9.1 Eisenstein Ideals
Let KD be an open compact subgroup of GU(3,1)(AQ) maximal at p and all primes outside Σ such that the Klingen-Eisenstein series we construct is invariant under KD(p). We consider the ring TD of reduced Hecke algebras acting on the space of Λ00D-adic nearly ordinary cuspidal forms with level group KD. It is generated by the Hecke operators Zv,0, Zv,0(i),Ti,v, Ti,v(j) defined before, together with the Up-operator and then taking the maximal reduced quotient. It is well known that one can interpolate the pseudo Galois characters attached to nearly ordinary cusp forms to get a pseudo-character RD of GK with values in TD (see [55, Section 7.2] for details). We define the idealID of TD to be generated by{t−λ(t)}t fort’s in the abstract Hecke algebra and λ(t) is the Hecke eigenvalue of t on ED,Kling. Then it is easy to see that the structure map Λ00D →TD/ID is surjective. Suppose the inverse image ofIDin Λ00DisED. We call it the Eisenstein ideal. It measures the congruences between the Hecke eigenvalues of cusp forms and Klingen-Eisenstein series. We have:
RD(modID)≡trρED,Kling(modED).
Now we prove the following lemma:
Lemma 9.1. Let P be a height 1 prime ofˆIur[[Γ00K]] which is not the pullback of a height 1 prime ofˆIur. Then
ordP(LΣf,ξ,K)≤ordP(ED).
Proof. Suppose t := ordP(LΣf,K,ξ) > 0. By the fundamental exact sequence Theorem 3.6 there is an H=ED,Kling− LΣf,ξ,KF for some Λ00D-adic form F such that His a cuspidal family. We write
` for the Λ00D-adic functional `(G) = hlθ?(G), π(g20)hi constructed in Subsection 8.5 on the space of Λ00D-adic forms. By our assumption onP we have proved that `(H) 6≡0(modP). Consider the Λ00D-linear map:
µ:TD→Λ00D,P/PtΛ00D,P given by: µ(t) =`(t.H)/`(H) for tin the Hecke algebra. Then:
`(t.H)≡`(tED)≡λ(t)`(ED)≡λ(t)`(H)(modPt)
so ID is contained in the kernel of µ. Thus it induces: Λ00D,P/EDΛ00D,P Λ00D,P/PtΛ00D,P which proves the lemma.
9.2 Galois Theoretic Argument
In this section, for ease of reference, we repeat the set-up and certain results from [55, Chapter 4]
with some modifications, which are used to construct elements in the Selmer group.
LetGbe a group andC a ring. Letr:G→AutC(V) be a representation ofGwithV 'Cn. This can be extended tor:C[G]→EndC(V). For anyx∈C[G], define: Ch(r, x, T) := det(id−r(x)T)∈ C[T].
Let (V1, σ1) and (V2, σ2) be two C representations of G. Assume both are defined over a local
henselian subring B ⊆C, we say σ1 and σ2 are residually disjoint modulo the maximal ideal mB if there exists x ∈B[G] such that Ch(σ1, x, T) mod mB and Ch(σ2, x, , T) mod mB are relatively prime in κB[T], where κB:=B/mB.
Let H be a group with a decomposition H = Go{1, c} with c ∈ H an element of order two normalizingG. For anyC representations (V, r) ofGwe write rcfor the representation defined by rc(g) =r(cgc) for allg∈G.
Polarizations:
Letθ:G→GLL(V) be a representation ofGon a vector spaceV over fieldL and letψ:H→L× be a character. We assume that θ satisfies theψ-polarization condition:
θc'ψ⊗θ∨.
By aψ-polarization ofθ we mean anL-bilinear pairing Φθ :V ×V →Lsuch that Φθ(θ(g)v, v0) =ψ(g)Φθ(v, θc(g)−1v0).
Let Φtθ(v, v0) := Φθ(v0, v), which is another ψ-polarization. We say that ψ is compatible with the polarization Φθ if
Φtθ=−ψ(c)Φθ. Suppose that:
(1)A0 is a pro-finiteZp algebra and a Krull domain;
(2) P ⊂A0 is a height one prime and A = ˆA0,P is the completion of the localization of A0 at P.
This is a discrete valuation ring.
(3)R0 is local reduced finite A0-algebra;
(4)Q⊂R0 is prime such thatQ∩A0=P and R= ˆR0,Q;
(5) there exist ideals J0 ⊂A0 and I0 ⊂R0 such that I0∩A0 =J0, A0/J0 =R0/I0, J = J0A, I = I0R, J0 =J∩A0 and I0=I∩R0;
(6)G and H are pro-finite groups; we have a subgroupG¯v0 ⊂G.
Set Up:
Suppose we have the following data:
(1) a continuous characterν :H →A×0;
(2) a continuous characterξ :G→A×0 such that ¯χ6= ¯νχ¯−c; Let χ0 :=νχ−c;
(3) a representationρ:G→AutA(V), V 'An, which is a base change from a representation over A0, such that:
a.ρc'ρ∨⊗ν,
¯
ρ is absolutely irreducible,
ρ is residually disjoint from χand χ0;
(4) a representationσ:G→AutR⊗AF(M), M '(R⊗AF)m withm=n+ 2, which is defined over the image ofR0 inR, such that:
a. σc'σ∨⊗ν,
b.trσ(g)∈R for all g∈G ,
c. for any v∈M,σ(R[G])v is a finitely-generated R-module
(5) a proper idealI ⊂Rsuch thatJ :=A∩I 6= 0, the natural mapA/J →R/I is an isomorphism, and
trσ(g)≡χ0(g) + trρ(g) +χ(g) mod I for all g∈G.
(6)ρ is irreducible and ν is compatible withρ.
(7) (local conditions for σ) There is a Gv¯0-stable sub-R⊗AF-module M¯v+0 ⊆ M such that M¯v+0 and Mv¯−0 := M/Mv¯+0 are free R ⊗AF modules. We also require that Mv¯+0 and M¯v−0 are disjoint modulo I. (In our applications this is always satisfied although the ¯Fp-representations of ¯ρf, ¯χ0, ¯χ are not necessarily mutually disjoint when restricting toGv¯0.)
(8) (compatibility with the congruence condition) Assume that for all x ∈ R[G¯v], we have con-gruence relation:
Ch(Mv¯+0, x, T)≡(1−T χ(x))mod I (then we automatically have:
Ch(Mv¯−0, x, T)≡Ch(V¯v0, x, T)(1−T χ0(x))mod I)
(9) For each F-algebra homomorphism λ : R⊗AF → K, K a finite field extension of F, the representation σλ : G → GLm(M ⊗R⊗F K) obtained from σ via λ is either absolutely irre-ducible or contains an absolutely irreirre-ducible two-dimensional sub K-representation σλ0 such that trσλ0(g)≡χ(g) +χ0(g) mod I.
One defines the Selmer groups XH(χ0/χ) := ker{H1(H, A∗0(χ0/χ)) → H1(G¯v0, A∗0(χ0χ))}∗. and XG(ρ0⊗χ−1) := ker{H1(G, V0⊗A0A∗0(χ−1))→H1(G¯v0, V0−⊗A0A∗0(χ−1))}∗. Let ChH(χ0/χ) and ChG(ρ0⊗χ−1) be their characteristic ideals asA0-modules.
Proposition 9.2. Under the above assumptions, if ordP(ChH(χ0/χ)) = 0 then ordP(ChG(ρ0⊗χ−1))≥ordP(J).
Proof. This can be proved in the same way as [55, Corollary 4.16]. The only difference is the Selmer condition at p, which we use the description of Section 3.3 to guarantee. Note that the part corresponding to ρ0 corresponds to the upper-left two by two block here while in [55] the ρf contains the highest and the lowest Hodge-Tate weights.
Before proving the main theorem we first prove a useful lemma, which appears in an earlier version of [55] .
Lemma 9.3. LetQ⊂I[[Γ00K]]be a height one prime such thatordQ(LΣf,K,ξ)≥1andordQ(Lχ
fξ¯0) = 0, thenordQ(LΣ
χfξ¯0) = 0.
Proof. Let θ=χf0ξ¯0. If ordQ(LΣχ
fξ¯0)≥1, then for some`∈Σ\{p}, Y
`∈Σ\{p}
(1−θ−1(γ+−1(1 +W))e`2−κ)∈Q,
wheree∈Zp be such that `=ω−1(`)(1 +p)e. Thus
θ(`)≡γ−e+ ω(`)κ−2(1 +p)e(2−κ)(modQ).
Thus there is some integer f such that
1≡(γ+(1 +W)−1(1 +p)κ−2)−f e(modQ)
which implies that for somep-power root of unityζ+,Qis contained in the kernel of anyφ0such that φ0(γ+(1 +W)−1) =ζ+(1 +p)2−κ. This implies, by [16, Theorem I] for the interpolation formula, that at the central critical point LK(fφ, θ1,1) = 0 where θ1 is some fixed CM character of infinity type (κ2,−κ2) andφany arithmetic point. But then we can specialize f to some pointφ00 of weight 4 (this is not an arithmetic point in our definition, but is an interpolation point, by loc.cit.). By temperedness for fφ00, the specialization is not 0. This is a contradiction.
Now we apply the above result to prove the theorem.
• H:=GQ,Σ,G=GK,Σ,cis the complex conjugation.
• A0 = Λ00D, A:= Λ00D,P.
• J0 :=ED, J :=EDA.
• R0 :=TD, I0 :=ID.
• Q⊂R0 is the inverse image ofP moduloED under TD→TD/ID = ΛD/ED.
• R:=Tf ⊗IΛ00D,ρ0 :=ρπfσ(ψ/ξ)c−κ+32 .
• V =V0⊗A0 A,ρ=ρ0⊗A0 A.
• χ=σψc,χ0 =σψcσ(ψ/ξ)0−κ. ν =χcχ0.
• M := (R⊗AFA)4,FA is the fraction field ofA.
• σ is the representation on M obtained as the pseudo-representation associated toTD, as in [55, Proposition 7.2.1].
Now we are ready to prove the main theorem in the introduction.
Proof. We first note that we need only to prove the corresponding inclusion for the Σ-primitive Selmer groups andL-functions since locally the sizes of the unramified extensions at primes outside p are controlled by the local Euler factors of the p-adic L-functions since Q∞ ⊆ K∞. (See [10, Proposition 2.4].)
Recall that we have enlarged our I at the beginning of Subsection 8.2 and the end of Subsection 8.4 which we denote as Jin this proof. We first prove the main theorem with ˆIur replaced by ˆJur. Under the assumption of Theorem 1.1, as in [55, Proposition 12.9] we know that by the discussion for the anticyclotomic µ-invariant at the end of Section 7.5, Lf,ξ,K is not contained in any height one prime which is the pullback of a prime in ˆIur[[Γ+K]]. Note that there might be height one primes dividing the Euler factors at non-split primes which are pullbacks of height one primes of ˆIur[[Γ+K]].
Such issue has been overlooked in [55]. These primes are treated in [58, Lemma 87, Theorem 101]
and can be treated in the same way here. In the following, we treat the height one primes which are not such pullbacks. By lemma 9.1 for any such height one primeP of ˆIur[[Γ00K]],
ordP(LΣf,K,ξ) = ordP(LΣf,K,ξ)≤ordP(ED).
Applying Proposition 9.2, we prove the first part of the theorem for ˆJur in place of I.
We replace ˆJur[[Γ00K]] by ˆIur[[ΓK]]. We write L for LΣf,ξ,K. Recall Fitting ideal respects base change. We claim that for any x ∈ Fitt(X), xL−1 ∈ˆIur[[ΓK]]. In fact, from what we proved for Fitt(V⊗ˆ
Iur[[ΓK]]ˆJur[[Γ00K]]) as ideals of ˆJur[[Γ00K]], we havexL−1 ∈ˆJur[[Γ00K]]∩Fˆ
Iur[[Γ00K]] whereFˆ
Iur[[ΓK]]
is the fraction field of ˆIur[[ΓK]]. Since ˆIur[[ΓK]] is normal and ˆJur[[Γ00K]] is finite over ˆIur[[ΓK]], we havexL−1 ∈ˆIur[[ΓK]]. Thus Fitt(X)⊆(L), which in turn implies that char(X)⊆(L). This proves Theorem 1.1.
Now assume we are under the assumption of Theorem 1.2. Note that in this caseLχξ¯0 = 1. Thus by the Lemma 9.3LΣf,ξ,Kis co-prime toLΣ
χξ¯0. SupposeP1, ..., Ptare the height one primes ofLΣf,K,ξ that are pullbacks of height one primes in ˆIur. Note that none of the primes passes throughφ0 since the two-variablep-adicL-function forf0 is not identically 0. We consider the ring ˆIurp,P1,...,Pt[[ΓK]] where the subscripts denote localizations. Then the argument as in the proof of Theorem 1.1 proves that there is a numberasuch that (LΣf,K,ξ)⊇(P1· · ·Pt)aFittˆ
Iur[[ΓK]](XΣ) as ideals of ˆIur[[ΓK]]. Specialize toφ00, using Proposition 2.4, we find
(LΣf
0,K,ξ)⊇FittOˆurL[[ΓK]]⊗L(Xf0).
This proves Theorem 1.2.
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Xin Wan, Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, China, 100190.
E-mail Address: xw2295@math.columbia.edu