THE ALGEBRAIC p-ADIC L-FUNCTION AND ISOGENY BETWEEN FAMILIES OF GALOIS REPRESENTATIONS
TADASHI OCHIAI
Abstract. In Theorem 1.6, we give a formula to compare the algebraic p-adic L- functions for two different lattices of a given family of Galois representations over a deformation ringR. This generalize a classical comparison formula by Schneider [Sc]
and Perrin-Riou [P], for whichRis the cyclotomic Iwasawa algebra.
Recall that we studied the two-variable Iwasawa theory for residually irreducible nearly ordinary Hida deformations in [O1], [O2] and [O3] and we acquired sufficient understanding through these works. By applying our formula to Hida’s nearly ordinary deformations, we understand better the two-variable Iwasawa theory for residually re- ducible cases, where the choice of lattices is not unique anymore.
Contents
1. Introduction 1
2. Euler-Poincar´e characteristic with large coefficients 6
3. Proof of Theorem 1.6 11
4. Two-variable Iwasawa Main conjecture for Hida deformations 14
References 17
1. Introduction
LetRbe a local domain which is finitely generated and torsion-free over a power series algebraZp[[X1,· · ·, Xn]] withp a fixed odd prime number and letKbe the fraction field of R. We fix embeddings Q → Qp and Q → C, where Q and Qp are the algebraic closures of the rational number fieldQand the p-adic fieldQp respectively. Throughout the paper, we denote by Dv the decomposition subgroup of the absolute Galois group G = Gal(Q/Q) at a primev.
Definition 1.1. Let V be a finite dimensionalK-vector space with K-linear G -action.
The representationρ:G −→AutK(V) is calledcontinuousifV has a finitely generated R-submodule T with the following properties:
1. T ⊗RK is isomorphic toV.
2. The R-submodule T is stable underG -action onV.
3. The action of G on T is continuous with respect to the M-adic topology on T, where Mis the maximal ideal ofR.
An R-submodule T of V satisfying the above properties is called a lattice (ofV). If T and T are lattices of the same continuous representation V, we say that T and T are
isogenious to each other. In general, a latticeT is not necessarily a free module overR. When a latticeT is free R-module, we call ita free lattice.
In this paper, we always assume that the action of G on T is unramified outside a finite set of primes Σ⊃ {p,∞}. We introduce several notations. Let R be the integral closure ofR inK. Note thatR is a finitely generated R-module. LetT =T ⊗RR and let A= T ⊗ R(R)∨ where (R)∨ is the Pontrjagin dual of R. When R =R, we denote Aby A. We denote by Homp(R, Qp) the set of non-trivialZp-algebra homomorphisms, which is naturally endowed ap-adic topology and is regarded as a p-adic rigid analytic space. For φ∈Homp(R, Qp), we define (T)φ to be the p-adic Galois representation of G in usual sense obtained by specializing T via φ.
Definition 1.2. Suppose also that we have a Dp-stable R-submodule F+AofA. Then, we define the Selmer group SelT as follows:
SelT = Ker
⎡
⎣H1(QΣ/Q,A)−→H1(Ip,F−A)×
l∈Σ\{p,∞}
H1(Il,A)
⎤
⎦, (1)
where F−A means A/ F+A, QΣ is the maximal algebraic extension of Q unramified outside Σ and the map in the definition is the natural localization map. Though the Selmer group SelT depends on the fixed Dp-stable R-submodule F+Aof A, we omit to note it in the notation SelT if there seems to be no confusion.
Remark 1.3. Since SelT is a discrete R-module, the Pontrjagin dual (SelT)∨ of SelT is naturally endowed with a structure of compactR-module. It is also not so hard to show that (SelT)∨ is a finitely generatedR-module (cf. [Gr2, §4]).
We recall the following conjecture which we find in the paper [Gr2] (under slightly different assumptions):
Conjecture 1.4. Let T be a lattice of an irreducible continuous K-linear representation V of G . We denote by d± the K-rank of the±-eigenspace of V under the action of the complex conjugation in G . Assume further that the following conditions are satisfied:
1. The action of G on T is unramified outside a finite set of primesΣ⊃ {p,∞}. 2. V has a Dp-stable K-subspace F+V ⊂ V with rankKF+V =d+ such that V/F+V is
free of rank d−. (Note that F+V induces the filtration F+T :=T ∩ F+V in T and the filtration F+A:= F+T ⊗ R(R)∨ in A).
3. There exists a dense subset S ⊂Homp(R, Qp) such that every φ∈ S satisfies the following conditions:
(a) For each φ ∈ S, there exists a pure motive Mφ over Q critical in the sense of [De2] such that (T)φ⊗pQp is isomorphic to the p-adic ´etale realization Vφ=H´et(Mφ⊗ Q,Qp).
(b) The Dp-equivariant isomorphism HdR(Mφ)⊗ pBdR−→∼ Vφ⊗ pBdR of p-adic Hodge theory induces an isomorphism Fil0(HdR(Mφ))⊗ p BdR −→∼ F+Vφ⊗ p
2
BdR, where HdR(Mφ) is the de Rham realization of Mφ and {FiliHdR(Mφ)} is the de Rham filtration.
(c) The zero locus N ={φ∈ S | L(Mφ,0) = 0} is contained in a certain Zariski closed subset of Homp(R,Qp).
Then, we conjecture that (SelT)∨ is a finitely generated torsion R-module.
For a noetherian integral domainR which is integrally closed in the fraction field, the localization R is a discrete valuation ring for every height-one prime p of R. Hence, for a finitely generated torsion R-module M, we have an invariant lengthR
M. We see that the invariant lengthR
((SelT)∨) makes sense if Conjecture 1.4 holds for T. On the other hand, an isomorphism class of lattices T in V is not unique when the residual representation is reducible. Thus, it is important to compare the difference between lengthR
((SelT)∨) and lengthR
((SelT)∨) when we take two lattices T and T. After preparing necessary notations, we will state our main result (Theorem 1.6) which calculate this difference.
Definition 1.5. Let us introduce the following conditions on a latticeT : (F ) The coinvariant quotient (T∗)G
É
is a pseudo-null R-module.
(FÔ) Ip acts non-trivially on every elements in F−T and Dp acts non-trivially on every elements in F+T(−1) and F−T(−1). The Pontrjagin dual of (F−A)Dp is a pseudo-null R-module.
(T) SelT is a cotorsionR-module.
(T) SelT∗(1) is a cotorsionR-module whereT∗(1) = HomR(T,R)⊗pZp(1).
Further, we assume the following condition for each prime v∈Σ\ {∞, p}:
(FÚ) Dv acts non-trivially on every elements in T(−1) and the modules ADv and (AIv)Dv are pseudo-nullR-modules.
Our main result is as follows:
Theorem 1.6. Suppose that the ringRis isomorphic toO[[X1,· · · , Xn]]with the integer ring of integers O of a finite extension of Qp and a certain natural number n. Suppose that T andT are free lattices of a continuous representation V of G over the fraction field K of R unramified outside a finite set of primes Σ ⊃ {p,∞}. Let us assume the conditions (F ),(FÔ),(T),(T) and (FÚ) for both of them.
Then, we have lengthR
((SelT)∨)−lengthR
((SelT)∨)
= lengthR
((T/T)G
Ê
)−lengthR
((F+T/F+T))
for every height-one prime p of R (Note that R=R by assumption).
As we will see in the following remark, the assumptions of this theorem are satisfied by a wide class of deformations. Hence, it ensures that we will have a plenty of applications of the theorem.
Remark 1.7. 1. It is not difficult to show that the conditions (T) and (T) are isogeny invariant (That is, if the property holds for one lattice T ⊂ V, it holds
for all lattices in V). The conditions (F ),(FÔ) and (FÚ) also might be isogeny invariant under fairly general situation. But, we do not pursue the question of isogeny invariance of these conditions and we assume them for both ofT and T. 2. Let T ∼= Z⊕dp be a p-adic Galois representation of G unramified outside a finite
set of primes Σ⊃ {p,∞}. We consider the cyclotomic Zp-extension Q∞ of Qwith Γ = Gal(Q∞/Q). We denote by Zp[[Γ]](χ) a rank-one Zp[[Γ]]-module on which G acts via the character χ: G −→ Zp[[Γ]]× obtained by tautological injection G Γ→ Zp[[Γ]]×. Then, we introduce the freeZp[[Γ]]-module T ⊗pZp[[Γ]](χ) on which G acts diagonally and we call it the cyclotomic deformation of T. The Selmer group SelT forT =T⊗pZp[[Γ]](χ) defined as in the definition (1) coincides with the cyclotomic Selmer group SelA(Q∞)⊂H1(QΣ/Q∞, A) given in [Gr1] where A =T ⊗pQp/Zp. When T is ordinary at p and SelA(Q∞)∨ is a torsion Zp[[Γ]]- module, Theorem 1.6 was obtained by Perrin-Riou in the paper [P].
3. We verify the conditions of the theorem in the following cases.
(a) Let Tp(E) be the p-Tate module of an elliptic curve E over Q with ordinary reduction atpand letT :=Tp(E)⊗pZp[[Γ]](χ). Note that a finitely generated Zp[[Γ]]-module is pseudo-null if and only if it is finite. In this situation, the conditions (T) and (T) always hold by results of Rubin [R1] (CM case) and Kato [Ka] (non CM case). Recall that the groupE(Q∞)p-tors := ∪
m≥1E(Q∞)[pm] is always finite as studied by Imai [I]. On the other hand, (T∗)G
É
is the Pon- trjagin dual of E(Q∞)p-tors by Shapiro’s lemma. Hence, T always satisfies the condition (F ). Let v ∈Σ\ {p,∞}. We see easily that Dv acts non-trivially on every elements of T(−1) = Tp(E)(−1)⊗pZp[[Γ]](χ). The group ADv is isomorphic to a finite number of copies of E(Qv,∞)p-tors = ∪
m≥1E(Qv,∞)[pm] whereQv is the completion ofQatvandQv,∞=Q∞Qv. Since it is easy to see thatE(Qv,∞)p-tors is a finite group, the Pontrjagin dual ofADv is a pseudo-null R-module. This implies immediately that the Pontrjagin dual of (AIv)Dv is a pseudo-null R-module. Thus, (FÚ) is satisfied for each v∈Σ\ {p,∞}. Let us discuss about the condition (FÔ). SinceIp acts non-trivially on any Tate-twist ofZp[[Γ]](χ),Ip acts on every elements in F−T(r) = F−Tp(E)(r)⊗pZp[[Γ]](χ) for any integer r. The group (F−A)Dp is isomorphic to ( ∪
m≥1F−E[pm])GÉp,∞
where Qp,∞ = Q∞Qp. This group is a finite group except when E has split- multiplicative reduction atpsince the action of Frobenius element is non-trivial.
Hence, the condition (FÔ) is satisfied except when E has split-multiplicative reduction at p. More generally, let T be a lattice of the p-adic representation V associated to an ordinary eigen cuspform f of weightk ≥ 2 in the sense of Deligne [De] and let T :=T⊗pZp[[Γ]](χ). The conditions (T)and(T)hold by the above mentioned results by Rubin and Kato. The conditions(F ),(FÔ) and (FÚ) for everyv∈Σ\ {p} hold in fairy general situations.
Thus, the assumptions of Theorem 1.6 are checked to be true in these cases and we give another proof of results by Schneider [Sc] (for abelian varieties) and Perrin-Riou [P] (for ordinary p-adic representations).
4
(b) Suppose that T is a free lattice of the two-variable nearly ordinary Hida defor- mation V ∼= K⊕2 associated to a Λ-adic ordinary eigen cuspform F. Here, K is isomorphic to a finite extension of the fraction field of Zp[[X, Y]]. Suppose that the ring of integers Rof K is isomorphic to O[[X1, X2]] with the integer ring of integersO of a finite extension ofQp and let us take a height-one prime I of R such that R/I is also a regular local ring. Then, a finitely generated R-module M is a torsion (resp. pseudo-null)R-module if M/IM is a torsion (resp. pseudo-null) R/I-module. The conditions (T) and (T) are proved by the control theorem at a height-one prime If specializing to the cyclotomic deformation of a cusp form f in the Hida family F and the above mentioned results by Kato and Rubin for the cyclotomic deformation off (The proof is the same as [O3, Proposition 4.9], where we prove it for T with irreducible resid- ual representation). Similarly, (F ),(FÔ) and (FÚ) are proved by specializing method. For(FÔ), if we have If corresponding to a cusp form f of weight two associated to an elliptic curve with split-multiplicative reduction at p,(FÔ) is not true overR/If as we saw above. The specialization method does not work for thisIf. However, if we chooseIf so that the weight off is greater than two, (FÔ) is always true over R/If. In this way, (F ),(FÔ),(T),(T) and (FÚ) are always true for two-variable nearly ordinary deformations.
By applying Theorem 1.6, we obtain the difference of the algebraic p-adicL-function for Hida deformations in Corollary 4.4 after preparing notations in §4. Note that the method in [P] is not applicable anymore to such more general deformations. We also remark that previous results by Schneider [Sc] and Perrin-Riou [P] for the cyclotomic deformations of elliptic curves can be recovered from our result in a completely different manner.
We would like to study the Iwasawa Main conjecture for Hida deformations (equality between the algebraic p-adicL-function and the analyticp-adic L-function). When the residual representation for F is irreducible, we had much progress (We refer to [O3, §2]
for a review of those as well as notations on Hida deformations which we will use in this paper). Thus, we are interested in the case where the reducible representation for F is reducible. In spite of the difficulty of ambiguity of the choice of lattice, it seems to be possible to say something around the main conjecture. First important ingredient for this is that we are able to calculate the difference of the algebraic p-adicL-functions for different choices of lattices by Corollary 4.4. Secondly, with help of [St] in which we find a conjectural answer on the minimal choice of lattice of the p-adic representation for elliptic modular forms of weight two, it seems reasonable to a canonical lattice TF
associated to F constructed in §4 gives the minimal one in the sense that the algebraic p-adicL-function forTF is the minimal one among those for all lattices associated toF. By combining these two points, we understand the Iwasawa Theory forF with reducible residual representation better than before.
For the proof of the main result (Theorem 1.6), the key tool is the Euler-Poincar´e characteristic formula (Theorem 2.1) for a family of Galois representation proved in§2.
The use of this generalized Euler-Poincar´e characteristic formula is the most important difference from the proof of [P], which makes us generalize the result in [P]. We also re- mark that the of proof of Theorem 2.1 is based on the specialization principle developed
in [O2] which allows us to recover the characteristic ideal of a given R-module M from the information on various specializations ofM. Though the proof of Theorem 2.1 occu- pies the whole of§2, the statement of the theorem is very natural and is understandable at a first glance.
Acknowledgements. First of all, the author would like to thank Ralph Greenberg.
The work of the author is greatly influenced by some of articles of Greenberg and sev- eral times of personal conversation. He also thanks to Robert Pollack for stimulating conversations. This paper has been typed during his stay at University of Paris 13 from February 2006 for two years. He is thankful to the arithmetic geometry group of Uni- versity of Paris 13 for their hospitality. Finally, he thanks to anonymous referee for suggestions for modifications of the article.
2. Euler-Poincar´e characteristic with large coefficients
In this section, we prove the Euler-Poincar´e characteristic formula for Galois coho- mologies with coefficients in a power series algebra Rgeneralizing the classical one with coefficients in the ring of integers of ap-adic field (cf. Theorem 2.1). The Euler-Poincar´e characteristic formula for a family of Galois representation is studied a little bit in [Gr1, Proposition 3]. However, the formula in [Gr1, Proposition 3] concerns only the R-rank of Galois cohomologies for only when R =Zp[[Γ]]. In this section, we rather study the characteristic ideal of R-cotorsion Galois cohomologies and we treat more general R’s.
The key tool for the proof of this section is “the specialization principle” established in the previous paper [O2]. The main result of this section (Theorem 2.1) will play an important role in the next section for the proof of Theorem 1.6.
Throughout the paper, we will denote the Galois group Gal(QΣ/Q) by GΣ for short.
For a finite abelian group A, we denote by A the order of A. Our main result in this section is as follows:
Theorem 2.1. Suppose that the ringRis isomorphic toO[[X1,· · · , Xn]]with the integer ring of integers O of a finite extension of Qp and with a certain natural number n. Let C be a discrete R-module such that C∨ is a finitely generated torsion R-module with GΣ-action. Then, we have the following formula for every height-one prime p of R:
1. We have the following Global Euler-Poincar´e characteristic formula:
0≤i≤2
(−1)ilengthR
(Hi(GΣ,C)∨)= lengthR
(C∨)−lengthR
((CGÊ)∨). 2. We have the following local Euler-Poincar´e characteristic formula:
⎧⎪
⎪⎨
⎪⎪
⎩
0≤i≤2
(−1)ilengthR
(Hi(Dv,C)∨) = 0 for v=p,
0≤i≤2
(−1)ilengthR
(Hi(Dp,C)∨) = lengthR
(C∨) for v=p.
To prove the above theorem, we recall necessary results from the paper [O2]. From now on throughout the section, we fix a power series O[[X1,· · ·, Xn]] and denote it by Λ(On) for short. We recall the following definition (cf. §3 of [O2]):
6
Definition 2.2. Let n≥1 be an integer.
1. A linear element l in an n-variable Iwasawa algebra Λ(On) = O[[X1,· · · , Xn]] is a polynomiall=a0+a1X1+· · ·anXn∈Λ(On)withai∈ Oof degree at most one such thatlis not divisible by a uniformizer π ofOand is not invertible in Λ(On). That is, lis a polynomial of degree at most one such that a0 is divisible byπ, but not all ai
are divisible byπ.
2. We denote by L(On) the set of all linear ideals of Λ(On). That is:
L(On) =
(l)⊂Λ(On) l is a linear element in Λ(On)
.
3. Let n ≥2. For a torsion Λ(On)-module M, we denote by L(On)(M) a subset of L(On) which consists of (l)⊂ L(On) satisfying the following conditions:
(a) The quotientM/(l)M is a torsion Λ(On)/(l)-module.
(b) The image of the characteristic ideal charΛ(n)
O (M)⊂Λ(On) in Λ(On)/(l) is equal to the characteristic ideal char
Λ(n)O /(l)(M/(l)M) ⊂Λ(On)/(l).
We will use the following theorem which was proved in [O2]:
Theorem 2.3 (Proposition 3.6 and Proposition 3.11 in [O2]). Let M and N be finitely generated torsionO[[X]]-modules. We have the following:
(1) The following conditions are equivalent:
(a) There exists an integer h≥0 such that charΛO(M)⊃(πh)charΛO(N).
(b) Let O be arbitrary complete discrete valuation ring which is finite flat over O. Then there exists a constant c depending only on MO and NO such that (MO/(l)MO) dividesc·(NO/(l)NO) for all but finitely many (l)∈ L(1)O. (2) As for the difference by the power of π, we have the following equivalence:
(a) Let M(π) (resp. N(π))be the localization ofM (resp. N)at the prime ideal (π).
Then we have length(ΛO)(π)(M(π))≤length(ΛO)(π)(N(π)).
(b) There exists a set of principal ideals
{(Em) |Em is an Eisenstein polynomial of degree m}m∈≥1
and a constant c depending only on M and N such that (M/(Em)M) divides c·(N/(Em)N) for all but finitely many m∈Z≥1.
Let n≥ 2 be an integer and let M and N be a finitely generated torsion Λ(On)-modules.
Then the following three statements are equivalent.
1. We have charΛ(n)
O (M)⊃charΛ(n) O (N).
2. There exists a complete discrete valuation ring O which is finite flat over O such that we have the inclusion
charΛ(n)O/(l)(MO/(l)MO)⊃char
Λ(n)O/(l)(NO/(l)NO) for all but finitely many (l)∈ L(On)(MO)∩ L(On)(NO).
Remark 2.4. 1. Not only the above theorem, but also a lot of basic properties on linear elements and linear specializations are developed in [O2,§3]. We refer to [O2,
§3] for some properties possibly used implicitly later in this paper.
2. Concerning known other results on the technique of specialization, the referee in- formed the author that, for the case of modules of rank two, there are a similar version of Theorem 2.3 (1) proved by Plater and an equivalent version of Lemma 2.5 (1) by Nekovar and Plater (cf. [NP]).
Under the above preparation, we return to the proof of Theorem 2.1. For a finitely generated Λ(On)-moduleM, we denote byMnullthe largest pseudo-null Λ(On)-submodule of M.
Lemma 2.5. Let C be a discrete R-module with continuous GΣ-action such that C∨ is a finitely generated torsion R-module. Suppose that R is isomorphic to Λ(On). For each complete discrete valuation ring O finite flat overO, the following statements follows:
1. Suppose that n= 1 (R is isomorphic to O[[X]]). The numerator and the denomi- nator of the ratio Hi(GΣ,CO[f])
Hi(GΣ,CO)[f] is bounded when f ∈ O[[X]] runs over elements prime to the characteristic ideal of (CO)∨. Similarly, for each v ∈ Σ\ {∞}, the ratio Hi(Dv,CO[f])
Hi(Dv,CO)[f] is bounded whenf ∈ O[[X]] varies as above.
2. Suppose that n≥2. Then, there exist pseudo-null R-modules YΣi(O) and ZΣi(O) such that the Pontrjagin duals of the kernel and the cokernel of Hi(GΣ,CO[l])−→
Hi(GΣ,CO)[l]are pseudo-nullΛ(On)/(l)-modules for everyl∈ L(On)((CO∨)null⊕YΣi(O)⊕ ZΣi(O)). Similarly, for each v ∈ Σ \ {∞}, there exist pseudo-null R-modules Yvi(O) and Zvi(O) such that the Pontrjagin duals of the kernel and the coker- nel of Hi(Dv,CO[l])−→ Hi(Dv,CO)[l] are pseudo-null Λ(On)/(l)-modules for every l∈ L(On)((CO∨)null⊕Yvi(O)⊕Zvi(O)).
Proof. For both of the case n= 1 and the case n≥2, we prove our statement only for the cohomology of the semi-global Galois groupGΣ. The case for the cohomology ofDv
is done in the same way and will be omitted.
First, let us discuss the first assertion (the case with n = 1). Let C0 be the O[[X]]- submodule ofC such that (C/C0)∨ is the largest finite submodule of C∨. Fix a complete discrete valuation ring O which is finite flat over O. Note that CO0 :=C0⊗OO is the O[[X]]-submodule ofCO :=C ⊗OOsuch that (CO/CO0)∨ is the largest finite submodule of CO∨. SinceHi(GΣ,CO/CO0) is a finite group for each i, we put:
ai(O) :=Hi(GΣ,CO/CO0).
For eachf ∈ O[[X]] prime to the characteristic ideal of (CO)∨, we consider the following commutative diagram:
· · · −−−→ Hi(GΣ,CO0) −−−→ Hi(GΣ,CO) −−−→ Hi(GΣ,CO/CO0) −−−→ · · ·
×f
⏐⏐
⏐⏐×f ⏐⏐×f
· · · −−−→ Hi(GΣ,CO0) −−−→ Hi(GΣ,CO) −−−→ Hi(GΣ,CO/CO0) −−−→ · · ·
8
By decomposing the two long exact sequences of the diagram into short exact sequences and by applying the snake lemma, we obtain the following inequality for eachi≥0 :
1≤ Hi(GΣ,CO)[f]
Hi(GΣ,CO0)[f] ≤ai(O). (2)
Since (CO0)∨ has no finiteO[[X]]-submodule andf is prime to the characteristic ideal of (CO0)∨, we have:
0−→ CO0[f]−→ CO0 −−→ C×f 0O −→0. We also put
bi(O) :=the largest finite O[[X]]-module ofHi(GΣ,CO0)∨. This gives us the following inequality for each i≥0 :
1≤ Hi(GΣ,CO0)[f]
Hi(GΣ,CO0[f]) ≤bi−1(O), (3)
where b−1(O) is defined to be 1. We have the following short exact sequence for each f ∈ O[[X]] prime to the characteristic ideal of C∨ :
0−→ CO0[f]−→ CO[f]−→(CO/CO0)[f]−→0.
Since the number ofO-submodules ofCO/CO0is finite, we note thatHi(GΣ,(CO/CO0)[f]) is bounded whenf varies. We define:
ci(O) := the maximum ofHi(GΣ,(CO/CO0)[f]) whenf varies.
We have the following inequality for each i≥0 : ci−1(O)
ci(O) ≤ Hi(GΣ,CO0[f])
Hi(GΣ,CO[f]) ≤ci−1(O), (4)
wherec−1(O) is defined to be 1. By adding three inequalities (2),(3) and (4), we obtain the following inequality :
ci−1(O)
ci(O) ≤ Hi(GΣ,CO)[f]
Hi(GΣ,CO[f]) ≤ai(O)bi−1(O)ci−1(O). (5)
Hence Hi(GΣ,CO)[f]
Hi(GΣ,CO[f]) is bounded when f ∈ O[[X]] runs elements prime to the charac- teristic ideal of (CO)∨. This proves the first assertion of the lemma.
Next, we consider the casen≥2. LetC0be a Λ(On)-submodule ofCsuch that (C/C0)∨ ∼= (C∨)null. We consider the following commutative diagram:
· · · −−−→ Hi(GΣ,CO0) −−−→ Hi(GΣ,CO) −−−→ Hi(GΣ,CO/CO0) −−−→ · · ·
×l
⏐⏐
⏐⏐×l ⏐⏐×l
· · · −−−→ Hi(GΣ,CO0) −−−→ Hi(GΣ,CO) −−−→ Hi(GΣ,CO/CO0) −−−→ · · · For finitely generated torsionR-modulesMandN, havingR-linear mapM −→Nwhose kernel and cokernel are pseudo-nullR-modules is an equivalence relation called pseudo- isomorphism. We writeM ∼N whenM andN are pseudo-isomorphic. By decomposing
the two long exact sequences of the above diagram into short exact sequences and by applying the snake lemma, we obtain the following statement for each i≥0:
(Hi(GΣ,CO)[l])∨ ∼(Hi(GΣ,CO0)[l])∨ ifl∈ L(On)(YΣi(O)), (6)
where
YΣi(O) =
Hi(GΣ,CO/CO0)∨⊕Hi−1(GΣ,CO/CO0)∨ ifi≥1, H0(GΣ,CO/CO0)∨ ifi= 0,
For a linear element l∈ Λ(On) prime to the characteristic ideal ofC∨, we have the short exact sequence 0−→ CO0[l]−→ CO0 −→ C×l O0 −→0. This implies that
Hi(GΣ,CO0[l])∨ ∼(Hi(GΣ,CO0)[l])∨ ifl∈ L(On)(ZΣi(O)), (7)
where
ZΣi(O) =
Hi−1(GΣ,CO0)∨)null ifi≥1,
0 ifi= 0.
Finally, the sequence 0−→ CO0[l]−→ CO[l]−→ (CO/CO0)[l]−→0 implies that we have the sequence for each i≥0 :
Hi(GΣ,CO0[l])∨ ∼Hi(GΣ,CO[l])∨ ifl∈ L(On)((CO∨)null) (8)
By combining (6), (7) and (8), we see that (Hi(GΣ,CO)[l])∨ and Hi(GΣ,CO[l])∨ is pseudo-isomorphic to each other for every l ∈ L(On)((CO∨)null⊕YΣi(O)⊕ZΣi(O)). This completes the proof of the second assertion for the cohomologies of GΣ. For the coho- mologies ofDv, the proof is done exactly in the same way by putting :
Yvi(O) =
Hi(Dv,CO/CO0)∨⊕Hi−1(Dv,CO/CO0)∨ ifi≥1, H0(Dv,CO/CO0)∨ ifi= 0, and
Zvi(O) =
Hi−1(Dv,CO0)∨)null ifi≥1,
0 ifi= 0.
Let us return to the proof of Theorem 2.1.
Proof of Theorem 2.1. As in the proof of Lemma 2.5, we prove our statement only for the cohomology of the semi-global Galois groupGΣ. The case for the cohomology ofDv is done in the same way and will be omitted.
We apply Theorem 2.3 to the following modules:
M =H0(GΣ,C)∨⊕H2(GΣ,C)∨⊕(CGÊ)∨, N =H1(GΣ,C)∨⊕ C∨
We will show that M is pseudo-isomorphic to N as a Λ(On)-module by induction with respect to n. Let us assume that n= 1 at first. For a complete discrete valuation ring O finite flat overO,MO (resp. NO) is isomorphic toH0(GΣ,CO)∨⊕H2(GΣ,CO)∨⊕ ((CO)GÊ)∨ (resp. H1(GΣ,CO)∨⊕(CO)∨). Let f be any element inO[[X]]. The groups
10
(MO/(f)MO) and (NO/(f)NO) are finite when f is prime to the characteristic ideal of (CO)∨. Lemma 2.5 immediately implies the following claim:
Claim 2.6. Let us introduce the following rational numbers:
rf := (MO/(f)MO)/(NO/(f)NO), sf :=
H0(GΣ,CO[f])⊕H2(GΣ,CO[f])⊕(COGÊ[f])
H1(GΣ,CO[f])⊕ CO[f] . Then the numerator and the denominator of rf/sf is bounded when f runs elements prime to the characteristic ideal of (CO)∨.
By the Global Euler-Poincar´e characteristic formula for finite GΣ-modules (cf. [NSW, Chap. VIII]), we have
0≤i≤2
Hi(GΣ, A)(−1)i = A AGÊ. (9)
This shows that sf = 1 for every f ∈ O[[X]] prime to the characteristic ideal of (CO)∨. Thus, rf = (MO/(f)MO)/(NO/(f)NO) is bounded when f varies in O[[X]]. By the first half of Theorem 2.3, we conclude that M andN are pseudo-isomorphic to each other.
Next, let n≥2 and assume that Theorem 2.1 is proved for n−1. We defineMO and NO to be
MO =MO ⊕YΣ0(O)⊕ZΣ0(O)⊕YΣ2(O)⊕ZΣ2(O), NO =NO⊕YΣ1(O)⊕ZΣ1(O).
By Lemma 2.5 and inductive assumption,MO /(l)MO is pseudo-isomorphic toNO /(l)NO
as Λ(On)/(l)-module for everyl∈ L(On)(MO)∩ L(On)(NO ). Hence, the last half of Theorem 2.3 implies thatMO is pseudo-isomorphic toNO as Λ(On)-module. Since MO (resp. NO ) is pseudo-isomorphic toM (resp. N) over Λ(On) by definition, this completes the proof for the cohomologies ofGΣ. For the cohomologies of Dv, we apply the local Euler-Poincar´e characteristic formula for finite Dv-modules (cf. [NSW, Chap. VII]) in place of the Global Euler-Poincar´e characteristic formula whenn= 1. Since the proof forn= 1 and n≥2 is done in the same way, we omit it.
3. Proof of Theorem 1.6
In this section, we will complete the proof of Theorem 1.6 by using the results in
§2. Throughout the section, we suppose that R is isomorphic to O[[X1,· · · , Xn]] with O the ring of integers of a finite extension of Qp and n a natural number. We fix an injection between two free latticesT −→ T
