An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters

0.38 MB·English

by  Tobias Berger

An Eisenstein ideal for imaginary quadratic fields and
the Bloch-Kato conjecture for Hecke characters
Tobias Berger
7
0
Abstract
0
2
For certain algebraic Hecke characters χ of an imaginary quadratic field F we define

n
an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of
a
J GL2/F. By finding congruences between Eisenstein cohomology classes (in the sense of

5 G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein
ideal in the Hecke algebra in terms of the special L-value L(0,χ). We further prove that
]
T its index is bounded from above by the order of the Selmer group of the p-adic Galois
N character associated to χ−1. This uses the work of R. Taylor et al. on attaching Galois
. representations to cuspforms of GL /F. Together these results imply a lower bound for
h 2
t the size of the Selmer group in terms of L(0,χ), coinciding with the value given by the
a
m Bloch-Kato conjecture.
[

Introduction
1
v
7 The aim of this work is to demonstrate the use of Eisenstein cohomology, as developed by Harder,
7 in constructing elements of Selmer groups for Hecke characters of an imaginary quadratic field F.
1
The strategy of first finding congruences between Eisenstein series and cuspforms and then using
1
0 the Galois representations attached to the cuspforms to prove lower bounds on the size of Selmer
7 groups goes back to Ribet [Rib76], and has been applied and generalized in [Wil90] in his proof of
0
the Iwasawa main conjecture for characters over totally real fields, and more recently by [SU02] and
/
h [BC04]amongstothers.Theseallusedforthecongruencesintegral structurescomingfromalgebraic
t
a geometry. In our case the symmetric space associated to GL /F is not hermitian and we therefore
2
m
use the integral structure arising from Betti cohomology. This alternative, more general approach
:
v was outlined for GL2/Q in [HP92].
i
X In [Har87] G. Harder constructed Eisenstein cohomology as a complement to the cuspidal co-
r homology for the groups GL2 over number fields and proved that this decomposition respects the
a
rational structure of group cohomology. The case interesting for congruences is when this decom-
position is not integral, i.e., when there exists an Eisenstein class with integral restriction to the
boundarythathasadenominator.In[Ber06a]suchclasses wereconstructedforimaginaryquadratic
fields and their denominator was bounded from below by the special L-value of a Hecke character.
In 3 we give a general set-up in which cohomological congruences between Eisenstein and cuspidal
§
classes can be proven (Proposition 9) and then apply this to the classes constructed in [Ber06a].
The result can be expressed as a lower bound for the index of the Eisenstein ideal of the title in a
Hecke algebra in terms of the special L-value. The main obstacle to obtaining a congruence from
the results in [Ber06a] is the occurrence of torsion in higher degree cohomology, which is not very
well understood (see the discussion in 3.4). However, we manage to solve this “torsion problem”
§
for unramified characters in 4.
§
2000 Mathematics Subject Classification 11F80, 11F33, 11F75
Keywords: Selmer groups, Eisenstein cohomology, congruences of modular forms, Bloch-Kato conjecture
Tobias Berger
The other main ingredient are the Galois representations associated to cohomological cuspidal
automorphic forms, constructed by R. Taylor et al. by means of a lifting to the symplectic group in
[Tay94]. Assuming the existence of congruences between an Eisenstein series and cuspforms we use
these representations in 2 to construct elements in the Selmer group of a Galois character. In fact,
§
we prove that its size is bounded from below by the index of the Eisenstein ideal.
These two results are combined in 5 to prove a lower bound on the size of Selmer groups of
§
Hecke characters of an imaginary quadratic field in terms of a special L-value, coinciding with the
value given by the Bloch-Kato conjecture.
To give a more precise account, let p > 3 be a prime unramified in the extension F/Q and
let p be a prime of F dividing (p). Fix embeddings F ֒ F ֒ C. Let φ ,φ : F∗ A∗ C∗
→ p → 1 2 \ F →
be two Hecke characters of infinity type z and z−1, respectively. Let be the ring of integers in
R
a sufficiently large finite extension of F . Let T be the -algebra generated by Hecke operators
p
R
acting on cuspidal automorphic forms of GL /F. For φ = (φ ,φ ) we define in 2 an Eisenstein
2 1 2
§
ideal I in T. Following previous work of Wiles and Urban we construct elements in the Selmer
φ
group of χ ǫ, where χ is the p-adic Galois characters associated to χ := φ /φ and ǫ is the p-adic
p p 1 2
cyclotomic character. We obtain a lower bound on the size of the Selmer group in terms of that
of the congruence module T/I . A complication that arises in the application of Taylor’s theorem
φ
is that we need to work with cuspforms with cyclotomic central character. This is achieved by a
twisting argument (see Lemma 8).
To prove the lower bound on the congruence module in terms of the special L-value (the first
step described above), we use the Eisenstein cohomology class Eis(φ) constructed in [Ber06a] in
the cohomology of a symmetric space S associated to GL /F. The class is an eigenvector for the
2
Hecke operators at almost all places with eigenvalues corresponding to the generators of I , and its
φ
restriction to the boundary of the Borel-Serre compactification of S is integral. The main result of
[Ber06a], which we recall in 5, is that the denominator δ of Eis(φ) H1(S,F ) is bounded from
p
§ ∈
below by Lalg(0,χ). As mentioned above, Proposition 9 gives a general setup for cohomological
congruences. It implies the existence of a cuspidal cohomology class congruent to δ Eis(φ) modulo
·
theL-valuesupposingthatthereexistsanintegralcohomology class withthesamerestriction tothe
boundaryas Eis(φ). Thelatter can bereplaced bytheassumption thatH2(S, ) = 0,andthis
c R torsion
result is given in Theorem 13. In 4 we prove that the original hypothesis is satisfied for unramified
§
χ, avoiding the issue of torsion freeness. We achieve this by a careful analysis of the restriction map
to the boundary∂S of the Borel-Serre compactification. Starting with a group cohomological result
for SL ( ) due to Serre [Ser70] (which we extend to all maximal arithmetic subgroups of SL (F))
2 2
O
we define an involution on H1(∂S, ) such that the restriction map
R
res
H1(S, ) ։ H1(∂S, )−,
R R
surjects onto the 1-eigenspace. We apply the resulting criterion to res(Eis(φ)) to deduce the
−
existence of a lift to H1(S, ).
R
Note that the restriction to constant coefficient systems and therefore weight 2 automorphic
forms is important only for 4. It was applied throughout to simplify the exposition. In particular,
§
the results of Theorems 10 and 13 extend (for split p) to characters χ with infinity type zm+2z−m
for m N . See [Ber06a] for the necessary modifications and results.
>0
∈
Combining the two steps we obtain in 5 a lower bound for the size of the Selmer group of χ ǫ
p
§
in terms of Lalg(0,χ) and relate this result to the Bloch-Kato conjecture. This conjecture has been
proveninourcase(atleastforclassnumber1)startingfromtheMainConjectureofIwasawatheory
for imaginary quadratic fields (see [Han97], [Guo93]). Similar results have also been obtained by
Hida in [Hid82] for split primes p and χ = χc using congruences of classical elliptic modular forms
between CM and non-CM forms. We seem to recover base changes of his congruences in this case
2
Eisenstein ideal for imaginary quadratic fields and Bloch-Kato
(see 3.6).
§
However,ourmethodofconstructingelementsinSelmergroupsusingcohomologicalcongruences
isverydifferentandshouldbemorewidelyapplicable.TheanalytictheoryofEisensteincohomology
has been developed for many groups, and rationality results are known, e.g. for GL by the work
n
of [FS98]. Our hope is that the method presented here generalizes to these higher rank groups.
To conclude, we want to mention two related results. In [Fel00] congruences involving degree 2
Eisenstein cohomology classes for imaginary quadratic fields were constructed but only the L-value
of the quadratic character associated to F/Q was considered. The torsion problem we encounter
does not occur for degree 2, but the treatment of the denominator of the Eisenstein classes is more
difficult.ForcasesoftheBloch-KatoconjecturewhentheSelmergroupsareinfinitesee[BC04].Their
method is similar to ours in that they use congruences between Eisenstein series and cuspforms,
however, they work with p-adic families on U(3) and do not use Eisenstein cohomology.
1. Notation and Definitions
1.1 General notation
Let F/Q be an imaginary quadratic extension and d its absolute discriminant. Denote the class-
F
group by Cl(F) and the ray class group modulo a fractional ideal m by Cl (F). For a place v of F
m
let F be the completion of F at v. We write for the ring of integers of F, for the closure of
v v
O O
in F , P for the maximal ideal of , π for a uniformizer of F , and ˆ for . We use
O v v Ov v v O vfiniteOv
the notations A,A and A ,A for the adeles and finite adeles of Q and F, respectively, and
f F F,f
Q
write A∗ and A∗ for the respective group of ideles. Let p be a prime of Z that does not ramify in
F
F, and let p be a prime dividing (p).
⊂ O
Denote by G the absolute Galois group of F. For Σ a finite set of places of F let G be the
F Σ
Galois group of the maximal extension of F unramified at all places not in Σ. We fix an embedding
F ֒ F for each place v of F. Denote the corresponding decomposition and inertia groups by G
v v
→
and I , respectively. Let g = G /I be the Galois group of the maximal unramified extension of
v v v v
F . For each finite place v we also fix an embedding F ֒ C that is compatible with the fixed
v v
→
embeddings i : F ֒ F and i :F ֒ C(= F ). For a topological G -module (resp. G -module)
v v ∞ ∞ F v
→ →
M write H1(F,M) for the continuous Galois cohomology group H1(G ,M), and H1(F ,M) for
F v
H1(G ,M).
v
1.2 Hecke characters
A Hecke character of F is a continuous group homomorphism λ : F∗ A∗ C∗. Such a character
\ F →
corresponds uniquely to a character on ideals prime to the conductor, which we also denote by λ.
Define the character λc by λc(x) = λ(x).
Lemma 1 (Lemma 3.1 of [Ber05]). If λ is an unramified Hecke character then λc = λ.
For Hecke characters λ of type (A ), i.e., with infinity type λ (z) = zmzn with m,n Z we
0 ∞
∈
define (following Weil) a p-adic Galois character
∗
λ : G F
p F p
→
associated to λ by the following rule: For a finite place v not dividing p or the conductor of λ,
put λ (Frob )= i (i−1(λ(π ))) where Frob is the arithmetic Frobenius at v. It takes values in the
p v p ∞ v v
integer ring of a finite extension of F .
p
Let ǫ : G Z∗ be the p-adic cyclotomic character defined by the action of G on the p-power
F → p F
roots of unity: g.ξ = ξǫ(g) for ξ with ξpm = 1 for some m. Our convention is that the Hodge-Tate
weight of ǫ at p is 1.
3
Tobias Berger
Write L(0,λ) for the Hecke L-function of λ. Let λ a Hecke character of infinity type za z b with
z
conductor prime to p. Assume a,b Z and a> 0 and b > 0. Put
∈ (cid:0) (cid:1)
b
2π
Lalg(0,λ) := Ω−a−2b Γ(a+b) L(0,λ).
√d ·
(cid:18) F(cid:19)
In most cases, this normalization is integral, i.e., lies in the integer ring of a finite extension of F .
p
See [Ber06a] Theorem 3 for the exact statement. Put
Lalg(0,λ) if val (Lalg(0,λ)) > 0
Lint(0,λ) = p
(1 otherwise.
1.3 Selmer groups
Let ρ : G ∗ be a continuous Galois character taking values in the ring of integers of a
F
→ R R
finite extension L of F . Write m for its maximal ideal and put ∨ = L/ . Let , L , and
p R ρ ρ
R R R
W = L / = ∨ be the free rank one modules on which G acts via ρ.
ρ ρ ρ ρ R F
R R ⊗ R
Following Bloch and Kato et al. we define the following Selmer groups: Let
ker(H1(F ,L ) H1(I ,L )) for v ∤p,
H1(F ,L )= v ρ → v ρ
f v ρ (ker(H1(Fv,Lρ) H1(Fv,Bcris Lρ)) for v p,
→ ⊗ |
where B denotes Fontaine’s ring of p-adic periods. Put
cris
H1(F ,W ) = im(H1(F ,L ) H1(F ,W )).
f v ρ f v ρ → v ρ
For a finite set of places Σ of F define
H1(F ,W )
SelΣ(F,ρ) = ker H1(F,W ) v ρ .
ρ → H1(F ,W )
v∈/Σ f v ρ !
Y
We write Sel(F,ρ) for Sel∅(F,ρ).
If p splits in F/Q and ρ = λ for a Hecke character λ of infinity type zazb with a,b Z
p
∈
(“ordinary case”) then we define
L if a< 0 (i.e., HT-wt of ρ > 0),
F+L = ρ
p ρ ( 0 if a> 0 (i.e., HT-wt of ρ6 0)
{ }
and
L if b < 0,
F+L = ρ
p ρ ( 0 if b > 0.
{ }
In the ordinary case we have H1(F ,L ) = H1(F ,F+L ) for v p (see [Guo96] p.361, [Fla90]
f v ρ v v ρ |
Lemma 2).
Lemma 2. Let ρ be unramified at v ∤p. If ρ(Frob ) ǫ(Frob ) mod p then
v v
6≡
SelΣ(F,ρ) = SelΣ\{v}(F,ρ).
Proof. By definition SelΣ\{v}(F,ρ) SelΣ(F,ρ) for any v. For places v as in the lemma we have
⊂
H1(F ,W ) = ker(H1(F ,W ) H1(I ,W )gv).
f v ρ v ρ → v ρ
It is clear that H1(I ,W )gv = Hom (Itame,W ) = Hom (Itame,W [mn]) for some n. By our
v ρ gv v ρ gv v ρ R
assumption therefore H1(I ,W )gv = 0 since Frob acts on Itame by ǫ(Frob ).
v ρ v v v
4
Eisenstein ideal for imaginary quadratic fields and Bloch-Kato
1.4 Cuspidal automorphic representations
We refer to [Urb95] 3.1 as a reference for the following: For K = K G(A ) a compact open
§ f v v ⊂ f
subgroup, denote by S (K ) the space of cuspidal automorphic forms of GL (F) of weight 2, right-
2 f 2
Q
invariant under K . For ω a finite order Hecke character write S (K ,ω) for the forms with central
f 2 f
character ω. This is isomorphic as a G(A )-module to πKf for automorphic representations π
f f
of a certain infinity type (see Theorem 3 below) with central character ω. For g G(A ) we have
f
L ∈
the Hecke action of [K gK ] on S (K ) and S (K ,ω). For places v with K = GL ( ) we define
f f 2 f 2 f v 2 v
O
π 0
T = [K v K ].
v f 0 1 f
(cid:18) (cid:19)
1.5 Cohomology of symmetric space
Let G = Res GL and B the restriction of scalars of the Borel subgroup of upper triangular
F/Q 2
matrices. For any Q-algebra R we consider a pair of characters φ= (φ ,φ ) of R∗ R∗ as character
1 2
×
a b
of B(R) by defining φ( ) = φ (a)φ (b). Put K = U(2) C∗ G(R). For an open compact
0 d 1 2 ∞ · ⊂
(cid:18) (cid:19)
subgroup K G(A ) we define the adelic symmetric space
f f
⊂
S = G(Q) G(A)/K K .
Kf \ ∞ f
Note that S has several connected components. In fact, strong approximation implies that the
Kf
fibers of the determinant map
S ։ π (K ) := A∗ /det(K )F∗
Kf 0 f F,f f
are connected. Any γ G(A ) gives rise to an injection
f
∈
G ֒ G(A)
∞
→
g (g ,γ)
∞ ∞
7→
and, after taking quotients, to a component Γ G /K S , where
γ\ ∞ ∞ → Kf
Γ := G(Q) γK γ−1.
γ f
∩
Thiscomponentisthefiberoverdet(γ).Choosingasystemofrepresentativesforπ (K )wetherefore
0 f
have
S = Γ H ,
Kf ∼ γ\ 3
[det(γ)a]∈π0(Kf)
where G /K has been identified with three-dimensional hyperbolic space H = R C.
∞ ∞ 3 >0
×
We denote the Borel-Serre compactification of S by S and write ∂S for its boundary.
Kf Kf Kf
TheBorel-Serre compactification S isgiven bytheunionofthecompactifications of its connected
Kf
components. For any arithmetic subgroup Γ G(Q), the boundary of the Borel-Serre compactifi-
⊂
cation of Γ H , denoted by ∂(Γ H ), is homotopy equivalent to
3 3
\ \
ΓBη H3, (1)
\
[η]∈P1(F)/Γ
a
where we identify P1(F) = B(Q) G(Q), take η G(Q), and put Γ = Γ η−1B(Q)η.
Bη
\ ∈ ∩
For X S and R an -algebra we denote by Hi(X,R) (resp. Hi(X,R)) the i-th (Betti)
⊂ Kf O c
cohomology group (resp. with compact support), and the interior cohomology, i.e., the image of
Hi(X,R) in Hi(X,R), by Hi(X,R).
c !
ThereisaHeckeactionofdoublecosets[K gK ]forg G(A )onthesecohomology groups(see
f f f
∈
π 0 π 0
[Urb98] 1.4.4 for the definition). We put T = [K v K ] and S = [K v K ].
§ πv f 0 1 f πv f 0 πv f
(cid:18) (cid:19) (cid:18) (cid:19)
5
Tobias Berger
The connection between cohomology and cuspidal automorphic forms is given by the Eichler-
Shimura-Harder isomorphism (in this special case see [Urb98] Theorem 1.5.1): For any compact
open subgroup K G(A ) we have
f f
⊂
S (K ) ∼ H1(S ,C) (2)
2 f → ! Kf
and the isomorphism is Hecke-equivariant.
One knows (see for example, [Ber06a] Proposition 4) that for any [1]-algebra R there is a
O 6
natural R-functorial isomorphism
H1(Γ H ,R)= H1(Γ,R), (3)
\ 3 ∼
where the group cohomology H1(Γ,R) is just given by Hom(Γ,R).
1.6 Galois representations associated to cuspforms for imaginary quadratic fields
Combining the work of Taylor, Harris, and Soudry with results of Friedberg-Hoffstein and Lau-
mon/Weissauer, one can show the following (see [BHR]):
Theorem 3. Given a cuspidal automorphic representation π of GL (A ) with π isomorphic to
2 F ∞
the principal series representation corresponding to
t t t
1 1 2
∗ | |
(cid:18)0 t2(cid:19) 7→ (cid:18)|t1|(cid:19)(cid:18) t2 (cid:19)
and cyclotomic central character ω (i.e. ωc = ω), let Σ denote the set of places above p, the primes
π
where π or πc is ramified, and primes ramified in F/Q.
Then there exists a continuous Galois representation
ρ :G GL (F )
π F 2 p
→
such that if v / Σ , then ρ is unramified at v and the characteristic polynomial of ρ (Frob ) is
π π π v
∈
x2 a (π)x+ω(P )Nm (P ),wherea (π)istheHeckeeigenvaluecorrespondingtoT .Theimage
v v F/Q v v v
−
of the Galois representation lies in GL (L) for a finite extension L of F and the representation is
2 p
absolutely irreducible.
Remark. i) TaylorrelatesπtoalowweightSiegelmodularformviaathetaliftandusestheGalois
representation attached to this form (via pseudorepresentations and theGalois representations
of cohomological Siegel modular forms) to find ρ .
π
ii) Taylor had some additional technical assumption in [Tay94] and only showed the equality of
Hecke and Frobenius polynomial outside a set of places of zero density. For this strengthening
of Taylor’s result see [BHR].
iii) Conjecture 3.2 in [CD06] describes for cuspforms of general weight a conjectural extension of
Taylor’s theorem.
Urban studied in [Urb98] the case of ordinary automorphic representations π, and together with
results in [Urb05] on the Galois representations attached to ordinary Siegel modular forms shows:
Theorem 4 (Corollaire 2 of [Urb05]). If π is unramified at p and ordinary at p, i.e., a (π) = 1,
p p
| |
then the Galois representation ρ is ordinary at p, i.e.,
π
Ψ
ρπ|Gp ∼= 01 Ψ∗2 ,
(cid:18) (cid:19)
where Ψ = 1, and Ψ = det(ρ ) = ǫ.
2|Ip 1|Ip π |Ip
For p inert we will need a stronger statement:
6
Eisenstein ideal for imaginary quadratic fields and Bloch-Kato
Conjecture 5. If π is unramified at p then ρ is crystalline.
π|Gp
This conjecture extends Conjecture 3.2 in [CD06] and would follow if one could prove the corre-
sponding statement for low weight Siegel modular forms.
2. Selmer group and Eisenstein ideal
In this section we define an Eisenstein ideal in a Hecke algebra acting on cuspidal automorphic
forms of GL and show that its index gives a lower bound on the size of the Selmer group of a
2/F
Galois character.
Let φ and φ be two Hecke characters with infinity type z and z−1, respectively. Let be
1 2
R
the ring of integers in the finite extension L of F containing the values of the finite parts of φ
p i
and Lalg(0,φ /φ ). Denote its maximal ideal by m . Let Σ be the finite set of places dividing
1 2 R φ
the conductors of the characters φ and their complex conjugates and the places dividing pd . Let
i F
K = K G(A ) be a compact open subgroup such that K = GL ( ) if v / Σ .
f v v ⊂ f v 2 Ov ∈ φ
Because of the condition on the central character in Theorem 3 we assume that there exists a
Q
finite order Hecke character η unramified outside Σ such that
φ
(φ φ η2)c = φ φ η2. (4)
1 2 1 2
Denote byTthe -algebra generated bytheHecke operatorsT ,v / Σ acting onS (K ,φ φ ).
v φ 2 f 1 2
R ∈
Call the ideal I T generated by
φ
⊆
T φ (P )Nm(P ) φ (P )v / Σ
v 1 v v 2 v φ
{ − − | ∈ }
the Eisenstein ideal associated to φ= (φ ,φ ).
1 2
Using the notation of 1.2, we define Galois characters
§
ρ = φ ǫ,
1 1,p
ρ = φ ,
2 2,p
ρ= ρ ρ−1.
1⊗ 2
Notice that ρ depends only on the quotient φ /φ . Let Σ be the set of places dividing p and those
1 2 ρ
where ρ is ramified. Our first main result is the following inequality:
Theorem 6.
val (#SelΣφ\Σρ(F,ρ)) > val (#(T/I )).
p p φ
Proof. We can assume that
T/I = 0.
φ
6
Let m T be a maximal ideal containing I . Taking the completion with respect to m we write
φ
⊂
n
S (K ,φ φ ) = VKf,
2 f 1 2 m πi,f
i=1
M
where V denotes the representation space of the (finite part) of a cuspidal automorphic represen-
πf
tation π.
By twisting the cuspforms by the finite order character η of (4) we can ensure that their central
character is cyclotomic. Hence we can apply Theorem 3 to associate Galois representations ρ :
πi⊗η
G GL (L ) to each π η,i = 1...n for some finite extensions L /F . Taking all of them
Σφ → 2 i i ⊗ i p
7
Tobias Berger
together (and untwisting by η) we obtain a continuous, absolutely irreducible Galois representation
n
ρ := ρ η−1 : G GL (T L).
T πi⊗η ⊗ Σφ → 2 m⊗R
i=1
M
Here we use that T L = n L , which follows from the strong multiplicity one theorem. We
m⊗R i=1 i
have an embedding
Q
n
T ֒ L
m i
→
i=1
Y
T ((a (π )),
v v i
7→
where a (π ) is the T -eigenvalue of π . The coefficients of the characteristic polynomial char(ρ )
v i v i T
lie in T and by the Chebotarev density theorem
m
char(ρ ) char(ρ ρ ) mod I .
T 1 2 φ
≡ ⊕
For any finite free T L-module , any T -submodule that is finite over T and
m m m
⊗ M L ⊂ M
such that L = is called a T -lattice. Specializing to our situation Theorem 1.1 of [Urb01]
m
L⊗ M
we get:
Theorem 7 (Urban). Given a Galois representation ρ as above there exists a G -stable T -
T Σφ m
lattice (T L)2 such that G acts on /I via the short exact sequence
L ⊂ m⊗ Σφ L φL
0 (N/I ) /I (T /I ) 0,
→ Rρ1 ⊗R φ → L φL → Rρ2 ⊗R m φ →
where N T L is a T -lattice with val (#T /I )6 val (#N/I N) < and no quotient of
m m p m φ p φ
⊂ ⊗ ∞ L
is isomorphic to ρ := ρ mod m .
1 1 R
Proof. Note that the R-algebra map R ։ Tm/Iφ is surjective and that L/IφL ∼= L ⊗R Tm/Iφ.
Hence this short exact sequence recovers the one in the statement of Theorem 1.1 of [Urb01]. For
the statement about val (#N/I N) see [Urb01] p. 519 and use that any -submodule of T /I or
p φ m φ
R
N/I N is a T -submodule.
φ m
See [Ber05] 7.3.2 for an alternative construction of such a lattice using arguments of Wiles
§
([Wil86] and [Wil90]).
Using the properties of the Galois representations attached to cuspforms listed in 1.6 we can
§
now conclude the proof of Theorem 6 by similar arguments as in [Ski04] and [Urb01]. To ease
notation we put := N/I and Σ := Σ .
φ φ
T
Identifying with Hom ( , ) and writing s : T /I T /I for the section
Rρ R Rρ2 Rρ1 Rρ2⊗ m φ → L⊗ m φ
as T /I -modules we define a 1-cocycle c :G by
m φ Σ ρ
→ R ⊗T
c(g)(m) = the image of s(m) g.s(g−1.m) in .
− Rρ1 ⊗T
Consider the -homomorphism
R
ϕ :Hom ( , ∨) H1(G ,W ), ϕ(f)= the class of (1 f) c.
R Σ ρ
T R → ⊗ ◦
We will show that
(i) im(ϕ) SelΣ\Σρ(F,ρ),
⊂
(ii) ker(ϕ) = 0.
From (i) it follows that
val (#SelΣ\Σρ(F,ρ)) > val (#im(ϕ)).
p p
8
Eisenstein ideal for imaginary quadratic fields and Bloch-Kato
From (ii) it follows that
val (#im(ϕ)) > val (#Hom ( , ∨))
p p R
T R
= val (# )
p
T
> val (#T /I ).
p m φ
For (i) we have to show that the conditions of the Selmer group at v p are satisfied: For split
|
p it suffices to prove that the extension in Theorem 7 is split when considered as an extension of
T [G ]-modules, because then the class in H1(G , ) determined by c is the zero class. In
m p p ρ
R ⊗T
this case the Hecke eigenvalues a (π ) p φ (p)+φ (p) 0 mod m , hence the cuspforms π η
p i 1 2 R i
≡ · 6≡ ⊗
are ordinary at p, so Theorem 4 applies and ρ is ordinary. Observing that the Hodge-Tate weights
T
at p of ρ and ρ are 0 and 1, respectively, the splitting of the extension as T [G ]-modules follows
1 2 m p
from comparing the basis given by Theorem 7 with the one coming from ordinarity.
For inert p we observe that by Conjecture 5 the ρ are all crystalline, which implies that the
πi
class determined by c is crystalline.
To prove (ii) we first observe that for any f Hom ( , ∨), ker(f) has finite index in since
R
∈ T R T
is a finite -module and so f Hom ( , ∨[mn]) for some n. Suppose now that f ker(ϕ).
T R ∈ R T R R ∈
We claim that the class of c in H1(G , /ker(f)) is zero. To see this, let X = ∨/im(f)
Σ ρ R
R ⊗ T R
and observe that there is an exact sequence
H0(G , X) H1(G , /ker(f)) H1(G , ∨).
Σ ρ R Σ ρ R Σ ρ R
R ⊗ → R ⊗ T → R ⊗ R
Since f ker(ϕ) and the second arrow in the sequence comes from the inclusion /ker(ϕ) ֒ ∨
∈ T → R
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 Hom ( , ) on which G acts trivially. By assumption, however, is a rank one module on
R ρ Σ ρ
R R R
which G acts non-trivially.
Σ
Suppose in addition that f is non-trivial, i.e., ker(f) ( . Note that any -submodule of is
T R T
actually a T -submodule since ։ T /I . Hence there exists a T -module A with ker(f) A
m m φ m
R ⊂ ⊂
such that /A = /m . From our claim it follows that the T [G ]-extension
T T ∼ R R m Σ
0 /m = /A /( A) / 0
→ Rρ1 ⊗RR R ∼ Rρ1 ⊗RT → L Rρ1 ⊗R → L L1 →
is split. But this would give a T [G ]-quotient of isomorphic to ρ , which contradicts one of the
m Σ L 1
properties of the lattice constructed by Urban. Hence ker(ϕ) is trivial.
The following Lemma will later provide us with the finite order character η of (4) used in the
twisting above.
Lemma 8. If χ = φ /φ satisfies χc = χ then there exists a finite order character η unramified
1 2
outside Σ such that (φ φ η2)c = φ φ η2.
φ 1 2 1 2
Proof. We claim that there exists a Hecke character µ unramified outside Σ such that
φ
χ = µµc.
Given such a character µ we then define η = (µφ )−1.
2
In the Lemma on p.81 of [Gre83] Greenberg defines a Hecke character µ : F∗ A∗ C∗ of
G \ F →
infinity type z−1 such that µc = µ and µ is ramified exactly at the primes ramified in F/Q. It
G G G
therefore suffices to prove the claim for the finite order character
χ′ := χµ2 = χµ (µc ).
G G G
By assumption we have that
χ′ 1 on Nm (A∗) A∗ A∗.
≡ F/Q F ⊂ Q ⊂ F
9
Tobias Berger
Thus χ′ restricted to Q∗ A∗ is either the quadratic character of F/Q or trivial. Since our finite
\ Q
order character has trivial infinite component, χ′ has to be trivial on Q∗ A∗ . Hilbert’s Theorem
\ Q
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∗ A, where
F →
A is the subset of A∗ of elements of the form x/xc and the map is x x/xc. If y A F∗ then
F 7→ ∈ ∩
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∗ C∗ which restricts to this character on A and χ′ = µ/µc (this argument is taken from
\ F →
the proof of Lemma 1 in [Tay94]).
By the following argument we can further conclude that the induced character vanishes on
A ∗ and therefore find µ on F∗ A∗/C∗ ∗ restricting to the character A C∗:
∩ v∈/ΣφOv \ F v∈/ΣφOv →
Writing U = ∗ for a prime ℓ in Q we have
Q F,ℓ v|ℓOv Q
Q H1(Gal(F/Q), ∗) = H1(Gal(F/Q),U ),
Ov ∼ F,ℓ
vY∈/Σφ ℓY∈/Σφ
where “ℓ / Σ ” denotes those ℓ Z such that v ℓ v / Σ . For the isomorphism we use that
φ φ
∈ ∈ | ⇒ ∈
v Σ v Σ .
φ φ
∈ ⇒ ∈
In fact, all these groups are trivial since all ℓ / Σ are unramified in F/Q and so
φ
∈
H1(Gal(F/Q),U )= H1(G , ∗) = 1.
F,ℓ ∼ v Ov
If y A ∗ then y has trivial normin ∗. But as shown,its firstGalois cohomology
∈ ∩ v∈/ΣφOv v∈/ΣφOv
groupistrivial sothereexists x ∗ A∗ suchthaty = x/xc.Sinceχ′ is unramifiedoutside
Q ∈ v∈/ΣφOv∩ QF
Σ the image of y under the induced character therefore equals χ′(x) = 1, as claimed above.
φ Q
3. Cohomological congruences
In [Ber06a] we constructed a class Eis(φ) in the cohomology of a symmetric space associated to
GL with integral non-zero restriction to the boundary of the Borel-Serre compactification that
2/F
is an eigenvector for the Hecke operators at almost all places. By a result of Harder one knows that
Eis(φ) is rational. The main result of [Ber06a] is a lower bound on its denominator (defined in (5)
below) in terms of the L-value of a Hecke character. In this section we show that if there exists
an integral cohomology class with the same restriction to the boundary as Eis(φ) then there exists
a congruence modulo the L-value between Eis(φ), multiplied by its denominator, and a cuspidal
cohomology class. This implies that the L-value bounds the index of the Eisenstein ideal from
below.
3.1 The Eisenstein cohomology set-up
Assume from now on in addition that p > 3. Recall the notation and definitions introduced in
Section 1.5. Let denote the ring of integers in the finite extension L of F obtained by adjoining
p
R
the values of the finite part of both φ and Lalg(0,φ /φ ). We write
i 1 2
H1(X, ) := H1(X, ) = im(H1(X, ) H1(X,L))
free
R R R →
for X = S or ∂S . For c H1(S ,L) define the denominator (ideal) by
Kf Kfe ∈ Kf
δ(c) := a : a c H1(S , ) . (5)
{ ∈ R · ∈ Kf R }
Recall the long exact sequence
e
... H1(S ,R) H1(S ,R) res H1(∂S ,R) H2(S ,R) ...
→ c Kf → Kf → Kf → c Kf →
10