行政院國家科學委員會專題研究計畫 成果報告
由 GL_2 與 GSp_4 所衍生的志村多樣體及其幾何(第 2 年)
研究成果報告(完整版)
計 畫 類 別 ： 個別型
計 畫 編 號 ： NSC 98-2115-M-004-007-MY2
執 行 期 間 ： 99 年 08 月 01 日至 100 年 07 月 31 日
執 行 單 位 ： 國立政治大學應用數學學系
計 畫 主 持 人 ： 余屹正
報 告 附 件 ： 國外研究心得報告
公 開 資 訊 ： 本計畫可公開查詢
中 華 民 國 100 年 12 月 23 日
中 文 摘 要 ： 我們研究某些志村曲線的特殊纖維之幾何,並且給了大域至局
部 Jacquet-Langlands 相容理論的直接證明。
中文關鍵詞： 志村曲線,消沒圈層,加羅瓦表示,p 進位單值化理論
英 文 摘 要 ： We study the geometry of the special fibers of
certain Shimura curves over a totally real field and
give a direct
proof of global-to-local Jacquet-Langlands
compatibility.
英文關鍵詞： Shimura curves； Vanishing cycles； Galois
RAMAKRISHNA-KHARE SYSTEMS AND MODULARITY LIFTING THEOREMS IN HIGHER WEIGHTS
YIH-JENG YU
Abstract. Following the ideas of Khare and Ramakrishna-Khare, we give a different approach to prove the modularity lifting theorem in higher weights without using Taylor-Wiles systems.
1. Introduction
The focus of this article is to give a different approach of proving modularity lifting theorems of Galois representations in Wiles [27] and Taylor-Wiles [26]:
Theorem. Let ρ : GQ → GL2(O) be an odd, continuous, absolutely irreducible, p-adic
Galois representation which is ramified at finitely many primes, and de Rham at p with Hodge-Tate weights (k− 1, 0) with k ≥ 2. Assume that the reduction modulo p of ρ is modular. Then ρ is isomorphic to an integral model of a p-adic representation ρf arising
from a newform f .
This is sometimes described as “R = bT”-theorems, where R is the universal deformation ring of the reduction ρ of ρ and bT is a certain localized Hecke algebra.
In this article, we generalize the approach introduced by Khare and Ramakrishna from weight 2 [15, 14] to higher weights k < p; we prove:
Theorem. Let N be a square-free positive integer, let p > 5 be a prime not dividing N .
Let f ∈ Sk(bΓ0(N )) be a cusp newform. Let 2≤ k < p be an integer. Let ρ = ρf : GQ →
GL2(F) be the modp Galois representation attached to f . Assume that ρ is irreducible,
minimally ramified at primes at primes dividing N p, and that ρ|Ip=
χk−1p ∗
0 1
!
with
∗ 6= 0, where χp is the mod p cyclotomic character. Then the universal deformation ring
R associated to ρ is canonically isomorphic to bT∅(≃ bT).
The essential point of this generalization is to replace the use of Jacobians of Shimura curves with the cohomology of certain sheaves over the curves in question.
The paper is organized as follows. We first review the basic properties of Shimura curves, and then take up the study of the bad reduction of the Shimura curves at a prime r dividing the level in question by the Tate-Oort theory. Along this line, we obtain an explicit description of the special fiber as the union of exactly two irreducible components
Date: 31, October, 2011.
2000 Mathematics Subject Classification. 11F80, 11G18, 14D05, 14F05, 14G22, 14G35, 14K10. Key words and phrases. Hecke algebras; Modular curves; Shimura curves; Vanishing cycles; Mon-odromy; Galois representations; Ramakrishna-Khare systems; ˇCerednik-Drinfel′d uniformizations.
of multiplicity 1 and r− 1 respectively. These two components cross transversally at the supersingular points and nowhere else. This description enables us calculate the vanishing cycles (Proposition 4.5) and the cohomology of Shimura curves (Proposition 4.7).
Suppose that ρ is an odd, continuous, absolutely irreducible Galois representation with values in GL2(F) where F is a finite field of characteristic p. Under certain hypotheses,
we impose the deformation conditions on such ρ, and show the existence of deformation ring RQ(Proposition 3.3) which is universal for the deformations of ρ unramified outside
S∪ Q and minimally ramified on S, where S is the set of primes at which ρ is ramified and Q is a set of auxiliary primes. According to Khare and Ramakrishna, we define for each α ⊆ Q a quotient Rα
Q of RQ which is universal for the deformations ρ of ρ such
that, for q∈ α, the local representation ρ |Gq is special, i.e., it is of the form
χp ∗
0 1 !
where χp is the p-adic cyclotomic character. With the result at hand, we shall identify
the tangent space of Rα
Q with a suitable Selmer group.
From now on, we suppose that ρ is modular with weight k < p. The first step (Proposition 5.1) is to show that there exists a set Q of auxiliary primes such that RQQ is isomorphic to W (F) and that the corresponding deformation ρQQ is modular. The existence of Q is due to Khare and Ramakrishna [15]. The proof of this isomorphism and the modularity of ρQQ use the fact that the tangent space of RQQ is trivial and the work of Diamond-Taylor [9]: there is a p-adic lifting of ρ that arises from the Q-new quotient of bTQ.
The second step (Theorem 5.2) is to show that the deformation ρQ parametrized by
RQis modular. As in the founder article of Wiles, we introduce a localized Hecke algebra
b
TQparametrizing a modular deformation of geometric origin and show that the canonical
homomorphism RQ → bTQ is actual an isomorphism. In the proof of Theorem 5.2, we
use a variant of Wiles’ numerical isomorphism criterion refined by Lenstra (Proposition 6.1). Thus, we are quickly reduced to study how a certain congruence module grows as one relaxes conditions of newness at primes in Q. Let π : bTQ → bTQQ ≃ RQQ ≃ W (F)
be the canonical homomorphism resulting from the first step, φ : RQ → bTQ, Φ =
ker(πφ)/ ker(πφ)2 and η = π(AnnT(ker(π))). The criterion consists with verifying the
equality|W (F)/η| = |Φ|.
The verification of this equality is the main part of this paper. First, we need a sophisticated calculations of Galois cohomology and identify Φ to certain Selmer groups; we thus obtain an upper bound for|Φ|. Then in the proof of theorem in weight 2, Khare used a result of Ribet-Takahashi [25] which generalized a calculation of Ribet [24] in his work on Serre’s conjecture. The idea of Ribet is to compare two Shimura curves such that two prime numbers q and q′ dividing the discriminant for one and dividing the level for the other. Hence in weight 2 we could compare the Jacobians of corresponding Shimura curves. For most of our work, we extend the level-lowering part by replacing Ribet’s method via Jacobians of certain Shimura curves with arguments using vanishing cycles on those curves (Proposition 7.6). This requires a study of Boutot-Carayol’s version of
ˇ
of isomorphism RQQ = bTQQ to RQ = bTQ is carried out by applying the level-lowering
(Proposition 8.2), and the numerical isomorphism criterion alluded to above.
The third and final step (Theorem 5.3) is to get rid of the set of auxiliary primes Q and yields that the ramified minimal universal deformation ρ∅ is modular. More precisely, we use the local-to-global principle of B¨ockle [1] to show the canonical morphism R∅ → bT∅
is an actual isomorphism.
The ideas developed here can be applied to the case over totally real fields, and this will be in our forthcoming work.
Acknowledgements. The author is deeply indebted to Jacques Tilouine, who sug-gested the problem, for many enlightening discussions. Moreover, he is very thankful to R. Livn´e for indicating the thesis of L. Yang [28], and to A. Mokrane who pointed out the article of L. Illusie [12]. He also thanks J. Yu of NCTS, and C.-F. Yu of Academia Sinica for their hospitality.
2. Background on Shimura curves
We first give a brief review of the basic properties of Shimura curves, following Buzzard [3]. We then take up the study of the reduction modulo a prime of the Shimura curves. As a preparation for our later work, we shall also present the Hecke correspondences. 2.1. Model of Shimura curves. Let B be an indefinite quaternion algebra over Q; let S be the set of places where it ramifies. Let D =Qℓ∈Sℓ, and let M be a square-free integer prime to D. Let OB be a maximal order of B.
For all ℓ /∈ S, we choose an isomorphism φℓ : B ⊗ Qℓ = Bℓ ∼
−→ M2(Qℓ) such that
φℓ(OBℓ) = M2(Zℓ). Let uM : (OB⊗ bZ)
× = bO×
B→ GL2(Z/M Z) induced by φℓ. We let
• bΓD0(M ) denote the preimage of ( a b c d ! ∈ GL2(Z/M Z) c = 0 ) under uM;
• bΓD1(M ) denote the preimage of ( a b c d ! ∈ GL2(Z/M Z) c = 0, d = 1 ) under uM.
We define an Eichler order of level M as follow: RM,D := ( x∈ OB φℓ(x)≡ ∗ ∗ 0 ∗ !
(mod ℓ) for all ℓ| M )
.
We write bRM,D = RM,D⊗ bZ =Qℓ(RM,D⊗ Zℓ). We see that bRM,D× = bΓD0 (M ).
Let Γ be an open compact subgroup of bB×. The Shimura curve XD(Γ) is defined by:
XD(Γ)(C) := B×\BA×/R×Γ· Ci
where Ciis the stabilizer of√−1 in (B ⊗R)×. In particular, if Γ = bR×M,D, we will denote
the corresponding Shimura curve by XD(M )(C).
Consider a torsion-free subgroup Γ of bOB× of level M . Let r be a prime, not dividing M D. Let Γ0 = Γ∩ bΓD0 (r) and Γ1 = Γ∩ bΓD1 (r). We study here the reduction modulo r
We fixed an isomorphism φr :OB⊗Zr≃ M2(Zr). Let e be the idempotent inOB/rOB
corresponding to 1 0 0 0
!
by φr. Following Buzzard [3], we define a Γ0(r)-structure,
(resp. Γ1(r)-structure, on a false elliptic curve A as a finite flat group scheme K1 of rank
r inside (1− e)A[r], resp. a (Drinfel′d) generator of this subgroup. For rigidification, we also introduce a Γ-level ν structure on A (that is, a full level structure ν of level N taken modulo Γ).
By the Corollary 4.2 of [3], the moduli problem on Zr-schemes S 7→ {isomorphism
classes of (A, ι, ν, K1)/S} is representable by a proper Zr-scheme which we denote by
XD(Γ0). Let us recall that this moduli problem is isomorphic to the problem (A, ι, ν, C)
where C is an isotropic subgroup of A[r] of order r2. (See the paragraph after Definition 3.1 in [3].) There is a universal triple (Au, ιu, Ku
1) defined over XD(Γ0). Recall the
Theorem 4.7 of [3]:
Proposition 2.1. (i) The scheme XD(Γ
0) is proper over Zr.
(ii) It is semistable over Zr, i.e. regular, and smooth away from the supersingular
points in characteristic r, with strictly henselian local ring at such a geometric point Zurr [[X, Y ]]/(XY − r); moreover, there are exactly two smooth irreducible
components, Xm and Xe, in the special fiber; they can be described as the Zariski
closure of the locus Xm,0 where K1 is of multiplicative type, resp. of Xe,0 where
K1 is ´etale.
(iii) The map π : XD(Γ
0)→ XD(Γ) forgetting the Γ0-structure is finite and flat.
We consider the moduli problem of Qr-schemes S 7→ {isomorphism classes of (A, ι, P )/S}
where P is a generator of K1; it is representable over Qr by the curve XD(Γ1)Qr. Let XD(Γ
1) be the normalization of XD(Γ0) in XD(Γ1)Qr.
Following [11] Proposition 3.3.6, we shall use Tate-Oort theory in order to prove: Proposition 2.2. (i) The model XD(Γ
1) of XD(Γ1)Qr is regular and flat over Zr. (ii) The map π10 : XD(Γ1) → XD(Γ0) is finite flat; the special fiber of XD(Γ1) is
a divisor with normal crossings, with exactly two irreducible components Ye = π−110(Xe) and Ym = π−1
10(Xm) with multiplicity 1 and r− 1 respectively, whose
underlying reduced subschemes are smooth.
(iii) The two components cross (transversally) at the supersingular points and nowhere
else.
Proof. Let us consider the finite group scheme C = K1u of rank r over the Zr-scheme
XD(Γ0) and C+the complement of the zero section in C.
If s∈ Xe,0, then C+is ´etale over XD(Γ
0) in a neighbourhood of s by Proposition 2.1.
If s ∈ Xm,0, then C+ is of multiplicative type, hence is isomorphic to µ
r on an ´etale
neighbourhood of s. Hence, the statement about multiplicities is obvious, as the map π10 is an isomorphism. Let s∈ Xm∩ Xe. By Proposition 2.1, the completed local ring
b OXD(Γ
0),s at s is isomorphic to R = Zr[[u, v]]/(uv− r) in such a way that
Tate-Oort theory [19] classifies finite flat group schemes of rank r over R; in particular the pull-back CRof C over Spf R is isomorphic to GR(x, y), the Tate-Oort group scheme
of rank r over R for some parameters x, y ∈ R, where, in the notations of Tate-Oort, a = νx, b = y and wr = νr = ab, where wr ∈ Zr is an explicit Gauss sum. Note that
GR(x, y) ≃ GR(x′, y′) if and only if x′x−1 is the (r− 1)st power of a unit in R. As in
[11] Corollary 3.3.5, we deduce from (♠) that x = αu and y = α−1v for some α∈ R×.
Therefore, CR ≃ GR(u, v) if and only if the unit α is a (r− 1)st power in R. Now
the extension R[(α)r−11 ] is finite ´etale over R. Since the problem is local in the ´etale topology, we may assume CR≃ GR(u, v).
Now we recall the Tate-Oort equations for GR(u, v) over R. Put X1 = νu, X2 = v,
then GR(u, v) = Spf(R[Y ][[X1, X2]]/(X1X2 − wr, Yr − X1Y )). (See p.13 of Tate-Oort
[19]).
Now the factorization YrX1Y = Y (Yr−1− X1) provides an embedding of the algebra
of formal functions on GR(u, v) into the ring R0× R∗, where
R0= Zr[[X1, X2]]/(X1X2− wr); R∗= Zr[[X2, Y ]]/(Yr−1X2− wr).
Dually, we have a surjective morphism
Spf(R0)⊔ Spf(R∗)→ GR(u, v).
The image of Spf(R0) corresponds to the zero section, while the image of Spf(R∗) is the
scheme-theoretic closure of C+. It follows that Spf(R∗) is a local model for C+ over a neighbourhood of the singular point s. Obviously, Spf(R∗) has the properties required. Moreover Spf(R∗) is normal, and it follows that Spf(R∗) is a formal local model for
XD(Γ1).
2.2. Local Hecke correspondences. The curve XD(Γ
1) is a fine moduli space for
triples x = (A, ν, P ), where, if D > 1, A is a false elliptic curve with a level Γ1 structure
ν and aOB-stable group scheme K1 of rank r inside (1−e)A[r] and a Drinfel′d generator
P of K1, and, if D = 1, A is a generalized elliptic curve and a generator P of a cyclic
subgroup K1 of order r. There is a universal triple (Au, ιu, K1u) defined over XD(Γ1);
let f : Au → XD(Γ 1).
Similarly, XD(Γ1∩ bΓD0 (ℓ)) classifies (x, C) where x is a triple as above and C is an
isotropic subgroup of order ℓ2 in A[ℓ] as noticed before Proposition 2.1.
For ℓ ∤ DM rp, we have two degeneracy maps αℓ and βℓ from XD(Γ1 ∩ bΓD0(ℓ)) to
XD(Γ1). XD(Γ 1∩ bΓD0(ℓ)) αℓ ww♦♦♦♦♦♦ ♦♦♦♦ ♦ β ℓ ''❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ XD(Γ1) XD(Γ1)
They are defined by αℓ((x, C)) = x and βℓ((x, C)) = (φ∗x) where φ : A→ A/C denotes
the quotient map and φ∗x = (A/C, φ∗ν, φ∗P ).
Recall that if we are given a lisse sheaf F on XD(Γ
1) with a morphism Aℓ: βℓ∗F →
(XD(Γ
1), F ). By contravariant functoriality, it induces an endomorphism Tℓof H•(XD(Γ1), F )
given by Tℓ= αℓ,∗◦ Aℓ,∗◦ βℓ∗.
In our situation, we take F = Symk−2Rf∗Zp. Γ1 acts on Symk−2Z2p from the left
by its p-component. We recall that the lisse sheaf associated to the corresponding representation of the fundamental group of XD(Γ1) is Symk−2Rf∗Zp. Let us consider the
morphism Aℓ: βℓ∗F → α∗ℓF induced by the left action of (1, . . . , 1,
1 0 0 ℓ
!
, 1, . . . , 1) on Symk−2Z2
p. Note that this action being through the p-component is trivial if ℓ6= p.
We define the ℓth Hecke correspondence T
ℓ as tℓ(Aℓ).
For a false elliptic curve A, (Z/ℓZ)× acts on A[ℓ] by multiplication. We thus have an action of (Z/ℓZ)× on Γ1-level structures on A. We let hai(A, K1, P ) = (A, K1, aP ) as
an endomorphism of XD(Γ 1).
These operators all commute with each other. We let T(Γ1) denote the Z-algebra
generated by Tℓ for all ℓ ∤ M Dp and the diamond operators.
3. Deformation rings and Hecke rings
Let F be a finite field of characteristic p > 5, and let W = W (F) be the ring of Witt vectors with coefficients in F, and letO be a totally ramified extension of W (hence its residue field is F). Consider the continuous absolutely irreducible modp Galois repre-sentation ρ : GQ → GL2(F). We write S to be the set of primes containing p, ∞ and
the primes at which ρ is ramified, and S′ = S\ {p}. Let Ad0 be the set of all trace zero two-by-two matrices over F with Galois action through ρ and by conjugation.
Suppose that ρ is modular and satisfies the following conditions:
• The Serre weight k := k(ρ) of ρ is greater than 2 and strictly less than p. • det(ρ) = χp the modp cyclotomic character.
• Ad0 is absolutely irreducible.
• ρ is semistable at every primes in S.
• Moreover, ρ is crystalline and ordinary at p.
Note that this implies the order of im(ρ) is divisible by p which gives the properties of ρ required in [23].
Remark 3.1. If p > k > 2, the condition being ordinary implies that being crystalline
by Perrin-Riou and Fontaine [20].
3.1. Deformation rings. We briefly recall the existence of certain deformation rings parametrizing liftings of ρ with given local conditions, referring [15] for the details.
Let Q be a finite set of primes disjoint from S such that for all q∈ Q, q 6≡ ±1 (mod p) and ρ(Frobq) has eigenvalues with ratio q. Consider the following covariant deformation
functor DQ from the category of complete noetherian local W -algebras to the category
of sets:
DQ: CNLW ❀ Sets
(A, ϕ) ❀ {ρ : GQ→ GL2(A)|ρ mod mA= ρ}/ ∼,
(DC1) det ρ =eγχk−1
p , where eγ is the Teichm¨uller lifting of S′-ramified character γ and
χp is the p-adic cyclotomic character;
(DC2) ρ is unramified outside S∪ Q; (DC3) ρ|Iℓ =h 1 1 0 1 ! i for all ℓ ∈ S′;
(DC4) for ℓ = p, ρ is ordinary and cristalline at p,
where ρ1 ∼ ρ2 if and only if there exists M ∈ ker(GL2(A) → GL2(F)) such that
ρ1 = M−1ρ2M .
Definition 3.2. For any representation satisfying conditions (DC1) – (DC4), we will call it minimally S-ramified.
Since Ad0 is absolutely irreducible, then by the Schlessinger’s criterion it is easy to see that:
Proposition 3.3. The functor DQ is pro-representable. We denote its universal couple
by (RQ, ρQ).
Remark 3.4. There is no condition at any primes q ∈ Q.
More generally for any subset α⊆ Q, we now consider the closed subfunctor of DQ:
DQα(A) ={ρ : GQ → GL2(A)|ρ mod mA= ρ}/ ∼
such that the conditions (DC1) – (DC4) hold and moreover (DC5) ρ|Gq ∼
χp ∗
0 1 !
for any q∈ α. (2.5)
Since the functor Dα
Q is relatively representable, hence we have the following:
Proposition 3.5. The functor DQα is pro-representable. We denote the corresponding universal couple by (RαQ, ραQ).
Remark 3.6. There is a sequence of natural surjections of local W -algebras RQ։ RQα ։
RQQ. If Q =∅, we denote the corresponding universal couple by (R∅, ρ∅), and call R∅ the
minimal deformation ring.
Remark 3.7. We could also consider similar deformation problems DQ,Oα over O instead of W . One checks easily that Rα
Q⊗W O is the universal deformation ring of DQ,Oα .
3.2. The local conditions. Let GS∪Q be the Galois group of the maximal extension
of Q in Q which is unramified outside S∪ Q. We introduce local conditions in order to define the Selmer group.
• For v ∈ S′, we let Lv = H1nr(Gv, Ad0) := ker H1(Gv, Ad0)→ H1(Iv, Ad0). • For v = p, we define Lp = ker H1(Gp, Ad0)→ H1(Ip, Ad0/Z) , where Z consists of 0 ∗ 0 0 ! .
• For v ∈ Q, Lv is spanned by the 1-cocycles class given by g(σv) = 0 0 0 0 ! , g(τv) = 0 1 0 0 !
modulo 1-coboundaries, where σv and τv generate the tame quotient of Gv and
satisfy σvτvσv−1= τvpv.
LetL be the collection of these local conditions, and define the Selmer group to be
H1L(GS∪Q, Ad0) := ker H1(G S∪Q, Ad0)→ M v∈S∪Q H1(Gv, Ad0)/Lv .
The importance of Selmer group stems from the fact that it is canonically isomorphic to the reduced tangent space of the deformation problem. The proof of the following proposition is routine (See [17]).
Proposition 3.8. Let mQ be the maximal ideal of RQQ. We have an isomorphism of
F-vector space:
H1L(GS∪Q, Ad0)≃ Hom(mQ/(p, m2Q), F).
3.3. Hecke rings and modular Galois representations. Let N ≥ 1 be a square-free integer. We let Γ = bΓ0(N ) be the subgroup of GL2(bZ) consisting of elements
a b
N c d !
with determinant 1. Let f ∈ Sk(bΓ0(N )) be a cusp eigen newform. We
fix a prime p > 5 not dividing N . Let K be the p-adic field generated by the Fourier coefficients of f , let O be the ring of integers of K, and let F be its residue field. Let T be the Z-algebra generated by Hecke operators Tℓ for all ℓ ∤ N acting on Sk(bΓ0(N )).
We consider the attached character λf : T → K and assume that f is ordinary; that
is, λf(Tp) ∈ OK×. We are interested in the Galois representation ρf : GQ → GL2(K)
attached to f . Let p = ker(λf) and let mf be the unique maximal ideal of T containing
p+ pT.
Suppose that (H1) 2≤ k < p;
(H2) mf ⊂ T is non-Eisenstein;
(H3) the residual representation ρf : GQ → GL2(F) is minimally ramified at primes
dividing N p; (H4) ρf |Ip= χk−1p ∗ 0 1 ! with∗ 6= 0.
Consider a finite set of primes Q = {q2, . . . , q2m} of odd cardinality such that ρf is
unramified on Q, qi 6≡ ±1 mod p for qi ∈ Q, and such that Tr(ρf(Frobqi)) =±(qi+ 1) for qi ∈ Q. Let D =Qq∈Qq. We write eN = pN D, and fix a prime q1 dividing pN .
For 0≤ s ≤ m, let Qs ={q1, q2, . . . , q2m−2s} and let Bs be the indefinite quaternion
algebra ramified on Qs := {q1, . . . , q2m−2s}. Let Ds = q1· · · q2m−2s and Ms = eNs/Ds
and for 0 ≤ s ≤ m. Choose an Eichler order RMs,Ds of level Ms in Bs. Denote the corresponding Shimura curve by XDs(M
Let A be a Zp-algebra, and let Λk(A) = Symk−2Z2p⊗ZpA. Notice that if k < p and A is a Zp-flat algebra, H1(XDs(Ms), Λk(A)) is a torsion-free A-module. If A = Zp, we
simply write it Λk or Λ. The ℓth Hecke correspondence Tℓ (defined in §2.2) defines an
endomorphism, still denoted by Tℓ, of H1(XDs(Ms), Λk(A)) for all ℓ ∤ eN . We define
the Hecke algebra TDs
Q to be the A-algebra generated by these endomorphisms Tℓ for all
primes ℓ ∤ eN . For A = W , we drop the A in the notation and we simply write TDs
Q .
We also have the minimal Hecke algebra T∅ generated over W by Hecke operators Tℓ
on the corresponding modular curve X(bΓ0(N p)) for all primes ℓ such that (ℓ, eN ) = 1.
For any 0≤ s ≤ m, we have obvious W -algebra homomorphisms TDs
Q → T∅→ T.
Note that they are surjective. We let mQbe the preimage of mf under the map TDQs → T,
and denote the completion of the Hecke algebra TDs
Q at mQ by bT Ds
Q . Note that for any
0≤ s ≤ m, the Hecke algebra TDs
Q is finite flat over W .
Lemma 3.9. (i) We have Galois representations ρDs
Q,mod: GQ → GL2(bTDQs) (resp. ρ∅,mod : GQ → GL2(bTmf))
which are unramified outside S∪ Q such that for ℓ 6∈ S ∪ Q,
Tr ρDs
Q,mod(Frobℓ) = Tℓ and Tr ρ∅,mod(Frobℓ) = Tℓ.
They arise by uniquely determined specializations of the universal representations
ρDs
Q : GQ → GL2(RDQs) (resp. ρ∅ : GQ → GL2(R∅)).
(ii) They satisfy ρDs
Q,mod∈ DQ,ODs (bTDQs) (resp. ρ∅,mod ∈ DO(bTmf)).
(iii) The local O-algebra homomorphisms defined by the universal property RDs
Q ⊗W O → bTDQs (resp. R∅⊗W O → bTmf)
are surjective.
Proof. By the irreducibility of the residual representations, we can apply the theorem of
Carayol [7] and Nyssen [18] to construct a representation using pseudo-representations on bTDs
Q and bT Ds
Q ⊗ Q. By definition of the representation, it satisfies the local conditions;
also by [7], the same holds for its integral structure. According to Carayol [6], the representations ρDs
Q,mod Gq and ρ∅,mod Gq are of the form ± χp ∗ 0 1 ! .
Hence, the existence of the specialization map follows from the universal property of deformation ring RDs
3.4. The main theorem. Our main theorem is:
Theorem 3.10. Let N be a square-free positive integer, let p > 5 be a prime not
dividing N . Let f ∈ Sk(bΓ0(N )) be a cusp newform. Let 2 ≤ k < p be an integer. Let
ρ = ρf : GQ → GL2(F) be the modp Galois representation attached to f . Assume it
satisfies conditions (H1) – (H4). Then the universal deformation ring R associated to
ρ is canonically isomorphic to bT∅(≃ bT).
Remark 3.11. The analogue result has been proven by Taylor-Wiles in weight 2 case
using the Taylor-Wiles systems method; it has been generalized by Ramakrishna [22] following a similar method. In weight 2, it has been reproved by Khare; for higher weights, we will prove the theorem stated above by following the ideas of Khare.
4. Review of vanishing cycles
In this section, we assume that X is a proper semistable curve over S = Spec Zq
such that the irreducible components of the special fiber are isomorphic to P1. Let
Qq an algebraic closure of Qq. We denote by Zq the normalization of Zq in Qq. Let
Iq= Gal(Qq/Qnrq ) be the inertia subgroup at q. Let S = Spec Zq, and let η (resp. s) the
generic point of S (resp. closed point of S). We consider the commutative diagram Xs i // X Xη j oo s //S oo η
Here, we still denote by X/S the base change to S of X. We denote byG the dual graph of Xs, and by Σi the i-simplices of G for i = 0, 1. We assume that a neighbourhood of
each point x∈ Σ1 in X is locally S-isomorphic to the subscheme of A2S = Zq[t1, t2] with
t1t2 = q6= 0.
In this section we follow the presentation of Illusie [12]. We also borrow from Jarvis [13] and Rajaei [21]. §3.1 – §3.3 provide a general framework which we apply to our situation in§3.4. We thus obtain the crucial proposition 3.3.
Let Y = Xs. For each x ∈ Σ1, let Y(x) denote the henselization of Y at x, and let
Cx denote the set of the two branches of Y at x. As in Illusie [12]§1.1, define Z(x) and
Z′(x) according to the following two dual exact sequences:
0−→ Z−→ Z(a) Cx −→ Z(x) −→ 0 0−→ Z′(x)−→ ZCx −→ Z −→ 0(b)
where (a) is the diagonal map, and (b) is the sum. Choosing an ordering for Cx for each
x∈ Σ1 defines a base δ′x= (1,−1) for Z′(x) and the dual base for Z(x) will be denoted
by δx. Let
4.1. Vanishing cycles. We have the following well-known results of vanishing cycles, for which we refer to SGA7 [10, 8], especially to the Expos´es I, XIII, XIV, XV by Deligne.
One has the Iq-equivariant Leray spectral sequence
Hm(X, Rnj∗Λ) =⇒ Hm+n(Xη, Λ).
By the Proper Base Change theorem, the morphism of functors i∗ induces an isomor-phism
Hm(X, Rnj∗Λ)≃ Hm(Xs, i∗Rnj∗Λ).
Let RnΨ(Λ) := i∗Rnj∗Λ be the nth-sheaf of vanishing cycles. Let RΦ(Λ) = Cone(Λ →
RΨ(Λ)) be the sheaf of nearby cycles. By [10], Expos´es XV,§3.1.2, the sheaves RnΦ(Λ) = 0 for n 6= 1, R0Ψ(Λ) = Λ, τ
≥1 R•Ψ(Λ) = R•Φ(Λ), and R1Φ(Λ) is a sheaf concentrated
on the set Σ1 of singular points of Xs.
Using these information, we obtain the following exact sequence of specialization 0−→ H1(Xs, Λ)(1) −→ Hsp 1(Xη, Λ)(1)−→β M x∈Σ1 (R1Φ(Λ))x(1) d2 −→ H2(Xs, Λ)(1) sp(1) −→ H2(Xη, Λ)(1)−→ 0. (4.1) Define X(Λ) to be: X(Λ) := ker M x∈Σ1 (R1Φ(Λ))x(1)→ ker(sp(1)) .
For x ∈ Σ1, Hix(Xs, RΨ(Λ)) = 0 for n 6= 1, 2, and for n = 2 we have the trace
isomorphism:
Tr : H2x(Xs, RΨ(Λ))−→ Λ(−1)≃
whereas for n = 1 we have
H1x(Xs, RΨ(Λ))−→ Λ(x).≃
So for any singular points x∈ Σ1, we get a vanishing cycle δx ∈ H1x(Xs, RΨ(Λ)).
Simi-larly, we have
Λ′(x)−→ R≃ 1Φ(Λ)x(1),
and δ′x∈ R1Φ(Λ
x)(1). These cycles are dual to each other with respect to the canonical
pairing on H1(Xη, Λ) to Λ(−1). That is, this pairing
R1Φ(Λ)x× H1x(Xs,(x), RΨ(Λ)) → Λ(−1)
(a, b) 7→ Tr(ab) is perfect between free rank one modules.
Recall that Y = Xs. Under the trace mapping, the free Λ-modules H1(Y, RΨ(Λ))(1)
and H2(Y, RΨ(Λ)) are respectively dual to H1(Y, RΨ(Λ)), and H0( eY , Λ) ( eY being the
normalization of Y ). The exact sequence dual to the specialization sequence would be 0−→ H0( eY , RΨ(Λ)) −→ H0( eY , Λ)−→ M x∈Σ1 H1x(Y, RΨ(Λ)) β′ −→ H1(Y, RΨ(Λ))−→ H1(Y, Λ)−→ 0, (4.2)
which is called the exact sequence of cospecialization. We define ˇ
X(Λ) = im(β′).
4.2. The variation morphism. Let tp : Iq→ Λ(1) be the p-component of the canonical
homomorphism Iq→Qq′6=qZq′(1). For x∈ Σ1 and σ∈ Iq, the variation morphism at x is defined by (see [8], Expos´e XV, §3.3.5):
Var(σ)x : R1Φ(Λ)x → H1x(Y, RΨ(Λ))
a 7→ −extp(σ)(aδx)δx
where (aδx)∈ Zp(−1) is the coordinate of a with respect to δ′x. Notice that here ex = 1
because we assume that X/S is semistable. The monodromy logarithm at x is defined to be
Nx: R1Φ(Λ)x(1)→ H1x(Y, RΨ(Λ))
Nx(tp(σ)a) = Var(σ)x(a), for a∈ R1Φ(Λ)x and σ∈ Iq.
We have the following commutative diagram: R1Φ(Λ) x(1) ≃ // Nx Λ′(x) δ′ x ❴ H1x(Y, RΨ(Λ)) ≃ //Λ(x) −δx.
This, together with the factorization of σ − 1 : H1(X
η, Λ) → H1(Xη, Λ) as sum of
Var(σ)x(a) ([8], Expos´e XIII,§2.4.6) implies that there is a unique homomorphism
N : H1(X, Λ)(1)→ H1(X, Λ),
N (tp(σ)a) = (σ− 1)(a), for a∈ H1(Xη, Λ) and σ∈ Iq
such that N makes the following diagram commutate H1(Xη, Λ)(1) = H1(Xs, RΨ(Λ))(1) β // N L x∈Σ1R 1Φ(Λ) x(1) L Nx H1(X η, Λ) = H1(Xs, RΨ(Λ)) Lx∈Σ1H 1 x(Xs, RΨ(Λ)). β′ oo
4.3. Monodromy pairing. If C denotes the set of irreducible components of Y = Xs,
the summation map ZC → Z has the image of Lx∈Σ1Z
′(x)→ ZC as its kernel, so we
can define a Z-module M via the following exact sequence
0→ M → M
x∈Σ1 Z′(x)
→ ZC → Z → 0, and dually we have
0→ Z → ZC → M
x∈Σ1
Z(x) → ˇM → 0,
where ZC → Z(x) is the composition of the projections ZC → ZCx and ZCx → Z(x). We define
by the composite homomorphism: M u∗ //Lx∈Σ 1Z ′(x) δ′ x ❴ ˇ M oo Lx∈Σ1Z(x) −δx
In fact, u is the symmetric bilinear form induced on M by the quadratic form−Pδx⊗δx
on LZ′(x). So in particular u
∗ is injective and u∗⊗ Q is an isomorphism.
We identify H2(Xs, Λ)(1) with ΛC via the trace isomorphism, and using the
isomor-phism between R1Φ(Λ)
x(1) and Λ′(x) for x∈ Σ1, we see that
M ⊗ Λ = im H1(X η, Λ)(1)→ M x∈Σ1 R1Φ(Λ)x(1) and ˇ M⊗ Λ = coker H0( eY , Λ)→ M x∈Σ1 H1x(Y, RΨ(Λ)) .
Define c and c′ by the commutative diagram:
H1(Xη, Λ)(1) c // // N β (( M ⊗ Λ // u∗⊗Λ L x∈Σ1R 1Φ(Λ) x(1) L Nx H1(Xη, Λ) ? _Mˇ ⊗ Λ c′ oo Lx∈Σ 1H 1 x(Y, RΨ(Λ)) oooo β′ hh
in which the horizontal composite maps are the maps β and β′.
4.4. Application I. Using the formalism of vanishing cycles, we have an injective map λ′q: X(Λ)→ ˇX(Λ),
and we deduce the following commutative diagram
H1(Xη, Λ)(1) c // // N β '' X(Λ) // λ′q L x∈Σ1R 1Φ(Λ) x(1) L Nx H1(X η, Λ) ? _X(Λ)ˇ c′ oo Lx∈Σ 1H 1 x(Y, RΨ(Λ)) oooo β′ gg
Taking cohomology with supports at x in the specialization exact sequence (4.1), we induce the following exact sequence
0 = H0x(Y, RΦ(Λ))→ H1x(Y, Λ) βx
−→ H1x(Y, RΨ(Λ))→ H1x(Y, RΦ(Λ)) γ
and define the map βx : H1x(Y, Λ)→ H1x(Y, RΨ(Λ)). By (4.1), we may regard H1(Y, Λ)
as a subset of H1(Xη, Λ). The morphism γ is injective because the composite
R1Φ(Λ)x −→ H≃ 1x(Y, RΦ(Λ))→ H2x(Y, Λ)
is injective (see Illusie [12], Lemme 1.5(b)). Hence, βx is an isomorphism. Moreover, we
have the following commutative diagram L x∈Σ1H 1 x(Y, Λ) L βx // L x∈Σ1H 1 x(Y, RΨ(Λ)) β′ H1(Y, Λ) sp //H 1(Y, RΨ(Λ)) ,
and this shows the image of β′ lands in this subspace H1(Y, Λ).
Lemma 4.1. We have that im(β′)⊂ H1(X s, Λ).
Let µ be the normalization map of µ : eY → Y over Zq. Define the sheaf G on Y by
the exact sequence of sheaves:
(4.3) 0→ Λ → µ∗µ∗Λ→ G → 0.
Note that G is a sheaf supported on Σ1. All its positive cohomology groups vanish.
Hence we get the following exact sequence
0→ H0(Y, Λ) → H0(Y, µ∗µ∗Λ)−→ Hθ 0(Y, G )−→ Hτ 1(Y, Λ)
→ H1(Y, µ∗µ∗Λ)→ 0 → H2(Y, Λ)→ H2(Y, µ∗µ∗Λ)→ 0.
(4.4) We define ˇ Yq(Λ) := H0(Y, G )/ im(θ), and Yq(Λ) := ker M x∈Σ1 (RΦ(Λ))x(1)→ ker(sp)(1) . By the monodromy pairing, we have
λq : Yq(Λ)→ ˇYq(Λ).
Since all irreducible components of X over k are rational curves, we have H1( eY , µ∗Λ) = 0; hence H1(Y, µ
∗µ∗Λ) = 0. Therefore, we get
Yq(Λ)≃ H1(Y, Λ).
Let eΣ1 = µ−1(Σ1). We may take the cohomology with supports in Σ1 corresponding
to (4.3) on Y : · · · → H0Σ1(Y, µ∗µ ∗Λ) → H0Σ1(Y, G )→ H 1 Σ1(Y, Λ)→ H 1 Σ1(Y, µ∗µ ∗Λ) → · · · . Lemma 4.2. We have the following canonical isomorphism induced by β′:
M
x∈Σ1
Proof. Note that the normalization map µ is finite. It follows that Hi Σ1(Y, µ∗µ ∗Λ) ≃ Hi e Σ1( eY , µ ∗Λ).
Since F is locally constant and each component of eY is smooth over s, it follows that Hie
Σ1( eY , µ
∗Λ) = 0 for i = 0, 1. Therefore, in the above exact sequence we have
H0
Σ1(Y, G )
≃
−→ H1
Σ1(Y, Λ). Since each of the maps βx are isomorphisms, we obtain H1Σ1(Y, Λ)≃ M
x∈Σ1
H1x(Y, RΨ(Λ)). As G is supported on Σ1, we see that H0Σ1(Y, G )≃ H
0(Y, G ). Hence, we get
M
x∈Σ1
H1x(Y, RΨ(Λ))≃ H1Σ1(Y, Λ)≃ H0Σ1(Y, G )≃ H0(Y, G ).
Recall that the map θ is given in the exact sequence (4.4).
Proposition 4.3. ˇX(Λ) ≃ H0(Y, G )/θ(H0(Y, µ
∗µ∗Λ)).
Proof. By Lemma 4.2, we have identifiedLx∈Σ
1H
1
x(Y, RΨ(Λ)) with H0(Y, G ). The rest
is to show that the image of H0(Y, G ) in H1(Y, Λ) is the same as the image of β′. Also note that we have proved that the image of β′ lies in H1(Y, Λ).
By Illusie [12]§1.5, we have the following “diagramme des 9” over Y(x) for the inclu-sions ix:{x} ֒→ Y(x) and jx : Ux = Y(x)\ {x} ֒→ Y(x) ix,∗Ri!xΛ // Λ // Rjx,∗j∗xΛ µ∗µ∗ix,∗Ri!xΛ // µ∗µ∗Λ // Rjx,∗j∗xΛ G G //0
So by [12], Lemme 5.4, the composite
γx: H0(Y(x), µ∗µ∗Λ)→ H0(Y(x), Rj∗j∗Λ)→ H1(Y(x), ix,∗Ri!xΛ)≃ H1x(Y, Λ)
is negative of the composite map
γx′ : H0(Y(x), µ∗µ∗Λ)→ H0(Y(x), G )→ H1(Y(x), ix,∗Ri!xΛ)≃ H1x(Y, Λ).
In particular, the image of ∂x: H0(Ux, Λ)≃ H0(Y(x), Rjx,∗jx∗Λ)→ H1x(Y, Λ) is the same as
that of τx: H0(Y(x), G )→ H1(Y(x), ix,∗Ri!Λ)≃ H1x(Y, Λ). But we have the commutative
diagram H0(U x, RΨ(Λ)) ∂′ x //H1 x(Y, RΨ(Λ)) β−1 x H0(Ux, Λ) ≃ OO ∂x //H1x(Y, Λ) which implies that the image of β−1
x is also the same as that of ∂x. Thus, βx−1 and τx
Consider the following commutative diagram H0Σ1(Y, G ) // τ L x∈Σ1H 0 x(Y, G ) ≃ (1) // L τx L x∈Σ1H 1 x(Y, RΨ(Λ)) // L β−1 x H1Σ1(Y, RΨ(Λ)) β′ H1(Y, Λ)oo σ Lx∈Σ1H1x(Y, Λ) Lx∈Σ 1H 1 x(Y, Λ) σ //H1(Y, Λ),
where (1) is given by Lemma 4.2. Thus we have im(β′) = im(τ ) and this is the same as
coker(θ).
Remark 4.4. Note that X(Λ) ⊂ Lx∈Σ1(R1Φ(Λ)x)(1), so it is torsion-free. Since Λ is
irreducible for k− 2 < p, we can also see that ˇX(Λ) has no p-torsion.
4.5. Application II: Descent. For our another application to descent from an auxil-iary level group bΓD
1(r) to a level prime to r, we shall need Langlands-Deligne-Carayol
theorem on the compatibility between local and global Langlands correspondance for B×. For this purpose, we will consider a regular scheme X = XD(Γ1) (cf. §2, Prop.
2.2) flat of finite type over Zr with smooth generic fiber Xη and special fiber Xs in this
section.
4.5.1. Calculation of vanishing cycles and monodromy. Let Λ be a Z/pm-module or a lisse sheaf of Qp-vector space over X. The special fiber is assumed to be ´etale locally
one of the following types:
(1) Xs is smooth. This corresponds to a neighbourhood of a point in Ym not in Ye.
Then the regularity of X implies X is smooth. In this case, RqΨΛ = 0 for q > 0, and RqΨΛ = Λ.
(2) X is of the form Spec Zr[X, Y±1]/(Xr−1Y − r). This corresponds to a
neigh-bourhood of a point in Ye not in Ym. Let X′= Spec Zr[x, y±1]/(xr−1− r), and
define a map π : X′ → X via the embedding
Zr[X, Y±1]/(Xr−1Y − r) → Zr[x, y±1]/(xr−1− r)
(X, Y ) 7→ (xy−1, yr−1) The morphism π is ´etale with Galois group (Z/rZ)×≃ µ
r−1 where ζ ∈ µr−1 acts
by multiplying both x and y by ζ. The special fiber Xs is a non-reduced divisor
with multiplicity r− 1; the associated reduced divisor Xs,redis defined by X = 0
and is smooth; it is isomorphic to Spec Fr[Y±1] which we view as (Gm)Fr. We first compute the vanishing cycle RqΨΛ in the ´etale neighbourhood of X′ of X. Let O = Zr[r−1√r] = Zr[x]/(xr−1− r). Then we write
(4.5) X′ = SpecO ×Spec ZrY
where Y = Spec Zr[y±1]. The second factor is smooth over Zr, hence (RqΨΛ)X′ is the pullback from RqΨ for the finite flat morphism SpecO → Spec Zr. The
morphism is of relative dimension zero, hence RqΨ = 0 for q > 0. Similarly, (R0ΨΛ)
Spec O is the pull-back of SpecO → Spec Zr and (R0Ψ)Spec Zr = Λ; hence (R0ΨΛ)Spec O ≃ Λr−1 as Z/pm-modules or Qp-vector space. Since the inertia
group of O over Zr acting on (R0ΨΛ)Spec O by µr−1 transitively, we thus have
(R0ΨΛ)Spec O is the group algebra Λ[µr−1].
It follows that RqΨΛ = 0 for q > 0, and that R0ΨΛ is a lisse p-adic sheaf of rank r − 1 on Xs,red that becomes constant over Xs,red′ . Moreover, since
Gal(X′/X) acts as inertia group on the first factor of (4.5), one sees that the canonical action of Gal(X′/X) on (R0ΨΛ)X′
s,red identifies the latter with the
group algebra Λ[Gal(X′/X)]. It follows that R0ΨΛ−→ π≃ ∗Λ.
The inertia group µr−1 acts on R0Ψ, and we have seen that the lift of this
action to Xs,red′ coincides with the action of Gal(X′/X). We write R0ΨΛ =M
χ
R0ΨΛ[χ],
the decomposition with respect to characters of the inertia group. We write L = R0Ψ, and L[χ] for the rank one local system R0ΨΛ[χ]. Let χ0 denote the
trivial character. Consider the embedding i : Xs,red= (Gm)Fr ֒→ A
1 = Spec F r[Y ]
as the complement of the origin Y = 0. The morphism π is totally ramified along Y = 0. It follows that
(4.6) R0i∗L[χ] = i!L[χ]; Rqi∗L[χ] = 0, q > 0 (χ6= χ0);
R0i∗L[χ0] = Λ; Rqi∗L[χ0] = 0, q > 0.
More generally, suppose X = Spec R[X, Y±1]/(Xr−1Y − r), where R is a smooth Zr-algebra of finite type. Then X is the fiber product
X = Spec R×Spec Zr Spec Zr[X, Y
±1]/(Xr−1Y − r),
where the first factor is smooth. We define
iX = 1× i : Xs,red= Spec R×Spec Zr Spec(Gm)Fr → Spec R ×Spec ZrSpec Fr[Y ]. Let X2denote the second factor above and let pr2denote the projection X → X2.
We see that
RqΨΛ = pr∗2RqΨX2Λ,
where RΨX2 denotes the vanishing cycle sheaves for the map from X2to Spec Zr. In particular, RqΨΛ = 0 for q > 0, while RqΨΛ breaks up under the action of the inertia subgroup of Gal(Qr/Qr) as the sum of rank one local system L[χ]:
L[χ0] = Λ, whereas L[χ] for nontrivial χ satisfies the analogue of (4.6):
R0iX,∗L[χ] = iX,!L[χ]; RqiX,∗L[χ] = 0, q > 0 (χ6= χ0).
(3) X is of the form Spec R[X, Y ]/(Xr−1Y − r), where R is a smooth Zr-algebra of
finite type. This corresponds to a neighbourhood of a point in Ye∩ Ym. We will calculate the stalks of RqΨΛ at a geometric point x of the singular locus X
sing of
q > 1; (R0ΨΛ)
x = Λ, (R1ΨΛ)x = Λ(−1), with trivial action of the inertia group
on Λ, (−1) denoting Tate twist. We write Ya = Ym∩ Ye. Let i
m : Ym → XD(Γ1), ie : Ye → XD(Γ1), and ia :
Ya→ XD(Γ
1) be the natural maps. Let Y0e denote the complement of Yain Ye, and let
je: (Y0e)red → (Ye)red be the open immersion. Then the vanishing cycle sheaves RqΨΛ
are calculated as follows:
Proposition 4.5. Let I denote the inertia subgroup of Gal(Qr/Qr). Then the action
of I on RΨqΛ factors through the map to (Z/rZ)× (which we identify to µr−1(Zr) by
Teichm¨uller lifting) given by the action on Q[ζr]. For a character χ of µr−1, let [χ]
denote the χ-isotypic component, and let χ0 denote the trivial character. Then
(i) R0ΨΛ[χ 0] = Λ.
(ii) R1ΨΛ = Λ[χ0] is a rank one local system supported on Ya, locally isomorphic at
any point of Ya to Λ(−1).
(iii) For χ6= χ0, R0ΨΛ is the extension by zero of a rank one lisse sheaf L[χ] supported
on Y0m. Moreover, the natural map im,!R0ΨΛ[χ] → Rim,∗Ψ0Λ[χ] is a
quasi-isomorphism.
(iv) RqΨΛ = 0 for q > 1.
Proof. Everything follows from the cases (1)–(3) discussed above except the global
triviality of R0ΨΛ[χ0]. But there is always an injection Λ → R0ΨΛ[χ0], so (i) follows
from the fact that all stalks of R0ΨΛ[χ0] are one-dimensional.
Since the tame vanishing cycle sheaves are concentrated in two degrees, the vanishing cycle spectral sequence degenerates into a long exact sequence
· · · → Hi(XD(Γ1)s, R0ΨΛ)→ Hi(XD(Γ1)η, Λ)
→ Hi−1(XD(Γ1)s, R1ΨΛ)→ Hi+1(XD(Γ1)s, R0ΨΛ)→ · · · .
Using Proposition 4.5 (ii), we rewrite this
(4.7) · · · → Hi(XD(Γ1)s, R0ΨΛ)→ Hi(XD(Γ1)η, Λ)→ Hi−1(Yreda , R1ΨΛ[χ0])→ · · · .
We deduce from (i) and (iii) of Proposition 4.5 that the first term in turn is calculated by a long exact sequence
· · · → Hi−1(Yreda , Λ)→ Hi(XD(Γ1)η, R0ΨΛ) → Hi(Ym, Λ)⊕ Hi((Ye)red, Λ)⊕ M χ6=χ0 Hic(Y0e, L[χ])→ · · · . (4.8)
Here and in (4.7) , we have replaced Yeand Yaby the associated reduced schemes, since the ´etale cohomology is insensitive to nilpotents.
The diamond operators act XD(Γ
1)η as well as on Ye and Ym, and thus induce
compatible actions on the spaces in the exact sequence (4.7) and (4.8). These are determined as follows:
Lemma 4.6. The diamond operators hai act on the outer terms of the exact sequence (4.8) as follows: The action acts via χ on L[χ], and acts trivially on H•((Ye)
red, Λ), on
Proof. The diamond operators acts trivially on (Ye)
red and (Ya)red, so it suffices to
determine their action on R0ΨΛ and R1ΨΛ.
For R0ΨΛ, by the discussion in (2), we see it suffices to determine the action of the diamond operators on H0(Spec(Q
r⊗Qr Qr[ζr]), Λ), via the identification of Qr[ζr] with the generic fiber of µr−1 and the latter with CRin (1)–(2). But the diamond operators
on µr−1 are tautologically given by the cyclotomic character.
For the action on R1ΨΛ, this is again local. But locally the discussion in (3) shows
that R1ΨΛ is a constant sheaf, so the triviality of the action of the diamond operators
is clear.
Proposition 4.7. Suppose χ 6= χ0, and denote by h i=χ the χ-isotypic component for
the action of the diamond operators. Then for any i, there is a canonical isomorphism of Gal(η/η)-modules
Hi(Ym, Λ)h i=χ⊕ Hic((Y0e)red, L[χ]) ≃
−→ Hi(XD(Γ1)η, Λ)h i=χ.
Proof. Indeed, in (4.8), the diamond operators act trivially on the term Hi((Ya)red, Λ)
and coincide with inertia on L[χ], inducing an isomorphism
Hi(XD(Γ1)s, R0ΨΛ)h i=χ−→ H≃ i(Ym, Λ)h i=χ⊕ Hic(Y0e, L[χ]).
Similarly, the diamond operators act trivially on the Hi−1((Ya)
red, R1ΨΛ[χ0]) term in
(4.8).
4.5.2. Descent. Let us consider the following condition on a prime r:
UR(r): r does not divide eN , r6≡ 1 mod p, and the ratio of the eigenvalues of ρ(Frobr) is not congruent to 1 or r±1mod p.
The existence of a prime r satisfying UR(r) follows from ˇCebotarev density theorem. We fix such a prime r in the sequel. For any 0 ≤ s ≤ m, let Γ1(Ms; r) = bR×Ms,Ds ∩
b ΓDs 1 (r) ⊂ bOBs and Γ0(Ms; r) = bR × Ms,Ds ∩ bΓ Ds
0 (r) ⊂ bOBs. Denote the corresponding Shimura curves by XDs(M
s; r). Let A be a Zp-algebra. Applying the Jacquet-Langlands
correspondence, we denote by TrQs
Q the Hecke algebra generated by Hecke
correspon-dences Tℓ on H1(XDs(Ms; r), Λk(A)) for all ℓ ∤ eN for primes ℓ ∤ eN . Note that if s = m,
we also have the minimal Hecke algebra T∅ generated by Hecke operators Tℓ on the
corresponding moduli curve X(bΓ0(N p)∩ bΓ1(r)) for primes ℓ such that (ℓ, eN ) = 1.
For each 0 ≤ s ≤ m, we set Γ1 = bR×Ms,Ds ∩ bΓD1s(r) and Γ0 = bRM×s,Ds ∩ bΓD0s(r),
and let XDs(Γ
1) and XDs(Γ0) be the corresponding Shimura curves. Let m(r) = m +
(Tr− αr− βr, rSr− αrβr). If π′ occurs in H1(XDs(Γ1), Λ)Tmr(r)≡αr is special, it occurs in
H1(XDs(Γ
0), Λ)Tmr(r)≡αr. By the weight monodromy conjecture for curves, the eigenvalues
of Frobr on ρπ′ ⊂ H1(XDs(Γ0), Λ)Tmr≡αr
(r) are of the form α
′
r, rα′r. (See also Carayol [6].)
However, we have
ρπ(Frobr)∼
αr 0
0 βr
!
which implies that αr/βr ≡ r±1 modulo p, and deduces a contradiction. Hence, πr′
(Z/rZ)× such that the diamond operator act on π′r by χ. In particular, π′ occurs in H1(XDs(Γ
1)η, Λ)h i=χm(r) . Then by Proposition 4.7, we see that
ρπ′ |I r∼ 1 0 0 χ ! .
Since r6≡ 1 mod p, this implies that ρπ′ = ρ is also ramified at r which is a contradiction. Therefore, ρπ′ is unramified.
Remark 4.8. Since the associated Galois representations of the Hecke algebras bTrDs
Q are
unramified at r, we still have the surjective specialization maps of local W -algebras RDs
Q → bTrDQ s for 0≤ s ≤ m. Hence we will ignore the auxiliary prime r in the sequel to
reduce our notations.
5. Isomorphisms between deformation rings and Hecke rings
By the Proposition 21 of [15], there exists a finite set of primes Q ={q2, . . . , q2m} of
odd cardinality such that for each q ∈ Q, q 6≡ ±1 mod p, Tr ρ(Frobq) = ±(q + 1), and
such that
H1L(GS∪Q, Ad0) = 0.
Then by Proposition 3.8, mQ = pRDQ; therefore, RDQ/pRDQ = F = W/pW and by
Nakayama’s lemma the structure morphism W → RDQ is surjective. Since k≤ p − 1, by Diamond-Taylor [9], there is a p-adic modular lifting of ρ which arises from a specializa-tion of the universal representaspecializa-tion ρQQ: GQ → GL2(RDQ) which gives a surjection from
RQD to W . Hence the structure morphism is also injective.
By Lemma 3.9, we have a surjective W -morphism RDQ ։ bTDQ, and since the algebra
b TD
Q is finite flat over W , thus ψ : RDQ → bTDQ is injective. Hence we deduce the following:
Proposition 5.1. Let 2≤ k < p. Then for D =Qq∈Qq we have isomorphisms of local W -algebras
W ≃ RQD ∼
−→ bTD Q.
We call such set Q a RK system or a RK set.
A proof of the following theorem which deduce it from Proposition 5.1, will be given in the Section 8.2.
Theorem 5.2. If Q is a RK system, we have an isomorphism RQ ≃ bTQ of complete
intersection rings.
An argument of B¨ockle implies that: Theorem 5.3.
R∅ ≃ bT∅
6. A numerical inequality
As in Wiles’ method, the proof of modularity is based on a numerical inequality relating the length of a congruence module to the cardinality of a Selmer group. We recall a numerical criterion due to Lenstra which refines a result of Wiles.
Proposition 6.1 (Numerical criterion). Let R, T ∈ CNLW. Suppose that T is finite flat as W -module and φ : R → T is a surjective local W -algebra homomorphism. Let π : T → W be a homomorphism of local W -algebras, and set Φ(R) = ker(πφ)/ ker(πφ)2
and ηT = π(AnnT(ker(π))). Then we have the following:
(i)
|W/ηT| ≤ |Φ(R)| .
(ii) Assume that ηT is not zero. Then the following are equivalent:
• The equality |W/ηT| = |Φ(R)| is satisfied.
• The rings R and T are complete intersections and φ is an isomorphism. We let φ : Rα
Q → bTαQ and π : bTQα → bTDQ. For any prime q ∈ Q, let tq generate the
unique Zp quotient of Iq. Then ρDQs(tq) is of the form
1 xq
0 1 !
for some xq∈ W \ {0}, and (xq) does not depend on the choice of an integral model for
ρDQ. Note that xq 6= 0. Indeed, if xq = 0 then ρDQ is unramified at q, and as RDQ ≃ bTDQ
we have Tr ρD
Q(Frobq) =±(q + 1) which contradicts the Ramanujan bound.
Lemma 6.2. Let q∈ Q and let n be the largest number such that ρD
Q |Gq is unramified.
Then we have:
(i) Let Z ⊂ Ad0(ρD
Q)be the upper triangular one dimensional subspace of nilpotent
elements which are preserved under the conjugation by ρDQ |Gq. The sequence 0→ Ad0(ρDQ)⊗ p−nZ/Z → Ad0(ρD
Q)⊗ Qp/Zp
Iq
→ Z ⊗ Qp/Zp → 0
is exact.
(ii) The inflation map
H1Gq/Iq, Ad0(ρDQ)⊗ Qp/ZpIq
∼
→ H1(Gq, Ad0(ρDQ)⊗ Qp/Zp)
is an isomorphism.
(iii) H1(Gq, Ad0(ρQD)⊗ Qp/Zp) is isomorphic to W/pn.
(iv) For all q∈ Q, Lq is trivial.
Proof. (i) The ramification first occurs modpn+1. The exactness of the sequence
follows.
(ii) Using the inflation-restriction sequence 0 → H1Gq/Iq, Ad0(ρDQ)⊗ Qp/ZpIq
→ H1(Gq, Ad0(ρDQ)⊗ Qp/Zp)
it is enough to show that H1(I
q, Ad0(ρDQ)⊗Qp/Zp) is trivial. Let M = Ad0(ρDQ)⊗
Qp/Zp. The exact sequence
0→ Pq → Iq→ Iq/Pq≃ Y ℓ6=q Zℓ(1)→ 0 induces 0→ H1(Iq/Pq, MPq)→ H1(Iq, M )→ H1(Pq, M ).
Note that Pq acts trivially on M because ρDQ is tame at q, hence H1(Pq, M ) =
Hom(Pq, M ) = 0.
(iii) By (i), we have the exact sequence of Gq/Iq-modules
0→ Ad0(ρDQ)⊗ p−nZ/Z → Ad0(ρD
Q)⊗ Qp/Zp
Iq
→ Z ⊗ Qp/Zp → 0.
Since q 6≡ ±1 mod p, these cohomology groups Hi(G
q/Iq, Z⊗ Qp/Zp) are trivial for i = 0, 1. Therefore, H1(Gq/Iq, Ad0(ρDQ)⊗ p−nZ/Z) ≃ H1 Gq/Iq, Ad0(ρDQ)⊗ Qp/Zp Iq . We have the following decomposition as Gq/Iq-modules
Ad0(ρDQ)⊗ p−nZ/Z ≃ W (−1)/pm⊕ W/pn⊕ W (−1)/pn
Denote W/pn by M1, W (−1)/pn⊕ W (−1)/pn by M2 and let M = M1⊕ M2.
Since q 6≡ ±1 mod p, we see that H0(G
q/Iq, M2) = 0 and that
H1(Gq/Iq, M1) ֒→ H1(Gq/Iq, M ).
In fact, H1(Gq/Iq, M1) and H1(Gq/Iq, M ) are isomorphic because their
dimen-sions are equal. The group Gq/Iq acts trivially on M1, so
H1(Gq/Iq, M1) = Hom(Gq/Iq, M1) = W/pn; therefore by (ii), H1(Gq/Iq, Ad0(ρQD)⊗ Qp/Zp) → H∼ 1 Gq/Iq, Ad0(ρDQ)⊗ Qp/Zp Iq = W/pn.
(iv) We have a short exact sequence:
0→ Z ⊗ Qp/Zp → Ad0(ρDQ)⊗ Qp/Zp → Ad0(ρDQ)/Z
⊗ Qp/Zp → 0.
Taking the Gq-cohomology, we get a long exact sequence
0 → W/pm→ W ⊗ Qp/Zp→ H1(Gq, Z⊗ Qp/Zp)
→ H1(Gq, Ad0(ρDQ)⊗ Qp/Zp)→ H1(Gq, Ad0(ρDQ)/Z
⊗ Qp/Zp)
We claim that W ⊗ Qp/Zp → H1(Gq, Z ⊗ Qp/Zp) is a surjection; indeed, by
the inflation-restriction we know H1(Gq, Z⊗ Qp/Zp) is isomorphic to H1(Iq, Z⊗
Qp/Zp)Gq/Iq and the later is just W ⊗ Qp/Zp. By this claim, we have
0→ H1(Gq, Ad0(ρDQ)⊗ Qp/Zp)→ H1(Gq, Ad0(ρDQ)/Z
⊗ Qp/Zp)
which implies Lq = 0.
By identifying Φ(Rα
Q) to the Selmer group, we thus obtain an upper bound for Φ(RαQ):
Proposition 6.3. For any subset α⊂ Q, we have Φ(Rα Q) ≤ Y q∈Q\α |W/(xq)| .
Proof. Let us abbreviate as above M = Ad0(ρDQ)⊗ Qp/Zp. We may identify Φ(RQ) to
the Selmer group. That is
HomW(Φ(RQ), Qp/Zp)≃ H1L(GS∪Q, M ).
Since RDQ is isomorphic to W , the kernel of this map H1(GS∪Q, M )→ M q∈S∪Q H1(Gq, M ) Lq is trivial. Hence, HomW(Φ(RQ), Qp/Zp) ֒→ M q∈Q H1(Gq, M ) Lq
By the previous lemma, we see
HomW(Φ(RQ), Qp/Zp) ֒→ M q∈Q W/pmq, and Φ(RD Q) ≤ Y q∈Q |W/(xq)| .
Moreover, since we have a surjective morphism Rα
D ։ RQD for any subset α ⊂ Q,
Φ(RαQ) is isomorphic to a subgroup of Qq∈Q\αW/(xq); hence the result is derived.
7. Ribet’s short exact sequence
By the cohomological formalism of vanishing cycles (as opposed to N´eron model the-orem), we can establish a higher weight version of character groups and Ribet’s short exact sequence for the curves.
7.1. Residual characteristic divides the level. We suppose that the prime q does not divide the discriminant D of the indefinite quaternion algebra B over Q. Let Γ = ΓqΓq = bΓ0(qM )∩ bΓD1(r) be a sufficiently small, open compact subgroup of bB×. We have
defined the Shimura curves XD(qM ; r) associated to Γ. Let V0(q) =
( a b c d ! ∈ GL2(Zq) c≡ 0 mod q ) . Let µ be the normalization map for the special fiber Y of X = XD(qM ; r) over
Zq. Recall that we have a natural morphism θ : H0(Y, µ∗µ∗Λ)→ H0(Y, G ). Let TqM ;r
denote the Hecke algebra generated over Zq by the endomorphisms Tℓ (ℓ ∤ M qr) of
H1(XD(qM : r), Λ). We will write Xq(qM ; r) resp. ˇXq(qM ; r) instead of X(F ) resp.
ˇ
X(F ) in order to emphasis the level structure.
Proposition 7.1. If m is a non-Eisenstein maximal ideal of TqM ;r, then we have:
(i) im(θ)m= 0, and we have a canonical isomorphism ˇXq(qM ; r)m≃
L
x∈Σ1Gx
(ii) In the exact sequence (4.1), after localization at m the map Lx∈Σ1 R1Φ(Λ)
x →
ker(sp) is the zero map, i.e. Xq(qM ; r)m≃
L x∈Σ1 R 1Φ(Λ) x m.
(iii) From (i) and (ii), we deduce ˇ Xq(qM ; r)m≃ M x∈Σ1 Gx m (1) ≃ M x∈Σ1 R1Φ(Λ)x m ≃ Xq(qM ; r)m,
where (1) is induced from the monodromy logarithm Nx at each x.
Proof. (i) We first study im(θ). Note that the normalization map µ : XD(qM ; r)⊗ Fq⊔ XD(qM ; r)⊗ Fq → XD(qM ; r)⊗ Fq is a finite morphism. We thus have an
isomorphism
H0(XD(qM ; r)⊗ Fq, µ∗µ∗F )≃ H0(XD(M ; r)⊗ Fq, F )2.
In Carayol [5] §2, we have that
π0(XD(M ; r)⊗ Fq) ≃ Q×+\(A∞)×/(Nm(Γ(M )∩ Γ1(r))× Z×q)
= (Z(q))×+\(Aq∞)×/ Nm(Γ(M )∩ Γ1(r)),
where we write Z(q) for Q∩ Zq. Then G(Aq∞) acts on the set of components
through via reduced norm (see [5] §1.3). But the maximal ideals in Tqq′M lying in the support of ˇX must correspond to the one-dimensional automor-phic representations, as cuspidal representations on quaternion groups admit infinite-dimensional components at almost every place, and thus do not factor through the norm (see [6] §4.4). However, the automorphic representations in im(θ) factor through the norm. By Lemma 4.2 and Proposition 4.3, we deduce
ˇ Xq(qM ; r)m≃ L x∈Σ1Gx m.
(ii) Since G concentrates at points, its cohomology groups vanish in degree greater that one. Hence, we also have an isomorphism induced from the normalization map µ:
H2(XD(qM ; r)⊗ Fq, F )≃ H2(XD(qM ; r)⊗ Fq, µ∗µ∗F ).
On the other hand, we may regard the latter group as H2(XD(M ; r)⊗ Fq, F )2,
and this is Poincar´e dual to the group H0(XD(M ; r)⊗ F
q, ˇF (1))2. Using similar
analysis in (i), the second point follows as before.
(iii) Notice that the homomorphisms Nx are isomorphisms for any regular model for
X over Zq. The third point (iii) follows from this together with the first and the
second assertions.
7.2. ˇCerednik-Drinfel’d uniformization theorem. Throughout this section, we fix a prime q. Let DM be a square-free integer. Suppose D is a product of an even number of primes, and that q | D. Let B be an indefinite quaternion algebra over Q with discriminant D.
Let B′ be the definite quaternion algebra ramified precisely at the primes dividing D′ = D/q, and the archimedean place. Let G′ be the group scheme over Z associated to
The q-adic upper half-plane is by definition
Ω = P1(Cq)\ P1(Qq).
It carries, in a natural way, the structure of a rigid Qq-space Ωrig. The rigid Qq-space
Ωrigdetermines a formal Zq-scheme up to admissible blowing ups in the special fiber [2].
There is a canonical choice of a formal Zq-scheme bΩ whose associated rigid Qq-space is
Ωrig.
From the definition of B′, we have an anti-isomorphism
(7.1) Bopp⊗QAq∞≃ B′⊗QAq∞.
We thus obtain a group isomorphism
(7.2) B×(Aq∞)≃ B′×(Aq∞)
after composition of (7.1) by the inversion g 7→ g−1. We write bΓ
0(M ) = Γ0q· ΓqM(= Γ
for short in this section), where Γ0q denotes the group of units in the maximal idealOBq and ΓqM is an open compact subgroup of B×(Aq). We may consider Γq
M as a subgroup
of B′×(Aq) via (7.2). Define
ZΓ = ΓqM\cB′ ×
/B′×.
The group GL2(Qq) acts naturally on the left on the Bruhat-Tits tree T , the rigid
analytic space Ωrig, the formal scheme bΩ, and on Z
Γ. Let Qnrq be the maximal unramified
extension of Qq and bZnrq the completion of the ring of integers in Qnrq . An element
g∈ GL2(Qq) acts on Qnrq and bZnrq via [Frob
−vq(det g)
q where [Frobq denotes the arithmetic
Frobenius automorphism.
Let XD(Γ) be the Shimura curve over Z
q corresponding to Γ. We denote by \XD(Γ)
the completion of XD(Γ) along its special fiber and by XD(Γ)rig the rigid analytic space over Qq associates with \XD(Γ).
Here comes the q-adic uniformization theorem (cf. [2]§5.2):
Theorem 7.2 ( ˇCerednik-Drinfel′d). There is a canonical isomorphism of formal schemes
over Zq
GL2(Qq)\[(bΩ b⊗ZqZb
nr
q )× ZΓ]−→ \≃ XD(Γ)
and a canonical isomorphism of rigid analytic spaces over Qq
GL2(Qq)\[(Ω ⊗QqQ
nr
q )× ZΓ]−→ X≃ D(Γ)rig.
7.3. Ribet’s short exact sequence. We let X = XD(M ; r) and assume that qq′| D. Let Γ⊂ bB× be the group of level M defining X. We write Γ = ΓqΓq, where Γq =OB×q and Γq is an open compact subgroup of B×(Aq). We also insist that Γq′ =O×
Bq′. Let GL2(Qq)+(resp. GL2(Qq)+) be the subset of elements in GL2(Qq) whose reduced norm
has even (resp. odd) valuation. In fact, GL2(Qq)+is a subgroup. Let B′ be the definite
quaternion of discriminant D/q obtaining from B by exchanging the local invariants at q and∞. The ˇCerednik-Drinfel’d uniformization theorem gives a description of the dual graphG of the special fiber Y of X at q. That is, it can be described as GL2(Qq)+\(T ×
We describe a bijection the set of edges of T
Ed(T ) ≃ GL2(Qq)+/V0(q)Q×q;
the set of vertices ofT
Ver(T ) ≃ PGL2(Qq)/ PGL2(Zq).
So the set of edges ofG is:
Ed(G) = GL2(Qq)+\Ed(T ) × ZΓ = GL2(Qq)+\ GL2(Qq)+/V0(q)Q×q × ZΓ = V0(q)Γq\ bB′×/B′×. Let us introduce V+:= GL2(Qq)+\ (PGL2(Qq)+/ PGL2(Zq)× ZΓ) and V− := GL2(Qq)+\ (PGL2(Qq)−/ PGL2(Zq)× ZΓ) .
Then the set of vertices ofG is:
Ver(G) = GL2(Qq)+\Ver(T ) × ZΓ
= GL2(Qq)+\ (PGL2(Qq)+⊔ PGL2(Qq)−) / PGL2(Zq)× ZΓ
= V+⊔ V−.
Define V := GL2(Oq)Γq\ bB′×/B′× = GL2(Oq)\ZΓ. Note that we have two
degener-acy maps α, β from Ed(G) to V corresponding to the inclusion of V0(q)Γqinto GL2(Oq)Γq
and the conjugation by Wq=
1 0 0 q
!
. We have bijections between V and V+, and V
and V−. Each edge e connects α(e) in F+ to β(e) in V−. In fact, we have
1 : V → V+, [x]7→ (1, x)
Wq: V → V−, [x]7→ (Wq, Wqx).
Using Honda-Tate theory, Carayol has described the set of singular points of XD/qq′
(qq′M ; r)
modq′ (i.e., the set Σqq′M ;r of supersingular points of XD/qq ′
(qq′M ; r)).
To simplify the notations and state Carayol’s result in the sequel, we make the fol-lowing assumptions: Let B1 be an indefinite quaternion algebra over Q with
discrimi-nant D1, p ∤ D1, M1 ≥ 4, and let XD1(M1; r) = G(Q)\G(A∞)× H∞/Z(A∞)Ξ where
Ξ = bΓD1
0 (M1)∩ bΓD11(r)⊂ G(A∞). We denote by K the restricted product of (B1⊗ Qv)×
for v6= p.
Proposition 7.3 (Carayol [4, 5]). Let ΣM1;rbe the set of supersingular points of XD1(M1; r) mod p. Then the group K× Q×
p acts transitively on ΣM1;r. For each x ∈ ΣM1;r, the
stabi-lizer of x is conjugate in K × Q×p to Z(Q)G′(Q) where G′ = B2× obtaining from B1 by
changing the local invariants at p and ∞ and Z(Q) is the closure of Z(Q) in Z(A∞).
Remark 7.4. For any M1 ≥ 4 as above, the set ΣM1;r is in bijection with double coset: ΣM1;r ≃ G
′(Q)\K × Q×
Let us apply these information to our cases with M1 = qq′M, D1= D/qq′, p = q′, B =
B1 and B′ = B2. Let Ξ = bΓ0(qq′M )∩ bΓ1(r); consider the modulo q′ reduction of
XD/qq′(qq′M ; r), notice that b Γ0(qq′M )∩ bΓ1(r) q′ × OB×′ q′ = V0(q) b Γ0(qq′M )∩ bΓ1(r) qq′ × O×B′ q′ ≃ V0(q) b Γ0(M )∩ bΓ1(r) qq′ × O×B′ q′ = V0(q)Γq,
and this gives a bijective correspondence between Ed(G) and the set of singular points of XD/qq′(qq′M ; r) mod q′.
(7.3) Ed(G) : the edges of
the dual graph of XD(M ; r) mod q ⇐⇒
Σqq′M ;r : singular points of XD/qq′
(qq′M ; r) mod q′
Similarly, for vertices of the dual graph of the Shimura curve XD(M ; r) we use
Carayol’s formula for L = bΓ0(q′M )∩ bΓ1(r). We find that
b Γ0(q′M )∩ bΓ1(r) q′ × OB×′ q′ ≃ GL2 (Oq)Γq.
The correspondence provides a bijection for V? (? =∅, +, −):
V? (? =∅, +, −) ⇐⇒
Σq′M ;r: singular points of XD/qq′(q′M ; r) mod q′ Therefore, the map 1∗ (resp. Wq,∗) will correspond to α (resp. β).
Note that the number of irreducible components of the normalization is equal to the number of the vertices of the dual graph of the special fiber. Hence, for the lisse sheaf Λ we let ζ be the composition of two Hecke-equivariant maps (1) and (2)
ζ : H0(Xs, µ∗µ∗Λ) (1) ≃ 1∗ M y∈Σq′ M;r Gy ⊕ Wq,∗ M y∈Σq′M;r Gy (2) ։ 1∗Xˇq′(q′M ; r)⊕ Wq,∗Xˇq′(q′M ; r),
where (1) follows from Proposition 7.1 and (2) follows Proposition 4.3. We let J = ker(ζ). Following from Proposition 7.1 H0(Xs, µ∗µ∗Λ)/J ≃ im(ζ) is a Tqq′M ;r-module. The Hecke-equivariant injection H0(Y, Λ)→ H0(Y, µ∗µ∗Λ) induces an injection of H0(Y, Λ)/(J∩
H0(Y, Λ) into H0(X
s, µ∗µ∗Λ)/J. Hence, this map induces a Tqq′M ;r-module structure on H0(Y, Λ)/(J ∩ H0(Y, Λ)).
Lemma 7.5. We have H0(Y, Λ)/J∩ H0(Y, Λ) m= 0.
Proof. Since the restriction of Λ to each irreducible component of Y is constant, H0(Y, Λ) is isomorphic to a direct sum of Λy’s, each corresponding to a connected component of
Y . Hence, as the connected components of X⊗ Q are defined over Qab, the
Gal(Q/Q)-action on H0(X ⊗ Q, Λ) factors through Gal(Qab/Q). This gives rise to a reducible
