§7.8 the Eisenstein ideal IS(Ψ ) and the associated local component (hordE , mE) of hord(K, Λ). For brevity, in what follows we write (R, Q) = (hordE , mE),I = IS(Ψ ), GK,S= Gal(KS/K) and
(8.2) δ1= Ψ−cε, δ2= 1, δ3= Ψ ε−1
(recall that ε : GK,S → Z×p is the p-adic cyclotomic character). Let XclsR be the subset of arithmetic points in XR := Homcont(R, Cp) ⊂ Spec R(Cp) such that to each x ∈ XclsR we can attach a p-ordinary cuspidal holomorphic automorphic representation πx with the ring homomorphism λπx : R → Cp. We shall call XclsR the space of classical weights. Thanks to the work of Rogawski [Rog92] and the techniques in [BR93], it is known that there exists a continuous p-adic semi-simple Galois representation Rp(πx) : G → GL3(Cp) attached to πx for each x ∈ XclsR
such that ρx := Rp(πx)(1) is conjugate self-dual (i.e. ρ∨x ρcx, where c is the complex conjugation) and Tr Rp(πx)(Frobw) = λπx(T1,w) for each split place w∈
SK, where T1,w are the Hecke operators defined in§7.7 (cf. [Mai08, Thm. 4.1.1] and [BC04,§3.2]).
Let iR,Λ: Spec R(Cp)→ Spec Λ(Cp) be the structure morphism. It is clear that XclsR contains i−1R,Λ(XclsΛ ), and hence XclsR is also Zariski dense in XR. Therefore, by a standard patching argument in the theory of pseudo-character, we can glue trace functions Tr ρx: GK,S→ Cpfor all x∈ XclsR and obtain a unique pseudo-character
T : GK,S → R such that T(Frobw) = T1,w· ε(Frobw) for split places w∈ SK∪ SpK
(cf. [SU13, Prop. 7.3]). By the very definition ofI = IS(Ψ ) in§7.8, we have T(Frobw) = T1,w· ε(Frobw)≡ λT1,w= (εΨ−c+ 1 + Ψ ε−1)(Frobw) (modI) for all but finitely Frobw at primes w split in K/F, and hence by Chebotarev’s density theorem,
T ≡ δ1+ δ2+ δ3(modI).
We further assume the following hypothesis holds
(Dist) ψ1+c≡ ωK2(mod m).
ThusT is a residually multiplicity free pseudo-character, i.e. the residual characters δi(mod mΛ) are pairwise distinct.
We shall construct an R[GK,S]-moduleL with certain good properties. This is the so-called lattice construction due to E. Urban who generalizes Wiles’ construc-tion of cocycles in [Wil90]. (See [Urb99], [Urb01] and [SU13,§4].) Here we use the language of generalized matrix algebra (GMA) in Bella¨ıche and Chenvier’s book [BC09, Chapter 1]. Recall that a GMA datum D = {ei, "i}i=1,2,3 for (T, δi) con-sists of orthogonal idempotents eitogether with isomorphisms "i: eiR[GK,S]ei→ R∼
such that "i⊗ R/I = δi ([BC09, Def.1.3.1]). Let F = Frac R be the totally frac-tional field of R. By [BC09, Thm. 1.4.4], we can associate a generalized matrix algebra withD. To be precise, there exists a three-dimensional F -vector space V with a basis{vi}i=1,2,3 and a genuine representation ρ : R[GK,S]→ EndF(V ) such that according to the basis{vi}i=1,2,3, ρ(R[GK,S]) is isomorphic to the generalized matrix algebra
ρ(R[GK,S]) =
⎡
⎣ R R12 R13
R21 R R23
R31 R32 R
⎤
⎦ ,
where Rijare fractional ideals in F such that RijRji⊂ I for i = j. For g ∈ R[GK,S], we write ρ(g) as
ρ(g) =
⎡
⎣a11(g) a12(g) a13(g) a21(g) a22(g) a23(g) a31(g) a32(g) a33(g)
⎤
⎦ , aij(g)∈ Rij.
Moreover, in virtue of the conjugate self-duality of Galois representations, we have T(g−c) =T(g), g−c:= cg−1c.
According to [BC09, Lemma 1.8.3] we can manage ρ such that
(8.3) ρ⊥(g) := ρ(g−c)t= wρ(g)w−1, w =
⎡
⎣ 1
1 1
⎤
⎦ .
Then (8.3) implies
(8.4) a12(g−c) = a23(g) and R12= R23.
LetL3:= Rv3and letL = R[GK,S]v3be the R[GK,S]-module generated byL3. By definition,L has the following decomposition as R-modules:
L = D ⊕ L3,D = R13v1⊕ R23v2.
By the relation RijRji ⊂ I, we have the following exact sequence as R[GK,S ]-modules:
(8.5) 0−→D ⊗ R/I−→L ⊗ R/I−→δ3⊗ R/I−→0.
Lemma 8.5. Every R[GK,S]-module quotient of L /QL ⊗ δ3−1 is an R/Q-module with the trivial G-action.
Proof. Suppose L has an irreducible quotient L /N → τ with τδ∼ 3−1 1. Put L = L /(N + D + QL ). Then we have δ3 ⊗ R/Q−→→L and τ−→→L. This impliesL = 0 and therebyL = N + D + QL . By the existence of idempotent e3 ∈ ρ(R[GK,S]) such that e3v3 = v3 and e3vi = 0 for i= 3, we deduce that v3 ∈ N . Because N is an R[GK,S]-module and v3 generatesL ⊗ k as an R/Q[GK,S ]-module, N + QL = L . Hence by the Nakayama lemma N = L , which is a
contradiction.
We set
(8.6) I1= R13,J1= R12R23+ IR13; I2= R23,J2= R21R13+ IR23.
For{i, j} = {1, 2}, let Mi= IL +Jivi+Rj3vjbe a sub-R-module inL . It can be verified easily thatMi is G-stable and L /Mi fits in the following exact sequence of R[GK,S]-modules:
(8.7) 0−→Ii/Ji⊗ 1−→L /Mi⊗ δ−1i −→R/I ⊗ δ3δi−1−→0,
where 1 is the trivial GK,S-action. The Pontryagin dual of (8.7) induces the exact sequence
H0(GK,S, (L /Mi)∗)−→ H0(GK,S, (Ii/Ji)∗) (8.8)
= (Ii/Ji)∗ Δ−→ Hi 1(GK,S, (R/I)∗⊗ δ3−1δi).
The natural homomorphism q : (R/I)∗ ∼→ (Λ/Eis(Ψ, S))∗ → Λ∗ gives rise to a homomorphism
(8.9)
ci:= q∗◦ Δi: (Ii/Ji)∗ Δ−→ Hi 1(GK,S, (R/I)∗⊗ δ3−1δi)−→ Hq∗ 1(GK,S, Λ∗⊗ δ−13 δi).
Indeed, for each f ∈ (Ii/Ji)∗= Homcont(Ii/Ji, Qp/Zp) we define the 1-cocycle ci(f ) : GK,S → Λ∗⊗ δ3−1δi by
ci(f )(g)(λ) = f (δ−13 (g)ai3(g)λ) (g∈ GK,S, λ∈ Λ).
Then ci(f ) is the class [ci(f )] in H1(GK,S, Λ∗⊗ δ3−1δi).
Lemma 8.6. The homomorphisms ci are injective.
Proof. Because δ3 ≡ δi(mod mΛ) for i = 1, 2, the map q∗ is injective. Hence by (8.8) it suffices to show H0(GK,S, (L /Mi⊗ δ−1i )∗) = 0. By Lemma 8.5, we have
HomGK,S(L ⊗ δ−1i , Qp/Zp) = 0.
Therefore we have
H0(GK,S, (L /M ⊗ δ−11 )∗) (8.10)
= HomGK,S(L /M ⊗ δ−11 , Qp/Zp) → HomGK,S(L ⊗ δ−1i , Qp/Zp) = 0.
8.4. p-adic Galois representations. In this subsection, we study local p-adic Galois representations associated to ordinary cohomological automorphic forms with separable weights and deduce that the maps ciin (8.9) actually factor through the corresponding Selmer groups. Let recp : (K ⊗QQp)× Dpab :=
w|pDwab
be the geometrically normalized local reciprocity law. We say a p-adic character
χ : Dp→ C×p is separable if χ(recp(zp)) = zkΣ+a(1p −c) for zp ∈ (OK⊗ Zp)×, where k > 2 is an integer and a =
σ∈Σaσσ ∈ Z≥0[Σ] is a d-tuple of non-negative integers such that
k + aσ− 1 > aσ+ 1 > 0 for every σ, σ ∈ Iw. We put
XsepΛ =
x∈ Homcont(ΓK, C×p)| x|Dp is separable
and Xcls,sepR = i−1R,Λ(XsepΛ )∩ XclsR . One can verify easily that Xcls,sepR is Zariski dense in XclsR .
Let x ∈ Xcls,sepR . Note that the Hecke character associated to x (the complex avatar of x) is unramified at all places above p and is of infinity type kΣ + a(1− c).
It follows that the corresponding automorphic representation πx is unramified at all places above p, and the L-parameter φπx,σ : WC = C× → GL3(C) of the holomorphic discrete series πx,σ at an archimedean place σ∈ Σ is given by
φπx,σ(z) = diag((z/z)0, (z/z)−1−aσ, (z/z)1−k−aσ).
On the other hand, consider the Galois representation ρx= Rp(πx)(1) associated to πx. Let w ∈ SpK and let Dw be the decomposition group of w in GK. From the fact that the Galois representation Rp(πx) is constructed out of the p-adic ´etale cohomology groups of certain compact Picard modular surfaces with good reduction at places above p, we can deduce that ρx,w := ρx|Dw is an ordinary p-adic Galois representation since πxis a p-ordinary cuspidal automorphic representation [Mai08, Thm. 4.1.1 (3)] (cf. [SU13, Lemma 7.2]). Suppose that w ∈ Σp. Then the local p-adic representation ρx,w (resp. ρx,w) is crystalline and has arithmetic Hodge-Tate weights
(0, 1 + aσ, k + aσ− 1)σ∈Iw(resp. (1− aσ− k, −aσ− 1, 0)σ∈Iw) (cf. [BC04, Prop 3.3]). The Newton and Hodge polygons of ρx,w meet at
(dw, 0), /
2dw, %
σ∈Iw
1 + aσ
0 ,
/
3dw, %
σ∈Iw
k + 2aσ
0
, dw= [Kw: Qp].
By [TU99, Lemma 7.2] (cf. [Mai08, Lemma 4.2.3]), the R-module L has a fil-tration {0} ⊂ F3,w ⊂ F1,w ⊂ F2,w = L of R-submodules, according to which ρw:= ρ|Dw has the matrix representation
ρw=
⎡
⎣δ3,w ∗ ∗ δ1,w ∗ δ2,w
⎤
⎦ .
For each i = 1, 2, 3, the specialization δi,w(x) of δi,w at x is a locally algebraic p-adic character of Dw by a theorem of Serre [Ser68, Corollary, page III-50], and δ3,w(x) has the maximal Hodge-Tate weights k + aσ−1 for each σ ∈ Iw. We remark thatF3,w= 0 as I = R and L /IL has a quotient isomorphic to δ3⊗R/I, whereas
the quotients F2,w/F1,w and F1,w/F3,w could be zero modules because L need not be a lattice. Let εw be the p-adic character of Iw induced by local class field theory εw: Iw→ Iwab
→ O∼ w×. Because the Hodge-Tate weights of{δi(x)}i=1,2,3 are distinct, by [Ser68, Thm. 2, page III-44] we find that
δ3,w (x)|Iw=
σ∈Iw
(σεw)k+aσ−1= δ3(x)|Iw
for every x∈ Xcls,sepR and δ3,w |Iw= δ3|Iw.
Lemma 8.7. Let P ∈ ht1(Λ) such thatIP ⊂ P RP. Assume that δiδ−1j |Iw≡ 1 (mod P ) for all i, j.
Then the natural map (F3,w)P → (L3)P is surjective.
Proof. The assumption implies that δi,w|Iw= δi|Iwmod P are distinct, so we can find t in RP[Iw] such that ρ(t)|LP/PLP is an idempotent, and according to the filtration{0} ⊂ F3,w⊂ F1,w⊂ F2,w=L , we have
ρ(t)|LP/PLP =
⎡
⎣1 ∗ ∗ 0 0 0
⎤
⎦ (mod P ).
For v∈ (L3)P, we find that ρ(t)v∈ (F3,w)P and ρ(t)v≡ v (mod P LP +DP), so the map (F3,w)P → (L3)P⊗RP/P is surjective. Therefore the map (F3,w)P →
(L3)P is surjective by Nakayama lemma.
Remark 8.8. We remark that the assumption in Lemma 8.7 is satisfied whenever ordP(LSp(Ψ, Σ)) > 0. Indeed, if δiδj−1|Iw ≡ 1 (mod P ) for some i, j and w ∈ Σp, then we can choose an integer n0 large enough such that (Γ+K)pn0 ⊂ Iw∩ Γ+K for all w∈ Σp. Thus
Ψ1+cε−2(γp+n0)≡ 1 (mod P )
and P = (T+− (ζ0ε(γ+)− 1)) for some ζ0 ∈ μpn0. It follows that we can choose a point y ∈ XΛ such that y(P ) = 0 and Ψy is a Hecke character of infinity type (1 + m)Σ + (1− m)Σc, m≥ 1. The specialization of LSp(Ψ, Σ) at y is LS(m + 1, θ) for some Hilbert modular form θ of parallel weight (2m + 1), which is non-zero by [Shi78, Prop. 4.16]. This in particular implies that ordP(LSp(Ψ, Σ)) = 0.
Corollary 8.9. If ordP(LSp(Ψ, Σ)) > 0, then there exists tP ∈ Λ − P such that c1: tP(I1/J1)∗⊂−→SelKS(Ψ−1−cε2)c=−1;
c2: tP(I2/J2)∗⊂−→SelKS(ΨD, Σc).
Proof. Injectivity of ci is proved in Lemma 8.6. By (8.4), for every g ∈ GK,S we have
a13(gc)a33(g) + a23(gc)a23(g) + a13(g)a33(gc) = 0, and thus
δ3(g)a13(gc) + a12(g)a23(g) + δ3(gc)a13(g)≡ 0 (mod IR13).
Let f ∈ (I1/J1)∗. Since f (R12R23+IR13) = 0, we find that
c1(f )(gc) = f (δ−13 (gc)a13(gc)) =−f(δ3−1(g)a13(g)) =−c1(f )(g).
This shows that c1(f )c =−c1(f ).
For every w ∈ Σp, by Lemma 8.7 and Remark 8.8 there exists uw ∈ (R13)P
and vw ∈ (R23)P such that (uw, vw, 1)t ∈ F3,w. Choose tP ∈ Λ − P such that uw = tPuw ∈ R13 and vw = tPvw ∈ R23 for all w ∈ Σp. By the construction ofF3,w, we have ρ(g)(uw, vw , tP)t= δ3(g)(uw, vw, tP)t for g∈ Iw, from which we deduce the relations:
δ−13 (g)a12(g)vw+ tPδ−13 (g)a13(g) = δ3−1δ1(g)uw− uw(mod IR13);
δ3−1(g)a21(g)uw+ tPδ−13 (g)a23(g) = δ3−1δ2(g)vw− vw(modIR23).
It follows that for every f ∈ (Ii/Ji)∗, i ∈ {1, 2}, the class [ci(tPf )] is trivial in H1(Iw, δ−13 δi⊗ Λ∗). The morphisms tPci thus factor through the corresponding
Selmer groups.