, i.e., for each w|p and u ∈ O∗w with u ≡ 1
mod fλOw we have λp(rec(u)) = uaub, where rec is the Artin reciprocity map. By a
116
theorem of Tate (cf. [Se68] III A7) this implies that the local Galois representations
λp|Gw are Hodge-Tate, i.e., F λ(λp|Gp)∼= F λ(ε
−a) and F λ(λp|Gp) ∼= F λ(ε
−b). The
exponents −a and −b are called the “Hodge-Tate weights” of the representation at
p and p, respectively.
Remark 7.11. We use here the arithmetic Frobenius normalization in the Artin
reciprocity map which implies that ε(rec(u)) = u−1 for u ∈ O∗p. This choice was
implicitly taken in our definition of the L-function L(s, λ) in Section 2.6, which
equals the Artin L-function L(s, λp) under this normalization.
For Hecke characters λ of infinity type zazb with a, b ∈ Z and V = Fλ(λp) we let
V +p =
V if a < 0 (or HT-wt > 0),
0 if a ≥ 0 (or HT-wt ≤ 0)
and
V +p =
V if b < 0,
0 if b ≥ 0.
The Hodge-Tate weights of the Galois characters we will be interested in are
summarized in the following table:
HT-wt at p p
ε 1 1
µ1,p -1 0
µ2,p 1 0
χp -2 0
χpε -1 1
χ−1p ε−1 1 -1
Example 7.12 (Continuation of Example 7.4). In our general setup we can
recover the class group example by taking M = Zp and Σ = w|p. As explained in
the earlier example we get Sel(Zp)∨ ∼= Cl(F )[p∞].
117
For a finite extension K over F the Galois group Gal(K/F ) acts on Cl(K). If we
want to study this finer structure, we can do the following: Let χ : ∆ = Gal(K/F ) →R∗ be a finite order character, with K an abelian extension of F of degree prime to
p. For a Zp[∆]-module B (e.g. Cl(K)[p∞]) denote by Bχ the χ-isotypical piece, i.e.
Bχ := b ∈ B ⊗Zp R : γ.b = χ(γ)b for every γ ∈ ∆. Take Σ sufficiently large such
that ∆ is a quotient of GΣ and extend χ trivially to GΣ. We claim that
Sel(Σ\w|p, R(χ))∨ ∼= Cl(K)[p∞]χ.
This can be seen as follows: In our general notation we have M = R(χ) and
Σ′ = Σ\w|p. Since χ has finite order, its Hodge-Tate weight is 0 and M+w = 0 for
all w|p. Therefore, by definition
Sel(Σ′, R(χ)) = ker(H1(GF ,M∗) →∏
v
H1(Iv,M∗)).
Since [K : F ] is prime to p the inflation-restriction sequence (see [Ru] Prop. B.2.5)
implies that H1(GF ,M∗) ∼= H1(GK ,M∗)∆. Together this shows that Sel(Σ′, R(χ)) =
Hom(Gal(HK/K),M∗)∆ = Hom(Cl(K)[p∞]χ, R∨) for the Hilbert class field HK of
K. The claim follows together with Lemma 7.6.
Definition 7.13. A Galois character ρ : GF → R∗ is called anticyclotomic if it
satisfies ρc = ρ−1, where ρc(σ) = ρ(cσc−1) with c ∈ GQ a lift of the non-trivial
automorphism of Gal(F/Q).
Example 7.14. If λ : F ∗\A∗F → C∗ is an anticyclotomic unitary Hecke character
(i.e. λc = λ = λ−1), then λp is anticyclotomic.
Lemma 7.15. We have Sel(ρ) ∼= Sel(ρc). Here the isomorphism is induced by con-
jugation of GF by a lift of the non-trivial automorphism of Gal(F/Q).
7.2 Statement and discussion of result
Let Σ0 be the set comprising the places above p and the places ramified in F/Q.
Using a method developed by Wiles, Skinner, and Urban, we will prove:
118
Proposition 7.16. With the same assumptions and notation as Theorem 6.3 and
Σ = Σ0, we have
valp(#Sel(M)∨) ≥ valp(#(Tχ/Iµ1,µ2))
for either M = Oχ(χpε) or M = Oχ((χpε)c) = Oχ(χ−1
p ε−1) (the two Selmer groups
are isomorphic by Lemma 7.15).
Remark 7.17. Note that χpε has negative Hodge-Tate weight at the place p and
positive at the place p, so Sel(Oχ(χpε)) will consist of cohomology classes unramified
at p, but possibly ramified at p.
Using Theorem 6.3, this immediately gives us lower bounds on the size of the
Selmer groups in terms of the special L-value:
Corollary 7.18. With the same assumptions and notation as Theorem 6.3 and Σ =
Σ0, we have
valp(#Sel(M)∨) ≥ valp(#(Oχ/(Lalg(0, χ))))
for M = Oχ(χpε) or M = Oχ((χpε)c) = Oχ(χ−1
p ε−1).
Remark 7.19. That Lalg(0, χ) gives bounds for these two Selmer groups is related
(via the anticyclotomic main conjecture) to the fact that Lalg(0, χ) gets interpolated
by two p-adic L-functions, see [AH] p. 12. See Remark 7.35 for the relation of our
results to consequences of the Main Conjecture of Iwasawa theory.
7.3 Proof of Proposition 7.16
We need to procure the ingredients of the following proposition, adapted for our
purposes from [S04] Proposition 6.1.17:
Proposition 7.20. Let ρ : GΣ → O∗χ be a continuous 1-dimensional representation
with positive Hodge-Tate weight at one of the primes lying above p, negative at the
other (call the latter prime w). Denote the module by M and write ρ for the reduction
modulo the maximal ideal of Oχ. Let T be a finite Oχ-algebra and I ⊂ T an ideal
such that the Oχ-algebra structure map surjects onto T/I.
119
Suppose we are given:
• a finite T -module L on which GΣ acts continuously and T -linearly and having
no T [GΣ] -quotient isomorphic to ρ.
• a T [GΣ]-submodule L1 ⊂ L such that GΣ acts trivially on L/L1 and L/L1∼=
T/I.
• a finite T -module T with FittT (T ) ⊂ I ⊂ AnnT (T ) and a T [GΣ]-identification
L1∼= M ⊗Oχ T .
We further require that the T [GΣ]-extension
0 → L1 → L → L/L1 → 0
be split when viewed as a T [Iw]-extension. Given this set-up,
valp(#Sel(M)∨) ≥ valp(#(T/I)).
Proof. We redo the proof of [S04] in our special case. As in Proposition 7.3 we fix
e ∈ L projecting onto a T -generator of L/L1 and define a 1-cocycle c : GΣ → M ⊗Tby
c(σ) = the image of e− σ(e) in L1.
Consider the Oχ-homomorphism
φ : HomOχ(T ,O∨χ) → H1(GΣ,M ⊗O∨
χ), φ(f) = the class of (1⊗ f) c.
We will show that
(i) im(φ) ⊂ Sel(M),
(ii) ker(φ)∨ = 0.
From (i) it follows that
valp(#Sel(M)∨) ≥ valp(#im(φ)∨).
120
From (ii) it follows that
valp(#im(φ)∨) ≥ valp(#Hom(HomOχ(T ,O∨χ),Qp/Zp))
= valp(#Hom(T ∨,Qp/Zp))
= valp(#T )
≥ valp(#T/I),
where the first equality comes from Lemma 7.6, and the last inequality from our
assumption on T and Proposition 7.1(iv). The latter implies that lengthT (T ) ≥lengthT (T/I). Since the action of T on both T and T/I is via T/I, which is a
quotient of Oχ, any Oχ-submodule of T or T/I is, in fact, a T -submodule. This
implies the corresponding inequality for the Oχ-lengths.
For (i) we observe that the assumption that the extension splits when considered
as an extension of T [Iw]-modules implies by Proposition 7.3 that the class cw in
H1(Iw,L1) determined by c is the zero class. This shows that im(φ) ⊂ Sel(M).
To prove (ii) we first observe that for any f ∈ HomOχ(T ,O∨χ), ker(f) has finite
index in T (like in the proof of Lemma 7.6 we see that f ∈ HomOχ(T ,O∨χ [pn]) for
some n). Suppose now that f ∈ ker(φ). We claim that the class of c in H1(GΣ, M⊗Oχ
T /ker(f)) is zero. To see this, let X = O∨χ/im(f) and observe that there is an exact
Since f ∈ ker(φ) and the second arrow in the sequence comes from the inclusion
T /ker(φ) → O∨χ induced by f , the image in the right module of the class of c in
the middle is zero. Our claim follows therefore if the module on the left is trivial.
But the dual of this module is a subquotient of HomOχ(M,Oχ) on which GΣ acts
trivially. By assumption, however, M has no nontrivial subquotients.
Suppose in addition that f is non-trivial, i.e. ker(f) ( T . Then there exists a
T -module A with ker(f) ⊂ A ⊂ T such that T /A ∼= Oχ/p (we use here again that
121
any Oχ-submodule of T is actually a T -submodule). From our claim it follows now
that the T [GΣ]-extension
0 → M ⊗Oχ Oχ/p ∼= M ⊗Oχ T /A → L/(M ⊗Oχ A) → L/L1 → 0
is split. But this would give a T [GΣ]-quotient of L isomorphic to ρ, which contradicts
our assumption. Hence ker(φ) (and therefore also ker(φ)∨) are trivial.
We will apply this Proposition for ρ = χpε (respectively ρ = (χpε)c), T a localiza-
tion of Tχ and I the ideal corresponding to Iµ1,µ2 . In the following we demonstrate
how to obtain L and L1 from the Galois representations associated to cuspidal au-
tomorphic representations by the work of Taylor et al.
7.3.1 Galois representations attached to cuspidal automorphic representations
Recall the Hecke-equivariant Eichler-Shimura-Harder isomorphism
eωH1! (SKf
,C) ∼= S0(Kf , ω,C)
(Theorem 2.6 and the end of Section 6.1). Here S0(Kf , ω,C) denotes the space
of cuspidal automorphic forms of GL2(F ) of weight 0, right-invariant under Kf ⊂G(Af ) with central character ω (see Section 2.7). This was isomorphic to ⊕π
Kf
f for
automorphic representations π of a certain infinity type with central character ω.
Combining the work of Taylor, Harris and Soudry with results of Friedberg-
Hoffstein and Laumon/Weissauer, one can show the following:
Theorem 7.21. Given a cuspidal automorphic representation π with π∞ isomorphic
to the principal series representation corresponding tot1 ∗
0 t2
7→
(t1|t1|
) ( |t2|t2
)
and cyclotomic central character ω (i.e. ωc = ω), let Σπ denote the set of places
above p, the primes where π or πc is ramified, and primes ramified in F/Q.
Then there exists a continuous Galois representation ρπ : Gal(F/F ) → GL2(F p)
such that if v /∈ Σπ, then ρπ is unramified at v and the characteristic polynomial of
122
ρf (Frobv) is x2− av(π)x + ω−1(Pv)NmF/Q(Pv), where av(π) is the Hecke eigenvalue
corresponding to Tv. The image of the Galois representation is actually inside GL2(L)
for a finite extension L of Fp and the representation is absolutely irreducible.
Remark 7.22. 1. Taylor relates π to Siegel modular forms via theta lifts and uses
the Galois representations associated to Siegel modular forms to find ρπ.
2. Taylor had some additional technical assumption in [T2] and only showed the
equality of Hecke and Frobenius polynomial outside a set of places of zero den-
sity. For this strengthening of Taylor’s result see [BHR].
3. Since Taylor’s convention for the Hecke operators differs from ours, the Galois
representations as stated above are twists of Taylor’s Galois representation by
the central character.
Urban studied in [U98] the case of ordinary automorphic representations π, and
together with results in [U04] on the Galois representations attached to ordinary
Siegel modular forms shows that for these π one has a particularly nice form for ρπ
when restricted to the decomposition group of p:
Theorem 7.23 (Corollaire 2 of [U04]). If π is a cuspidal automorphic represen-
tation with cyclotomic central character and is ordinary at p (i.e., π unramified at p
and |ap(π)|p = 1), then the Galois representation ρπ is ordinary and
ρπ|Dp∼=
Ψ1 ∗
0 Ψ2
,
where Ψ2|Ip = 1, and Ψ1|Ip = det(ρπ)|Ip = ε.
As indicated at the start of this section the connection to our work in the previous
chapters will come via the Eichler-Shimura-Harder isomorphism eωH1! (SKH,s
f,C) ∼=
S0(KH,sf , ω,C). The compact open subgroup KH,s
f that we defined for our unramified
Hecke character χ after choosing a factorization χ = µ1/µ2 is given by GL2(Ov) at
all places v /∈ S for some finite set of places S at which the µi are ramified. Note
that for each π with πKH,s
f
f 6= 0 the set Σπ in Taylor’s theorem is a subset of the set
123
comprising the places above p, the places in S and their complex conjugates, and
the places ramified in F/Q. Note that if we take the factorization used in Corollary
4.18 then S coincides with the places ramified in F/Q and hence we have that for
each π with πKH,s
f
f 6= 0 the set Σπ is contained in the set Σ0 used in Proposition 7.16.
To apply Taylor’s result we need the central character ω = µ1µ2 of the cuspforms
arising in Theorem 6.3 to be cyclotomic. The anticyclotomic characters µi used in
the definition of the Eisenstein cohomology class will in general not be such that their
product is cyclotomic. It is possible, however, to factor χ = η1/η2 with (η1η2)c = η1η2
by the following Lemma (take η1 = µ and η2 = (µc)−1):
Lemma 7.24. Let χ : F ∗\A∗F → C∗ be a Hecke character of infinity type z2mz2n, for
m,n ∈ Z, such that χc = χ. Then there exists a Hecke character µ : F ∗\A∗F → C∗
of infinity type zmzn such that χ = µµc. Furthermore, if χ is unramified away from
Σ0 then we can find such a µ ramified only at places in the set Σ0.
Proof. We first reduce to the case of finite order characters (i.e. m = n = 0). We
again use Greenberg’s character µG : F ∗\A∗F → C∗ of infinity type z−1 such that
µcG = µG and µG is ramified exactly at the primes ramified in F/Q (cf. Lemma
3.18). Given χ as in the Lemma it suffices to prove the Lemma for the character
χµ2mG µ2n
G = χµmG (µc
G)mµnG(µc
G)n, which has trivial infinite component.
For finite order characters we argue as follows: By assumption we have that
χ ≡ 1 on NmF/Q(A∗F ) ⊂ A∗
Q ⊂ A∗F .
Thus χ restricted to Q∗\A∗Q is either the quadratic character of F/Q or trivial.
Since our finite order character has trivial infinite component, χ has to be trivial on
Q∗\A∗Q. Hilbert’s Theorem 90 then implies that there exists µ such that χ = µ/µc.
To control the ramification we analyze this last step closer: χ factors through
A∗F → A, where A is the subset of A∗
F of elements of the form x/xc and the map is
x 7→ x/xc. If y ∈ A ∩ F ∗ then y has trivial norm and so by Hilbert’s Theorem 90,
y = x/xc for some x ∈ F ∗. Thus the induced character A → C∗ vanishes on A∩F ∗.
This implies that there is a continuous finite order character µ : F ∗\A∗F → C∗ which
124
restricts to this character on A and χ = µ/µc (this argument is taken from the proof
of Lemma 1 in [T2]). If χ is unramified, we can similarly conclude that the induced
character vanishes on A ∩∏v/∈Σ0
O∗v and therefore find µ on F ∗\A∗
F /∏
v/∈Σ0O∗
v re-
stricting to the character A → C∗: Writing UF,` =∏
v|`O∗v for a prime ` in Q we
have
H1(Gal(F/Q),∏
` unramified in F/Q
UF,`) →∏
`
H1(Gal(F/Q), UF,`)
and
H1(Gal(F/Q), UF,`) ∼= H1(Gv,O∗v) = 1
since Fv/Q` is unramified. If y ∈ A ∩ ∏v/∈Σ0
O∗v then y has trivial norm in each
O∗v, v /∈ Σ0. By what we just showed this implies that there exists x ∈ ∏
v/∈Σ0O∗
v∩A∗F
such that y = x/xc. The image of y under the induced character therefore equals
χ(x) = 1, as claimed above.
To associate now Galois representations to the cuspforms π congruent to our
Eisenstein cohomology class (i.e., with Hecke eigenvalues av(π) congruent to the
eigenvalues of the Eisenstein cohomology class), we twist the forms by η2/µ2. Since
µ1 = χµ2 this gives cuspforms with central character η1η2, so we can apply Theorem
7.21. Then we “untwist” the resulting Galois representation ρπ⊗η2/µ2 by this finite
order character to get a Galois representation ρ′ with trace(ρ′(Frobv)) = av(π) for
v /∈ Σ0. We will in the following suppress this twisting process and just denote the
end product ρ′ by ρπ.
To apply Urban’s result to the cuspforms congruent to the Eisenstein cohomology
class we have to check that the eigenvalue at p of the latter is a p-adic unit:
Lemma 7.25. The Hecke eigenvalue ap((µ1, µ2)) of Lemma 3.11 lies in O∗χ.
Proof. Denoting the place corresponding to p by v0 we have ap((µ1, µ2)) = µ−11,v0
(πv0)+
pµ−12,v0
(πv0). By Lemma 3.21 the first summand has valuation at v0 equal to 1 (the
infinity type of µ1 is z), the second summand equal to 0, so their sum lies in O∗χ.
125
7.3.2 Constructing the lattice
By the Eichler-Shimura-Harder isomorphism Tχ from Definition 6.1 is isomorphic
to the Oχ-subalgebra T of EndOχ(S0(KH,sf , ω,C)) generated by the Hecke operators
Tv for v /∈ S, where S is the set of places ramified in F/Q. Recall that S0(KH,sf ,C) ∼=
⊕πKH,s
f
f . Consider all the cuspidal automorphic representations with central character
ω = µ1µ2 with πKH,s
f
f 6= 0 and denote them by π1, . . . , πm. There are finitely many
such because H1! (SKH,s
f,C) has finite dimension.
As explained in the previous section we have associated Galois representations ρπi
for i = 1, . . . ,m with trace(ρπi(Frobv)) = av(πi) and det(ρπi
(Frobv)) = ω−1(Pv)Nm(Pv)
for v /∈ Σ0, where Pv denotes the maximal ideal of Ov and Σ0 = S ∪ w|p.Let Li be a finite extension of Fχ such that ρπi
: GΣ0 → GL2(Li), L := ∪iLi, and
A :=∏
i Li. We will now show that we can embed T in A.
We define the following Oχ-algebra map:
T → A : Tv 7→ (av(πi))i.
We claim that this map is injective: By definition, T → ⊕iEndOχ(VKH,s
fπi ), where we
denote by Vπ the representation space of π. Since Tv acts on πi by av(πi), the image
in each summand is given by the Oχ-algebra generated by the av(πi)’s. Note that, in
fact, T → ∏iOLi
since av(πi) ∈ OLi. We can therefore view each OLi
, i = 1, . . . ,m
and∏
iOLias a T-module.
For later, we remark that T⊗Oχ Fχ = A. This follows from T⊗Oχ L =∏
i L by
comparing dimensions. For the latter, observe that on the one hand, dimL(T⊗Oχ L)
is clearly less than or equal to m. On the other hand, each homomorphism T → OL
arising from the projection onto one of the m factors of∏
iOLi⊗OL gives rise to a
minimal prime, and these are distinct by (strong) multiplicity one, so the dimension
of T⊗Oχ L =∏
minimal primes P(T/P)⊗OLL is also bounded below by m.
We also want to remark that ⊕mi=1trace(ρπi
)(σ) ∈ T for all σ ∈ GΣ0 . This follows
from the Chebotarev density theorem (which tells us that the Frobenius elements
of unramified primes in a Galois extension are dense in the Galois group) and the
126
continuity of the ρπisince T is a finite Oχ-algebra.
From now on we will assume that T/I 6= 0, where I is the ideal corresponding to
Iµ1,µ2 ⊂ Tχ (there is nothing to prove otherwise in Proposition 7.16!). Let P be the
maximal ideal of T containing I. We now consider the completions of T and∏
iOLi
at P.
Lemma 7.26. If OLiis not in the kernel of
∏iOLi
→ (∏
iOLi)P then ρss
πi
∼= µ−11,p ⊕
µ−12,pε.
Proof. A factor OLiis not in the kernel of this localization if and only if
0 6= POLi⊂ mOLi
.
Since we have Tv − µ−11 (Pv) − µ−1
2 Nm(Pv) ∈ P for all v /∈ S we must have that
av(πi)− µ−11 (Pv)− µ−1
2 Nm(Pv) lies in the maximal ideal of OLi. We deduce that πi
has Hecke eigenvalues congruent to those of the Eisenstein cohomology class.
By definition we get that the characteristic polynomial of ρπi(Frobv) for v /∈ Σ0
is x2 − (µ−11,p(Frobv) + µ−1
2,pε(Frobv))x + ω−1p ε(Frobv). By the Chebotarev density
theorem any element of GΣ0 can be approximated by such Frobenius elements. Since
ρπiis continuous we have that ρπi
factors through a finite Galois extension and
that for any element σ in this extension the characteristic polynomial is given by
x2 − (µ−11,p(σ) + µ−1
2,pε(σ))x + ω−1p ε(σ). But since this agrees with the characteristic
polynomial of the representation µ−11,p ⊕ µ−1
2,pε the claim follows from:
Theorem 7.27 (Brauer-Nesbitt). If ρ1, ρ2 are two finite dimensional representa-
tions of a finite group G acting on vector spaces V1, V2 over a field then
ρss1∼= ρss
2 ⇔ characteristic poly(ρ1(g)) = characteristic poly(ρ2(g)) for all g ∈ G.
Denote the “surviving” index set by J (which is non-trivial since by assumption
TP/I ∼= T/I 6= 0) and write AP =∏
i∈J Li ⊂ A.
We will now follow the method of [W86] and [W90], with modifications by Skin-
ner in [S02b], to construct the finite TP-modules L1 ⊂ L in Proposition 7.20. In
127
Proposition 7.16 we consider two cases: ρ = χpε or ρ = χ−1p ε−1. In the following we
deal with the first case; the modifications necessary for the second being obvious. So
from now on ρ = χpε and we denote the place p at which it has negative Hodge-Tate
weight by w. Let ρi = ρπi⊗ µ1,p. By the preceding lemma ρss
i∼= 1⊕ ρ for i ∈ J .
We consider the TP-module WP = AP ⊕ AP. Fix σ0 ∈ Iw such that ρ(σ0) 6≡ 1
mod p (this is possible since the Hodge-Tate weight of ρ at w is -1). We fix a basis
of the representations ρi for i ∈ J such that
• ρi(σ0) =
αi 0
0 βi
with αi, βi ∈ Oχ, (αi), (βi) ∈ TP (and αi 6≡ βi ≡ 1 mod p),
• ρi : GΣ0 → GL2(OLi),
• ρi|Dw =
Ψ
(i)2 0
∗ Ψ(i)1
with Ψ
(i)1 unramified.
For the first condition we note that by Hensel’s Lemma the distinct eigenvalues
of ρi(σ0) lift to distinct eigenvalues of ρi(σ0) in Oχ. That it is possible to find a
Galois stable lattice is a standard argument using the compactness of GΣ0 and the
continuity of ρi. The third condition uses Theorem 7.23 on the ordinarity of ρπiand
the fact that µ1,pε is unramified at w.
Now put a Galois action on WP via ρ := ⊕i∈Jρi. The two actions commute and
from now on we consider WP as a TP[GΣ0 ]-module.
Definition 7.28. A lattice L in WP is a finitely generated TP-module such that
L⊗Oχ Fχ = WP. By a stable lattice we mean a Galois stable lattice.
We first note that∏
i∈J(OLi⊕ OLi
) ⊂ WP is a stable lattice. We modify it as
follows: Write ρ(σ) =
aσ bσ
cσ dσ
. Since ⊕i∈Jtrace(ρi)(σ) ∈ TP for any σ ∈ GΣ0 we
get that aσ + dσ ∈ TP and aσαi + dσβi ∈ TP. Together with αi 6≡ βi mod p we
deduce aσ, dσ ∈ TP.
The cσ lie in∏
i∈J OLiby assumption. Because Oχ is a discrete valuation ring,
any two lattices are commensurable, so there exists an x ∈ Oχ such that xcσ ∈ TP
128
for all σ. Replacing ρ by
x−1 0
0 x
ρ
x 0
0 x−1
we can ensure that ρ stabilizes the
lattice L0 :=∏
i∈J1xOLi
⊕ xTP.
In order to find a lattice L that satisfies the requirements of the Proposition, we
apply the sequence of Lemmas from [W86] and [W90]:
Lemma 7.29. Suppose L is any stable lattice in WP. Then any irreducible TP[GΣ0 ]-
quotient V of L/I satisfies either V ∼= TP/P with trivial GΣ0-action (“type 1”)or
with GΣ0-action via ρ (“type ρ”).
Proof. The ρπisatisfy that the characteristic polynomials of ρπi
(Frobv) are x2 −av(πi)x + ω−1(Pv)NmF/Q(Pv). This implies that we have
ρ(Frobv)2 − Tvµ1(Pv)ρ(Frobv) + ρ(Frobv) = 0 on L for v /∈ Σ0.
By the definition of I we deduce that (Frobv − 1)(Frobv − ρ(Frobv)) annihilates
L/I. The statement follows by another application of the Chebotarev density and
Brauer-Nesbitt theorems.
Definition 7.30. A finite TP[GΣ0 ]-module is said to be of type ρ (resp. type 1) if
all irreducible subquotients are of type ρ (resp. type 1).
We seek a stable lattice L ⊂ L0 having a filtration
0 → (type ρ) → L/I → ( type 1) → 0
and such that L/I has no type ρ quotients.
Lemma 7.31. Given any stable lattice L there exists a stable sublattice L′ such that
L/L′ has type ρ, and if L′′ ⊂ L is a stable sublattice such that L/L′′ has type ρ, then
L′ ⊂ L′′.
Proof. This is proved exactly as Proposition 3.2 of [W86]. Set L′ =⋂L′′ ⊂
L stable |L/L′′ type ρ. If L′ is not a lattice then L/L′ ⊗Oχ Fχ 6= 0 and any ir-
reducible constituent of L/L′⊗Oχ Fχ (as a GΣ0-module) must be isomorphic to some
129
ρi by the irreducibility of the ρπi. Set (L/L′)(1) = x|σ0x = x ⊂ L/L′. By the
form of ρi(σ0) this is non-zero. But L/L′ → ∏L′′ L/L′′, and each of these is of type
ρ. Since ρ(σ0) 6≡ 1 mod p, (L/L′)(1) maps to 0 under this embedding, so we get a
contradiction.
Note that L′ in the Lemma has no type ρ quotient by the minimality property
(“ρ-deprived sublattice”). Set L = L′0.
Proposition 7.32 (Proposition 5.4 of [W90]). Let E be a finite TP/I[GΣ0 ]-
module. Suppose that E has no type ρ quotient. Let Eρ be the maximal type ρ
submodule. Then E/Eρ is of type 1.
Let E := L/I and Eρ ⊂ E the maximal type ρ submodule. By the Proposition
we deduce that E1 := E/Eρ has type 1. Finally we conclude:
Proposition 7.33. There is an exact sequence of TP[GΣ0 ]-modules
0 → Eρ → E := L/I → E1 → 0,
where Eρ is of type ρ, E1∼= TP/I is of type 1, and no TP[GΣ0 ]-quotient of E is
isomorphic to ρ.
Proof. It only remains to prove E1∼= TP/I. Instead of [W86] Lemma 3.4 and
[W90] Prop. 5.5. we follow [S02b] in using the element σ0 to do this. We denote
L0 =∏
i∈J1xOLi
⊕ xTP =: L(ρ)0 ⊕ L
(1)0 . Similarly decompose L = L(ρ) ⊕ L(1) and
E = E(ρ) ⊕ E(1).
On L(ρ)0 /I the element σ0 must act either trivially or by ρ(σ). Since αi 6≡ 1 mod p
it acts via ρ. Similarly, σ0 acts trivially on L(1)0 /I. Since L0/L is of type ρ this implies
L(1)0 ⊂ L. Since clearly L(1) ⊂ L
(1)0 , we have L
(1)0 = L(1) ∼= TP.
Now E1 = E/Eρ has type 1, so E(ρ) ⊂ Eρ and E(1) ³ E1. Since E1 = L(1)0 /I
is type 1 we also get E(1) → E1. This concludes the proof that E1 = L(1)0 /I ∼=
TP/I.
130
For Proposition 7.20 we now take ρ = χpε, T = TP, and I the ideal generated
by the Eisenstein ideal in TP. It is clear from the definitions that the Oχ-algebra
structure map surjects onto T/I ∼= Tχ/Iµ1,µ2 . We rename L1 := Eρ and L := E. The
proof of the previous Proposition shows that Eρ equals L(ρ)/I. Since L(ρ) is a faithful
TP-module, we obtain FittTP(L(ρ)) = (0), from which it follows that FittTP
(Eρ) ⊂ I,
as desired. That 0 → L1 → L → L/L1 → 0 is split as a sequence of TP[Iw]-modules
(even TP[Dw]-modules!) follows from the form of ρ|Dw . Applying Proposition 7.20
this now proves Proposition 7.16.
7.4 Dealing with ramification at places other than w
It is possible to prove a lower bound for the smaller Selmer groups Sel(Σ′,M) for
non-trivial Σ′ ⊂ Σ0\w|p after imposing an additional condition (independent of
the character χ) on our prime p:
Proposition 7.34. Assume in addition that ` 6≡ ±1 mod p for ` | dF . Then for
Σ′ = Σ0\w|p = v|dF we have
valp(#Sel(Σ′,M)∨) ≥ valp(#(Oχ/(Lalg(0, χ))))
for M = Oχ(χpε) or M = Oχ((χpε)c) = Oχ(χ−1
p ε−1).
Proof. By the definition of Sel(Σ′,M) we have to show that the extension L = E
constructed in the previous section is split when viewed as a TP[Iv]-extension for
v|dF . By construction ρ acts on L by
ρ ∗
0 1
mod I. Since TP/I ∼= Oχ/pn for
some n as Oχ-algebras this corresponds to acting by
ρ ∗
0 1
∈ ∏
i∈J GL2(OLi/pn).
We will show for each ρi that, in fact, the inertia groups Iv with v|dF act trivially. By
assumption ρ(Iv) = 1, so ρi(Iv) = 1 ∗
0 1
mod pn. Suppose now that ρi|Iv 6≡ 1
mod pn. Then there must exist x ∈ Itamev such that ρi(x) ≡
1 b
0 1
with b /∈ pn.
131
Let σv ∈ Gv be any lift of Frobv. Then ρi(σv) ≡ρ(σv) ∗
0 1
and so ρi(σvxσ−1
v ) =
ρi(xqv) ≡
1 bqv
0 1
(for qv = #Ov/Pv) is also congruent to
1 ρ(σv)b
0 1
. Since ρ
is anticyclotomic and v is fixed under complex conjugation we get ρ(σv) = ρ(σcv) =
ρ−1(σv), or ρ(σv) = ±1. Under the additional assumption on p the congruence for
ρi(σvxσ−1v ) cannot exist, contradicting our assumption of a non-trivial action of Iv.
Remark 7.35. Conjecturally, the p-valuations of the two sides in Proposition 7.34
are equal. When #Cl(F ) = 1, this has been proved by very different methods in
[Gu93] using the 2-variable Main Conjecture of Iwasawa theory proved by Rubin in
[Ru2].
Since the notation in [Gu93] is very different we briefly explain the translation
to our setup: For Ψ the Großencharakter attached to an elliptic curve defined over
Q with complex multiplication by O, Guo shows in [Gu93] that for 0 < j < k
and p − 1 > k the p-valuation of Lalg(0, ΨjΨ−k) equals the p-valuation of the strict
Selmer group of the 1-dimensional Galois representation (ΨkΨ−j
)p. In the class
number one case the only unramified Hecke character of infinity type z2 is χ = µ−2G
(see Lemma 3.18 for the definition of µG) and there exists an elliptic curve defined
over Q with complex multiplication by O and associated Großencharakter µG. The
proof in [Gu93] extends to k = j = 1 for which the Selmer group in [Gu93] agrees
with Sel(Σ′,Oχ((χpε)−1)) in Proposition 7.34 if Ψ = µG. Applying the functional
equation and using µcG = µG one can show that the special L-value in [Gu93] has the
same p-valuation as Lalg(0, χ).
BIBLIOGRAPHY
[AH] A. Agboola and B. Howard, Anticyclotomic Iwasawa theory of CM elliptic curves,Preprint available at arXiv:math.NT/0302319.
[AS] A. Ash, G. Stevens, Cohomology of arithmetic groups and congruences between systemsof Hecke eigenvalues, J. Reine Angew. Math. 365 (1986), 192-220.
[BHR] T. Berger, G. Harcos, D. Ramakrishnan, `-adic representations associated to modularforms over imaginary quadratic fields, in preparation.
[BN] J. Blume-Nienhaus, Lefschetzzahlen fur Galois-Operationen auf der Kohomologie arith-metischer Gruppen, Bonner Mathematische Schriften 230, Bonn (1992).
[BJ] A. Borel, L. Ji, Compactifications of symmetric and locally symmetric spaces, to appearin Birkhauser, 2005.
[BS] A. Borel and J.P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973),436-491.
[BW] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representa-tions of reductive groups, Princeton University Press, Princeton (1980).
[Bour] N. Bourbaki, Algebre commutative, Hermann, 1961-1983.
[B67] G. Bredon, Sheaf theory, McGraw-Hill, New York (1967).
[B92] D. Bump, Automorphic Forms and Representations, Cambridge University Press, Cam-bridge (1996).
[Cas] W. Casselman, On some result of Atkin and Lehner, Math. Ann. 201 (1973), 301-314.
[CW] J. Coates, A. Wiles, Kummer’s criterion for Hurwitz numbers, Algebraic Number Theory(Kyoto Internat. Sympos.), Res. Inst. Math. Sci., Univ. Kyoto (1976), 9-23.
[Co] B. Conrad, Modular Forms, Cohomology, and the Ramanujan Conjecture, Preprint.
[dS] E. de Shalit, Iwasawa theory of elliptic curves with complex multiplication, Perspectivesin Math. 3, Academic Press, 1987.
[EGM] J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on hyperbolic space, Springer-Verlag, New York (1997).
[F] D. Feldhusen, Nenner der Eisensteinkohomologie der GL(2) uber imaginar quadratischenZahlkorpern, Bonner Mathematische Schriften 330, Bonn (2000).
[Fi] T. Finis, Arithmetic properties of a theta lift from GU(2) to GU(3), Thesis Dusseldorf(1999).
[Fi2] T. Finis, Divisibility of anticyclotomic L-functions and theta functions with complexmultiplication, available at www.mathnet.or.kr, to appear in Ann. of Math.
132
133
[Gel] S. Gelbart, Automorphic forms on adele groups, Ann. Math. Stud. 83, Princeton Univ.Press, Princeton (1975).
[Gh] E. Ghate, An Introduction to congruences between modular forms, Current trends innumber theory, Hindustan Book Agency (2002), 39-58.
[Go] R. Godement, Topologie algebrique et theorie des faisceaux, Hermann, Paris (1958).
[G67] M. Greenberg, Lectures on algebraic topology, W.A. Benjamin, New York (1967).
[G85] R. Greenberg, On the critical values of Hecke L-functions for imaginary quadratic fields,Invent. Math. 79 (1985), no.1, 79-94.
[G89] R. Greenberg, Iwasawa theory for p-adic representations, in: Algebraic number theoryin honor of K. Iwasawa, J. Coates et al., eds., Adv. Stud. in Pure Math. 17, Boston:Academic Press (1989), 97-137.
[Gu93] L. Guo, General Selmer groups and critical values of Hecke L-functions, Math. Ann. 297(1993) no. 2, 221-233.
[Ha79] G. Harder, Period Integrals of Cohomology Classes which are represented by Eisensteinseries, Proc. Bombay Colloquium 1979, Springer (1981), 41-115.
[Ha82] G. Harder, Period Integrals of Eisenstein Cohomology Classes and special values of someL-functions, in: Number Theory related to Fermat’s Last Theorem, ed. by N. Koblitz,Progr. Math. 26, Birkhauser (1982), 103-142.
[HaGL2] G. Harder, Eisensteincohomology of arithmetic groups. The case GL2, Invent. Math. 89(1987), no.1, 37-118.
[HaCAG] G. Harder, Cohomology of arithmetic groups, Preprint.
[Ha02] G. Harder, The arithmetic properties of Eisenstein classes (Chapter 6 of [HaCAG]),Preprint.
[HP] G. Harder, R. Pink, Modular konstruierte unverzweigte abelsche p-Erweiterungen vonQ(ζp) und die Struktur ihrer Galoisgruppen, Math. Nachr, 159 (1992), 83-99.
[HC] Harish-Chandra, Automorphic forms on semisimple Lie groups, Lect. Notes in Maths.62, Springer, New York (1968).
[Hart] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag, NewYork (1977).
[Hi82] H. Hida, Kummer’s criterion for the special values of Hecke L-functions of imaginaryquadratic fields and congruences among cusp forms, Invent. Math. 66 (1982), 415-459.
[Hi93] H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge UniversityPress, Cambridge (1993).
[Hi03] H. Hida, Anticyclotomic Main Conjectures, Preprint downloadable atwww.math.ucla.edu/ hida.
[Hi04a] H. Hida, Non-vanishing modulo p of Hecke L-values, in: “Geometric Aspects of Dwork’stheory”, Walter de Gruyter (2004).
[Hi04b] H. Hida, Non-vanishing modulo p of Hecke L-values and application, Preprint download-able at www.math.ucla.edu/ hida.
[HT] H. Hida, J. Tilouine, Anti-cyclotomic Katz p-adic L-functions and congruence modules,Ann. Sci. Ecole Norm. Sup. (4) 26 (1993), no.2, 189-259.
134
[Ka] C. Kaiser, Die Nenner von Eisensteinklassen fur gewisse Kongruenzuntergruppen,Diplomarbeit Bonn (1990).
[K76] N. M. Katz, p-adic interpolation of real analytic Eisenstein series, Ann. of Math. (2) 104(1976), 459-571.
[K78] N. M. Katz, p-adic L-functions for CM-fields, Invent. Math. 49 (1978), 199-297.
[Ko] H. Konig, Eisenstein-Kohomologie von SL2(Z[i]), Bonner Mathematische Schriften 222,Bonn (1991).
[La] S. Lang, Algebraic Number Theory, Addison-Wesley, Reading (1970).
[Langl] R. P. Langlands, On the functional equations satisfied by Eisenstein series, Lect. Notesin Maths. 544, Springer-Verlag, New York (1976).
[Ma99] J. P. May, A concise course in algebraic topology, The University of Chicago Press,Chicago (1999).
[MW] B. Mazur, A. Wiles, Class fields of abelian extensions of Q, Invent. Math. 76 (1984),179-330.
[MR] C. Mclachlan, A. W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts inMath. 219, Springer-Verlag, New York (2003).
[Mil] J. Milne, Class field theory, Course notes, www.jmilne.org/math.
[Miy] T. Miyake, Modular Forms, Springer-Verlag, New York (1986).
[Miy2] T. Miyake, On automorphic forms on GL2 and Hecke operators, Ann. of Math. (2) 94(1971), 174-189.
[P] J. Parson, Constructing congruences between modular forms, Preprint 2005.
[PR] D. Prasad, A. Raghuram, Kirillov Theory for GL2(D) where D is a division algebra overa non-archimedean local field, Duke Math. J. 104 (2000), No.1, 19-44.
[Ri] K. Ribet, A modular construction of unramified p-extensions of Q(µp), Invent. Math. 34(1976), 151-162.
[Ru] K. Rubin, Global units and ideal class groups, Invent. Math. 89 (1987), 511-526.
[Ru2] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields,Invent. Math. 103 (1991), 25-68.
[Ru3] K. Rubin, Euler Systems, Annals of Mathematics Studies 147, Princeton University Press,Princeton (2000).
[Schw] J. Schwermer, Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lect.Notes in Maths. 988, Springer-Verlag, New York (1983).
[Se68] J.-P. Serre, Abelian `-adic representations and elliptic curves, W. A. Benjamin Inc., NewYork (1968).
[Se70] J.-P. Serre, Le probleme de groupes de groupes de congruence pour SL2, Ann. of Math.(2), 92 (1970), 489-527.
[Se79] J.-P. Serre, Local fields, Graduate Texts in Mathematics 67, Springer-Verlag, New York(1979).
[S02a] C. Skinner, The Eisenstein ideal again, Preprint.
135
[S02b] C. Skinner, Modular Forms and Arithmetic, Lecture Course 2002.
[S04] C. Skinner, Selmer groups, Preprint.
[SW] C. Skinner, A. Wiles, Ordinary representations and modular forms, Proc. Natl. Acad.Sci. USA, Vol. 94 (1997), 10520-10527.
[T] R.L. Taylor, On Congruences between modular forms, Thesis, Princeton (1998).
[T2] R.L. Taylor, l-adic representations associated to modular forms over imaginary quadraticfields II, Invent. Math. 116 (1994), 619-643.
[Th] F. Thaine, On the ideal class groups of real abelian number fields, Ann. of Math. 128(1988), 1-18.
[Ti] J. Tilouine, Sur la conjecture principale anticyclotomique, Duke Math. J. 59, no. 3 (1989),629-673.
[U95] E. Urban, Formes Automorphes Cuspidales pour GL2 sur un corps quadratique imag-inaire. Valeurs speciales de fonctions L et congruences, Composition Math. 99 (1995),no.3, 283-324.
[U98] E. Urban, Module de congruences pour GL(2) d’un corps imaginaire quadratique ettheorie d’Iwasawa d’un corps CM biquadratique, Duke Math. J. 92 (1998), no.1, 179-220.
[U01] E. Urban, Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J. 106 (2001),485-525.
[U04] E. Urban, Sur les representation p-adiques associees aux representations cuspidales deGSp4/Q, to appear in: Proceedings of Conference on Automorphic Forms at Center EmileBorel 2000.
[Wa] L. Washington, Galois cohomology, in: G. Cornell, J. Silverman, G. Stevens, ModularForms and Fermat’s Last Theorem, Springer-Verlag, New York (1997).
[We55] A. Weil, On a certain type of characters of the idele-class group of an algebraic numberfield, Proceedings of the international symposium on number theory, Tokyo and Nikko,Science Council of Japan, Tokyo(1956), 1-7.
[We71] A. Weil, Dirichlet series and automorphic forms, Lecture Notes in Mathematics 189,Springer-Verlag, New York (1971).
[Wes] U. Weselmann, Eisensteinkohomologie und Dedekindsummen fur GL2 uber imaginar-quadratischen Zahlkorpern, J. Reine Angew. Math. 389 (1988), 90-121.
[W86] A. Wiles, On p-adic representations for totally real fields, Ann. of Math. (2) 123 (1986),no. 3, 407-456.
[W90] A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990),no. 3, 493-540.
[Z] S. Zucker, On the boundary cohomology of locally symmetric varieties, Vietnam Journalof Mathematics 25 (1997), no.4, 279-318.