aa r X i v : . [ m a t h . N T ] M a y ON EXTRA ZEROS OF P -ADIC L -FUNCTIONS: THE CRYSTALLINE CASEDenis Benois April 2013
Abstract.
We formulate a conjecture about extra zeros of p -adic L -functions at near central pointswhich generalizes the conjecture formulated in [Ben2]. We prove that this conjecture is compatiblewith Perrin-Riou’s theory of p -adic L -functions. Namely, using Nekov´aˇr’s machinery of Selmer com-plexes we prove that our L -invariant appears as an additional factor in the Bloch-Kato type formulafor special values of Perrin-Riou’s module of L -functions. Nous avons toutefois suppos´e pour simplifier queles op´erateurs − ϕ et − p − ϕ − sont inversibleslaissant les autres cas, pourtant extrˆemementint´eressant pour plus tard. Introduction to Chapter III of [PR2]
Table of contents
Introduction §
1. Preliminaries1.1. ( ϕ, Γ)-modules1.2. Crystalline representations §
2. The exponential map2.1. The Bloch-Kato exponential map2.2. The large exponential map §
3. The L -invariant3.1. Definition of the L -invariant3.2. L -invariant and the large exponential map §
4. Special values of p -adic L -functions4.1. The Bloch-Kato conjecture4.2. p -adic L -functions §
5. The module of p -adic L -functions5.1. The Selmer complex5.2. The module of p -adic L -functionsAppendix. Galois cohomology of p -adic representations DENIS BENOIS
Introduction0.1. Extra zeros.
Let M be a pure motive over Q . Assume that the complex L -function L ( M, s )of M extends to a meromorphic function on the whole complex plane C . Fix an odd prime p . It isexpected that one can construct p -adic analogs of L ( M, s ) interpolating p -adically algebraic partsof its special values. This program was realised and the corresponding p -adic L -functions wereconstructed in many cases, but the general theory remains conjectural. In [PR2], Perrin-Riouformulated precise conjectures about the existence and arithmetic properties of p -adic L -functionsin the case then the p -adic realisation V of M is crystalline at p . Let D cris ( V ) denote the filteredDieudonn´e module associated to V by the theory of Fontaine. Let D be a subspace of D cris ( V )of dimension d + ( V ) = dim Q p V c =1 stable under the action of ϕ . One says that D is regular if onecan associate to D a p -adic analog of the six-term exact sequence of Fontaine and Perrin-Riou(see [PR2] for exact definition). Fix a lattice T of V stable under the action of the Galois groupand a lattice N of a regular module D . Perrin-Riou conjectured that one can associate to thisdata a p -adic L -function L p ( T, N, s ) satisfying some explicit interpolation property. Let r denotethe order of vanishing of L ( M, s ) at s = 0 and let L ∗ ( M,
0) = lim s → s − r L ( M, s ) . Then at s = 0the interpolation property writeslim s → L p ( T, N, s ) s r = E ( V, D ) R V,D ( ω V,N ) L ∗ ( M, R M, ∞ ( ω M ) . Here R M, ∞ ( ω M ) (resp. R V,D ( ω V,N )) is the determinant of the Beilinson (resp. the p -adic regula-tor) computed in some compatible bases ω M and ω V,V and E ( V, D ) is an Euler-like factor givenby E ( V, D ) = det(1 − p − ϕ − | D ) det(1 − ϕ | D cris ( V ) /D ) . If either D ϕ = p − = 0 or ( D cris ( V ) /D ) ϕ =1 = 0 we have E ( V, D ) = 0 and the order of vanishing of L p ( N, T, s ) should be > r.
In this case we say that L ( T, N, s ) has an extra zero at s = 0 . The samephenomenon occurs in the case then V is semistable and non-crystalline at p . An architypicalexample is provided by elliptic curves having split multiplicative reduction [MTT]. Assume that0 is a critical point for L ( M, s ) and that H ( M ) = H ( M ∗ (1)) = 0 . In [Ben2] using the theoryof ( ϕ, Γ)-modules we associated to each regular D an invariant L ( V, D ) ∈ Q p generalising bothGreenberg’s L -invariant [G] and Fontaine-Mazur’s L -invariant [M]. This allows to formulate aquite general conjecture about the behavior of p -adic L -functions at extra zeros in the spirit of[G]. To the best of our knowledge this conjecture is actually proved in the following cases:1) Kubota-Leopoldt p -adic L -functions [FG], [GK]. Here the L -invariant can be interpretedin terms of Gross p -adic regulator[Gs].2) Modular forms of even weight [K], [GS], [S]. Here the L -invariant coincides with Fontaine-Mazur’s L ( f ).3) Modular forms of odd weight [Ben3]. The associated p -adic representation V is eithercrystalline or potentially crystalline at p and we do need the theory of ( ϕ, Γ)-modules to definethe L -invariant.4) Symmetric square of an elliptic curve having either split multiplicative reduction [Ro] or agood ordinary reduction (Dasgupta, work in progress). Here V is ordinary and the L -invariantreduces to Greenberg’s construction [G].5) Symmetric powers of CM-modular forms [HL]. In this paper we generalise the conjecture from [Ben2] to the non critical point case. Assume that V is crystalline at p . Then the weight argument shows that E ( V, D ) can vanish only if wt ( M ) = 0 or −
2. In particular, we expect that the interpolation
XTRA ZEROS 3 factor does not vanish at s = 0 if wt ( M ) = − p -adic L -function can not have anextra zero at the central point in the good reduction case. To fix ideas assume that wt ( M ) − M has no subquotients isomorphic to Q (1) . Then D is regular if and only if theassociated p -adic regultor map r V,D : H f ( V ) −→ D cris ( V ) / (Fil D cris ( V ) + D )is an isomorphism. The semisimplicity of ϕ : D cris ( V ) −→ D cris ( V ) (which conjecturally alwaysholds) allows to decompose D into a direct sum D = D − ⊕ D ϕ = p − . Under some mild assumptions (see 3.1.2 and 4.1.2 below) we associate to D an L -invariant L ( V, D ) which is a direct generalization of the main construction of [Ben2]. The Beilinson-Deligne conjecture predicts that L ( M, s ) does not vanish at s = 0 and that L ( M ∗ (1) , s ) has azero of order r = dim Q p H f ( V ) at s = 0 . We propose the following conjecture:
Extra zero conjecture.
Let D be a regular subspace of D cris ( V ) and let e = dim Q p ( D ϕ = p − ) . Then1) The p -adic L -function L p ( T, N, s ) has a zero of order e at s = 0 and L ∗ p ( T, N, R V,D ( ω V,N ) = − L ( V, D ) E + ( V, D ) L ( M, R M, ∞ ( ω M ) .
2) Let D ⊥ denote the orthogonal complement to D under the canonical duality D cris ( V ) × D cris ( V ∗ (1)) −→ Q p . The p -adic L -function L p ( T ∗ (1) , N ⊥ , s ) has a zero of order e + r where r = dim Q H f ( V ) at s = 0 and L ∗ p ( T ∗ (1) , N ⊥ , R V ∗ (1) ,D ⊥ ( ω V ∗ (1) ,N ⊥ ) = L ( V, D ) E + ( V ∗ (1) , D ⊥ ) L ∗ ( M ∗ (1) , R M ∗ (1) , ∞ ( ω M ∗ (1) ) . In the both cases E + ( V, D ) = E + ( V ∗ (1) , D ⊥ ) = det(1 − p − ϕ − | D − ) det(1 − p − ϕ − | D cris ( V ∗ (1))) . Remarks. E + ( V, D ) is obtained from E ( V, D ) by excluding zero factors. It can be also writtenin the form E + ( V, D ) = , E ∗ p ( V,
1) det Q p (cid:18) − p − ϕ − − ϕ | D − (cid:19) where E p ( V, t ) = det(1 − ϕt | D cris ( V )) is the Euler factor at p and E ∗ p ( V, t ) = E p ( V, t ) (cid:18) − tp (cid:19) − e .
2) Assume that H f ( V ) = 0. Since H f ( V ∗ (1)) should also vanish by the weight reason, ourconjecture in this cases reduces to the conjecture 2.3.2 from [Ben2].3) The regularity of D supposes that the localisation H f ( V ) −→ H f ( Q p , V ) is injective.Jannsen’s conjecture (precised by Bloch and Kato) says that the p -adic realisation map H f ( M ) ⊗ The last condition is not really essential and can be suppressed DENIS BENOIS Q p −→ H f ( V ) is an isomorphism. The composition H f ( M ) −→ H f ( Q p , V ) of these two maps isessentially the syntomic regulator. Its injectivity seems to be a difficult open problem. In the last part of the paper we showthat our extra zero conjecture is compatible with the Main Conjecture of Iwasawa theory as for-mulated in [PR2]. The main technical tool here is the descent theory for Selmer complexes [Ne2].We hope that the approach to Perrin-Riou’s theory based on the formalism of Selmer complexescan be of independent interest.For a profinite group G and a continuous G -module X we denote by C • c ( G, X ) the standardcomplex of continuous cochains. Let S be a finite set of primes containing p . Denote by G S the Galois group of the maximal algebraic extension of Q unramified outside S ∪ {∞} . Set R Γ S ( X ) = C • c ( G S , X ) and R Γ( Q v , X ) = C • c ( G v , X ), where G v is the absolute Galois group of Q v . Let Γ be the Galois group of the cyclotomic p -extension Q ( ζ p ∞ ) / Q ) , Γ = Gal( Q ( ζ p ∞ ) / Q ( ζ p )and ∆ = Gal( Q ( ζ p ) / Q ). Let Λ(Γ) = Z p [[Γ]] denote the Iwasawa algebra of Γ . Each Λ(Γ)-module X decomposes into the direct sum of its isotypical components X = ⊕ η ∈ ˆ∆ X ( η ) and we denoteby X ( η ) the component which corresponds to the trivial character η . Set Λ = Λ(Γ) ( η ) . Let H denote the algebra of power series with coefficients in Q p which converge on the open unitdisk. We will denote again by H the associated large Iwasawa algebra H (Γ ) . In this paper weconsider only the trivial character component of the module of p -adic L -functions because itis sufficient for applications to trivial zeros, but in the general case the construction is exactlythe same. We keep notation and assumptions of section 0.2 Assume that the weak Leopoldtconjecture holds for ( V, η ) and ( V ∗ (1) , η ) . We consider global and local Iwasawa cohomology R Γ Iw ,S ( T ) = R Γ S (Λ(Γ) ⊗ Z p T ) ι ) and R Γ Iw ( Q v , T ) = R Γ( Q v , (Λ(Γ) ⊗ Z p T ) ι ) where ι is thecanonical involution on Λ(Γ) . Let D be a regular submodule of D cris ( V ) . For each non archimedianplace v we define a local condition at v in the sense of [Ne2] as follows. If v = p we use theunramified local condition which is defined by R Γ ( η )Iw ,f ( Q v , N, T ) = R Γ ( η )Iw ,f ( Q v , T ) = h T I v ⊗ Λ ι f v −→ T I v ⊗ Λ ι i where I v is the inertia subgroup at v and f v is the geometric Frobenius. If v = p we define R Γ ( η )Iw ,f ( Q v , N, T ) = ( N ⊗ Λ)[ − . The derived version of the large exponential map Exp
V,h , h ≫ R Γ ( η )Iw ,f ( Q v , N, T ) −→ R Γ ( η )Iw ( Q p , T ) ⊗ H . Therefore we have a diagram R Γ ( η )Iw ,S ( T ) ⊗ Λ H / / ⊕ v ∈ S R Γ ( η )Iw ( Q v , T ) ⊗ Λ H (cid:18) ⊕ v ∈ S R Γ ( η )Iw ,f ( Q v , N, T ) (cid:19) ⊗ Λ H . O O Let R Γ ( η )Iw ,h ( D, V ) denote the Selmer complex associated to this data. By definition it sits in thedistinguished triangle R Γ ( η )Iw ,S ( D, V ) −→ (cid:18) R Γ ( η )Iw ,S ( V ) ⊕ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ,f ( Q v , D, V ) (cid:19)(cid:19) ⊗ H −→ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ( Q v , V ) (cid:19) ⊗ H . (0.1) XTRA ZEROS 5
Define∆ Iw ,h ( N, T ) = det − (cid:18) R Γ ( η )Iw ,S ( T ) ⊕ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ,f ( Q v , N, T ) (cid:19)(cid:19) ⊗ det Λ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ( Q v , T ) (cid:19) . Our results can be summarized as follows (see Theorems 5.1.3, 5.2.5 and Corollary 5.2.7).
Theorem 1.
Assume that L ( V, D ) = 0 . Theni) The cohomology R i Γ ( η )Iw ,h ( D, V ) are H -torsion modules for all i .ii) R i Γ ( η )Iw ,h ( D, V ) = 0 for i = 2 , and R Γ ( η )Iw ,h ( D, V ) ≃ (cid:0) H ( Q ( ζ p ∞ ) , V ∗ (1)) ∗ (cid:1) ( η ) ⊗ Λ H . iii) The complex R Γ ( η )Iw ,h ( D, V ) is semisimple i.e. for each i the natural map R i Γ ( η )Iw ,h ( D, V ) Γ −→ R i Γ ( η )Iw ,h ( D, V ) Γ is an isomorphism. Assume that L ( V, D ) = 0 . Let K be the field of fractions of H . Then Theorem 1 togetherwith (0.1) define an injective map i V, Iw ,h : ∆ Iw ,h ( N, T ) −→ K and the module of p -adic L -functions is defined as L ( η )Iw ,h ( N, T ) = i V, Iw ,h (∆ Iw ,h ( N, T )) ⊂ K . Let γ be a fixed generator of Γ . Choose a generator f ( γ −
1) of the free Λ-module L ( η )Iw ,h ( N, T )and define a meromorphic p -adic function L Iw ,h ( T, N, s ) = f ( χ ( γ ) s − , where χ : Γ −→ Z ∗ p is the cyclotomic character. Theorem 2.
Assume that L ( V, D ) = 0 . Then1) The p -adic L -function L Iw ,h ( T, N, s ) has a zero of order e = dim Q p ( D ϕ = p − ) at s = 0 .
2) One has L ∗ Iw ,h ( T, N, R V,D ( ω T,N ) ∼ p Γ( h ) d + ( V ) L ( V, D ) E + ( V, D )
III ( T ∗ (1)) Tam ω M ( T ) H S ( V /T ) H S ( V ∗ (1) /T ∗ (1)) , where III ( T ∗ (1)) is the Tate-Shafarevich group of Bloch-Kato [BK] and Tam ω M ( T ) is the productof local Tamagawa numbers of T . Remarks.
1) Using the compatibility of Perrin-Riou’s theory with the functional equation weobtain analogous results for the L p ( T ∗ (1) , N ⊥ , s ) (see section 5.2.9).2) It D cris ( V ) ϕ =1 = D cris ( V ) ϕ = p − = 0 the phenomenon of extra zeros does not appear, L ( V, D ) = 1 and Theorem 2 was proved in [PR2], Theorem 3.6.5. We remark that even in thiscase our proof is different. We compare the leading term of L ∗ Iw ,h ( T, N, s ) with the trivialisation
DENIS BENOIS i ω M ,p : ∆ EP ( T ) −→ Q p of the Euler-Poincar´e line ∆ EP ( T ) (see [F3]) and show that in compatiblebases one has L ∗ Iw ,h ( T, N, R V,D ( ω V,N ) ∼ p Γ( h ) d + ( V ) L ( V, D ) E + ( V, D ) i ω M ,p (∆ EP ( T )) (0.2)(see Theorem 5.2.5). Now Theorem 2 follows from the well known computation of i ω M ,p (∆ EP ( T ))in terms of the Tate-Shafarevich group and Tamagawa numbers ([FP], Chapitre II).3) Let E/ Q be an elliptic curve having good reduction at p. Consider the p -adic representation V = Sym ( T p ( E )) ⊗ Q p , where T p ( E ) is the p -adic Tate module of E. It is easy to see that D = D cris ( V ) ϕ = p − is one dimensional. In this case some versions of Theorem 2 were provedin [PR3] and [D] with an ad hoc definition of the L -invariant. Remark that p -adic L -functionsattached to the symmetric square of a newform were constructed by Dabrowski and Delbourgo[DD].4) Another approach to Iwasawa theory in the non-ordinary case was developped by Pottharstin [Pt1], [Pt2]. Pottharst uses the formalism of Selmer complexes but works with local conditionscoming from submodules of the ( ϕ, Γ)-module associated to V rather then with the large expo-nential map. This approach has many advantages, in particular it allows to develop an interestingtheory for representations which are not necessarily crystalline. Nevertheless it seems that thelarge exponential map is crucial for the study of extra zeros at least in the good reduction case.5) The Main conjecture of Iwasawa theory [PR2], [C2] says that the analytic p -adic L -function L p ( N, T, s ) multiplied by a simple explicit Γ-factor depending on h can be written in the form L p ( N, T, s ) = f ( χ ( γ ) s −
1) for an appropriate generator f ( γ −
1) of L ( η )Iw ,h ( N, T ) . Therefore themain conjecture implies Bloch-Kato style formulas for special values of L p ( N, T, s ) . We remarkthat the Bloch-Kato conjecture predicts that L ∗ ( M, R M, ∞ ( ω M ) ∼ p III ( T ∗ (1)) Tam ω M ( T ) H S ( V /T ) H S ( V ∗ (1) /T ∗ (1))and therefore Theorem 2 implies the compatibility of our extra zero conjecture with the Mainconjecture. Note that this also follows directly from (0.2) if we use the formalism of Fontaineand Perrin-Riou [F3] to formulate Bloch-Kato conjectures. The organisation of the paper is as follows. In § ϕ, Γ)-moduleswhich is the main technical tool in our definition of the L -invariant. We also give the derivedversion of computation of Galois cohomology in terms of ( ϕ, Γ)-modules. This follows easily fromthe results of Herr [H1] and Liu [Li] and the proofs are placed in Appendix. Similar results canbe found in [Pt1], [Pt2]. In § L -invariant is constructed is section 3.1. Insection 3.2 we relate this construction to the derivative of the large exponential map. This resultplays a key role in the proof of Theorem 2. The extra zero conjecture is formulated in §
4. In § Acknowledgements.
I am very grateful to Jan Nekov´aˇr and Daniel Delbourgo for severalinteresting discussions and comments concerning this work.
XTRA ZEROS 7 §
1. Preliminaries1.1. ( ϕ, Γ) -modules.1.1.1. The Robba ring (see [Ber1],[C3]). In this section K is a finite unramified extension of Q p with residue field k K , O K its ring of integers, and σ the absolute Frobenius of K . Let K analgebraic closure of K , G K = Gal( ¯ K/K ) and C the completion of K. Let v p : C −→ R ∪ {∞} denote the p -adic valuation normalized so that v p ( p ) = 1 and set | x | p = (cid:16) p (cid:17) v p ( x ) . Write B ( r, p -adic annulus B ( r,
1) = { x ∈ C | r | x | < } . As usually, µ p n denotes the group of p n -th roots of unity. Fix a system of primitive roots of unity ε = ( ζ p n ) n > , ζ p n ∈ µ p n such that ζ pp n = ζ p n − for all n . Set K n = K ( ζ p n ), K ∞ = S ∞ n =0 K n , H K = Gal( ¯ K/K ∞ ), Γ = Gal( K ∞ /K )and denote by χ : Γ −→ Z ∗ p the cyclotomic character.Set e E + = lim ←− x x p O C / p O C = { x = ( x , x , . . . , x n , . . . ) | x pi = x i ∀ i ∈ N } . Let ˆ x n ∈ O C be a lifting of x n . Then for all m > x p n m + n converges to x ( m ) =lim n →∞ ˆ x p n m + n ∈ O C which does not depend on the choice of liftings. The ring e E + equipped withthe valuation v E ( x ) = v p ( x (0) ) is a complete local ring of characteristic p with residue field ¯ k K .Moreover it is integrally closed in his field of fractions e E = Fr( e E + ).Let e A = W ( e E ) be the ring of Witt vectors with coefficients in e E . Denote by [ ] : e E −→ W ( e E )the Teichmuller lift. Any u = ( u , u , . . . ) ∈ e A can be written in the form u = ∞ X n =0 [ u p − n ] p n . Set π = [ ε ] − A + K = O K [[ π ]] and denote by A K the p -adic completion of A + K [1 /π ].Let e B = e A [1 /p ], B K = A K [1 /p ] and let B denote the completion of the maximal unramifiedextension of B K in e B . Set A = B ∩ e A , e A + = W ( E + ), A + = e A + ∩ A and B + = A + [1 /p ] . Allthese rings are endowed with natural actions of the Galois group G K and Frobenius ϕ .Set A K = A H K and B K = A K [1 /p ] . Remark that Γ and ϕ act on B K by τ ( π ) = (1 + π ) χ ( τ ) − , τ ∈ Γ ϕ ( π ) = (1 + π ) p − . For any r > e B † ,r = (cid:26) x ∈ e B | lim k → + ∞ (cid:18) v E ( x k ) + prp − k (cid:19) = + ∞ (cid:27) . Set B † ,r = B ∩ e B † ,r , B † ,rK = B K ∩ B † ,r , B † = ∪ r> B † ,r A † = A ∩ B † and B † K = ∪ r> B † ,rK .It can be shown that for any r > p − B † ,rK = ( f ( π ) = X k ∈ Z a k π k | a k ∈ K and f is holomorphic and bounded on B ( r, ) . Define B † ,r rig ,K = ( f ( π ) = X k ∈ Z a k π k | a k ∈ K and f is holomorphic on B ( r, ) . DENIS BENOIS
Set R ( K ) = ∪ r > p − B † ,r rig ,K and R + ( K ) = R ( K ) ∩ K [[ π ]] . It is not difficult to check that these ringsare stable under Γ and ϕ. To simplify notations we will write R = R ( Q p ) and R + = R + ( Q p ) . As usual, we set t = log(1 + π ) = ∞ X n =1 ( − n +1 π n n ∈ R Note that ϕ ( t ) = pt and τ ( t ) = χ ( γ ) t , τ ∈ Γ . ( ϕ, Γ) -modules (see [F2], [CC1]). Let A be either B † K or R ( K ) . A ( ϕ, Γ)-module over Ais a finitely generated free A -module D equipped with semilinear actions of ϕ and Γ commutingto each other and such that the induced linear map ϕ : A ⊗ ϕ D −→ D is an isomorphism. Sucha module is said to be etale if it admits a A † K -lattice N stable under ϕ and Γ and such that ϕ : A † K ⊗ ϕ N −→ N is an isomorphism. The functor D
7→ R ( K ) ⊗ B † K D induces an equivalencebetween the category of etale ( ϕ, Γ)-modules over B † K and the category of ( ϕ, Γ)-modules over R ( K ) which are of slope 0 in the sense of Kedlaya’s theory ([Ke] and [C5], Corollary 1.5). ThenFontaine’s classification of p -adic representations [F2] together with the main result of [CC1] leadto the following statement. Proposition 1.1.3. i) The functor D † : V D † ( V ) = ( B † ⊗ Q p V ) H K establishes an equivalence between the category of p -adic representations of G K and the categoryof etale ( ϕ, Γ) -modules over B † K . ii) The functor D † rig ( V ) = R ( K ) ⊗ B † K D † ( V ) gives an equivalence between the category of p -adic representations of G K and the category of ( ϕ, Γ) -modules over R ( K ) of slope .Proof. see [C4], Proposition 1.7. ( ϕ, Γ) -modules (see [H1], [H2], [Li]). Fix a generator γ of Γ. If D is a( ϕ, Γ)-module over A , we denote by C ϕ,γ ( D ) the complex C ϕ,γ ( D ) : 0 f −→ D −→ D ⊕ D g −→ D −→ f ( x ) = (( ϕ − x, ( γ − x ) and g ( y, z ) = ( γ − y − ( ϕ − z. Set H i ( D ) = H i ( C ϕ,γ ( D )) . A short exact sequence of ( ϕ, Γ)-modules0 −→ D ′ −→ D −→ D ′′ −→ −→ H ( D ′ ) −→ H ( D ) −→ H ( D ′′ ) −→ H ( D ′ ) −→ · · · −→ H ( D ′′ ) −→ . Proposition 1.1.5.
Let V be a p -adic representation of G K . Then the complexes R Γ( K, V ) , C ϕ,γ ( D † ( V )) and C ϕ,γ ( D † rig ( V )) are isomorphic in the derived category of Q p -vector spaces D ( Q p ) . Proof.
This is a derived version of Herr’s computation of Galois cohomology [H1]. The proof isgiven in the Appendix, Propositions A.3 and Corollary A.4.
XTRA ZEROS 9
Recall that Λ denotes the Iwasawa algebra of Γ , ∆ = Gal( K /K )and Λ(Γ) = Z p [∆] ⊗ Z p Λ. Let ι : Λ(Γ) −→ Λ(Γ) denote the involution defined by ι ( g ) = g − ,g ∈ Γ . If T is a Z p -adic representation of G K , then the induced module Ind K ∞ /K ( T ) is isomorphicto (Λ(Γ) ⊗ Z p T ) ι and we set R Γ Iw ( K, T ) = R Γ( K, Ind K ∞ /K ( T )) . Write H i Iw ( K, T ) for the Iwasawa cohomology H i Iw ( K, T ) = lim ←− cor Kn/Kn − H i ( K n , T ) . Recall that there are canonical and functorial isomorphisms R i Γ Iw ( K, T ) ≃ H i Iw ( K, T ) , i > , R Γ Iw ( K, T ) ⊗ L Λ(Γ) Z p [ G n ] ≃ R Γ( K n , T )(see [Ne2], Proposition 8.4.22). The interpretation of the Iwasawa cohomology in terms of ( ϕ, Γ)-modules was found by Fontaine (unpublished but see [CC2]). We give here the derived version ofthis result. Let ψ : B −→ B be the operator defined by the formula ψ ( x ) = p ϕ − (cid:0) Tr B /ϕ ( B ) ( x ) (cid:1) . We see immediately that ψ ◦ ϕ = id . Moreover ψ commutes with the action of G K and ψ ( A † ) = A † . Consider the complexes C Iw ,ψ ( T ) : D ( T ) ψ − −−−→ D ( T ) ,C † Iw ,ψ ( T ) : D † ( T ) ψ − −−−→ D † ( T ) . Proposition 1.1.7. i) The complexes R Γ Iw ( K, T ) , C Iw ,ψ ( T ) and C † Iw ,ψ ( T ) are naturally iso-morphic in the derived category D (Λ(Γ)) of Λ(Γ) -modules.Proof.
See Proposition A.7 and Corollary A.8. ( ϕ, Γ) -modules of rank . Recall the computation of the cohomology of ( ϕ, Γ)-modulesof rank 1 following Colmez [C4]. As in op. cit. , we consider the case K = Q p and put R = B † rig , Q p and R + = B +rig , Q p . The differential operator ∂ = (1 + π ) ddπ acts on R and R + . If δ : Q ∗ p −→ Q ∗ p isa continuous character, we write R ( δ ) for the ( ϕ, Γ)-module R e δ defined by ϕ ( e δ ) = δ ( p ) e δ and γ ( e δ ) = δ ( χ ( τ )) e δ . Let x denote the character induced by the natural inclusion of Q p in L and | x | the character defined by | x | = p − v p ( x ) . Proposition 1.1.9.
Let δ : Q ∗ p −→ Q ∗ p be a continuous character. Then:i) H ( R ( δ )) = (cid:26) Q p t m if δ = x − m , k ∈ N ii) dim Q p ( H ( R ( δ ))) = (cid:26) δ ( x ) = x − m , m > δ ( x ) = | x | x m , k > , . iii) Assume that δ ( x ) = x − m , m > . The classes cl( t m , e δ and cl(0 , t m ) e δ form a basis of H ( R ( x − m )) .iv) Assume that δ ( x ) = | x | x m , m > . Then H ( R ( | x | x m )) , m > is generated by cl( α m ) and cl( β m ) where α m = ( − m − ( m − ∂ m − (cid:18) π + 12 , a (cid:19) e δ , (1 − ϕ ) a = (1 − χ ( γ ) γ ) (cid:18) π + 12 (cid:19) ,β m = ( − m − ( m − ∂ m − (cid:18) b, π (cid:19) e δ , (1 − ϕ ) (cid:18) π (cid:19) = (1 − χ ( γ ) γ ) b Proof.
See [C4], sections 2.3-2.5. cris and B dR (see [F1], [F4]). Let θ : A + −→ O C be the map given by theformula θ ∞ X n =0 [ u n ] p n ! = ∞ X n =0 u (0) n p n . It can be shown that θ is a surjective ring homomorphism and that ker( θ ) is the principal idealgenerated by ω = p − P i =0 [ ǫ ] i/p . By linearity, θ can be extended to a map θ : ˜ B + −→ C . The ring B +dR is defined to be the completion of ˜ B + for the ker( θ )-adic topology: B +dR = lim ←− n ˜ B + /ker ( θ ) n . This is a complete discrete valuation ring with residue field C equipped with a natural action of G K . Moreover, there exists a canonical embedding ¯ K ⊂ B +dR . The series t = ∞ P n =0 ( − n − π n /n con-verges in the topology of B +dR and it is easy to see that t generates the maximal ideal of B +dR . TheGalois group acts on t by the formula g ( t ) = χ ( g ) t. Let B dR = B +dR [ t − ] be the field of fractionsof B +dR . This is a complete discrete valuation field equipped with a G K -action and an exhaustiveseparated decreasing filtration Fil i B dR = t i B +dR . As G K -module, Fil i B dR / Fil i +1 B dR ≃ C ( i ) and B G K dR = K. Consider the
P D -envelope of A + with a respect to the map θ A PD = A + (cid:20) ω , ω , . . . , ω n n ! , . . . (cid:21) and denote by A +cris its p -adic completion. Let B +cris = A +cris ⊗ Z p Q p and B cris = B +cris [ t − ] . Then B cris is a subring of B dR endowed with the induced filtration and Galois action. Moreover, itis equipped with a continuous Frobenius ϕ , extending the map ϕ : A + −→ A + . One has ϕ ( t ) = p t. (see [F5], [Ber1], [Ber2]).Let L be a finite extension of Q p . Denote by K its maximal unramified subextension. A filteredDieudonn´e module over L is a finite dimensional K - vector space M equipped with the followingstructures: XTRA ZEROS 11 • a σ -semilinear bijective map ϕ : M −→ M ; • an exhaustive decreasing filtration (Fil i M L ) on the L -vector space M L = L ⊗ K M. A K -linear map f : M −→ M ′ is said to be a morphism of filtered modules if • f ( ϕ ( d )) = ϕ ( f ( d )) , for all d ∈ M ; • f (Fil i M L ) ⊂ Fil i M ′ L , for all i ∈ Z . The category MF ϕL of filtered Dieudonn´e modules is additive, has kernels and cokernels butis not abelian. Denote by the vector space K with the natural action of σ and the filtrationgiven by Fil i = (cid:26) K, if i , , if i > . Then is a unit object of MF ϕL i.e. M ⊗ ≃ ⊗ M ≃ M for any M .If M is a one dimensional Dieudonn´e module and d is a basis vector of M , then ϕ ( d ) = αv forsome α ∈ K . Set t N ( M ) = v p ( α ) and denote by t H ( M ) the unique filtration jump of M. If M isof an arbitrary finite dimension d , set t N ( M ) = t N ( d ∧ M ) and t H ( M ) = t H ( d ∧ M ) . A Dieudonn´emodule M is said to be weakly admissible if t H ( M ) = t N ( M ) and if t H ( M ′ ) t N ( M ′ ) for any ϕ -submodule M ′ ⊂ M equipped with the induced filtration. Weakly admissible modules form asubcategory of MF L which we denote by MF ϕ,fL . If V is a p -adic representation of G L , define D dR ( V ) = ( B dR ⊗ V ) G L . Then D dR ( V ) is a L -vector space equipped with the decreasing filtration Fil i D dR ( V ) = (Fil i B dR ⊗ V ) G L . Onehas dim L D dR ( V ) dim Q p ( V ) and V is said to be de Rham if dim L D dR ( V ) = dim Q p ( V ) . Analogously one defines D cris ( V ) = ( B cris ⊗ V ) G L . Then D cris ( V ) is a filtered Dieudonn´e moduleover L of dimension dim K D cris ( V ) dim Q p ( V ) and V is said to be crystalline if the equalityholds here. In particular, for crystalline representations one has D dR ( V ) = D cris ( V ) ⊗ K L. Bythe theorem of Colmez-Fontaine [CF], the functor D cris establishes an equivalence between thecategory of crystalline representations of G L and MF ϕ,fL . Its quasi-inverse V cris is given by V cris ( D ) = Fil ( D ⊗ K B cris ) ϕ =1 . An important result of Berger ([Ber 1], Theorem 0.2) says that D cris ( V ) can be recovered fromthe ( ϕ, Γ)-module D † rig ( V ) . The situation is particularly simple if If L/ Q p is unramified. In thiscase set D + ( V ) = ( V ⊗ Q p B + ) H K and D +rig ( V ) = R + ( K ) ⊗ B + K D + ( V ) . Then D cris ( V ) = (cid:18) D +rig ( V ) (cid:20) t (cid:21)(cid:19) Γ (see [Ber2], Proposition 3.4). §
2. The exponential map2.1. The Bloch-Kato exponential map ([BK], [Ne1], [FP]).
Let L be a finite extension of Q p . Recall thatwe denote by MF ϕL the category of filtered Dieudonn´e modules over L. If M is an object of MF ϕL ,define H i ( L, M ) = Ext i MF ϕL ( , M ) , i = 0 , . Remark that H ∗ ( L, M ) can be computed explicitly as the cohomology of the complex C • ( M ) : M f −→ ( M L / Fil M L ) ⊕ M where the modules are placed in degrees 0 and 1 and f ( d ) = ( d (mod Fil M L ) , (1 − ϕ ) ( d ))([Ne1],[FP]). Remark that if M is weakly admissible then each extension 0 −→ M −→ M ′ −→ −→ is weakly admissible too and we can write H i ( L, M ) = Ext i MF ϕ,fL ( , M ) . Let
Rep cris ( G K ) denote the category of crystalline representa-tions of G K . For any object V of Rep cris ( G K ) define H if ( K, V ) = Ext i Rep cris ( G K ) ( Q p (0) , V ) . An easy computation shows that H if ( K, V ) = H ( K, V ) , if i = 0,ker ( H ( K, V ) −→ H ( K, V ⊗ B cris )) , if i = 1,0 , if i > . Let t V ( K ) = D dR ( V ) / Fil D dR ( V ) denote the tangent space of V . The rings B dR and B cris arerelated to each other via the fundamental exact sequence0 −→ Q p −→ B cris f −→ B dR / Fil B dR ⊕ B cris −→ f ( x ) = ( x (mod Fil B dR ) , (1 − ϕ ) x ) (see [BK], § V andtaking cohomology one obtains an exact sequence0 −→ H ( K, V ) −→ D cris ( V ) −→ t V ( K ) ⊕ D cris ( V ) −→ H f ( K, V ) −→ . The last map of this sequence gives rise to the Bloch-Kato exponential mapexp
V,K : t V ( K ) ⊕ D cris ( V ) −→ H ( K, V ) . Following [F3] set R Γ f ( K, V ) = C • ( D cris ( V )) = h D cris ( V ) f −→ t V ( K ) ⊕ D cris ( V ) i . From the classification of crystalline representations in terms of Dieudonn´e modules it followsthat the functor V cris induces natural isomorphisms r iV,p : R i Γ f ( K, V ) −→ H if ( K, V ) , i = 0 , . The composite homomorphism t K ( V ) ⊕ D cris ( V ) −→ R Γ f ( K, V ) r V,p −−→ H ( K, V )coincides with the Bloch-Kato exponential map exp
V,K ([Ne1], Proposition 1.21). Γ f ( K, V ) −→ R Γ( K, V ) . Let g : B • −→ C • be a morphism of complexes.We denote by Tot • ( g ) the complex Tot n ( g ) = C n − ⊕ B n with differentials d n : Tot n ( g ) −→ Tot n +1 ( g ) defined by the formula d n ( c, b ) = (( − n g n ( b ) + d n − ( c ) , d n ( b )) . It is well knownthat if 0 −→ A • f −→ B • g −→ C • −→ f induces a quasi XTRA ZEROS 13 isomorphism A • ∼ → Tot • ( g ) . In particular, tensoring the fundamental exact sequence with V , weobtain an exact sequence of complexes0 −→ R Γ( K, V ) −→ C • c ( G K , V ⊗ B cris ) f −→ C • c ( G K , ( V ⊗ ( B dR / Fil B dR )) ⊕ ( V ⊗ B cris )) −→ R Γ( K, V ) ∼ → Tot • ( f ) . Since R Γ f ( K, V ) coincides tautologicallywith the complex C c ( G K , V ⊗ B cris ) f −→ C c ( G K , ( V ⊗ ( B dR /F B dR )) ⊕ ( V ⊗ B cris ))we obtain a diagram R Γ( K, V ) ∼ / / Tot • ( f ) R Γ f ( K, V ) O O g g which defines a morphism R Γ f ( K, V ) −→ R Γ( K, V ) in D ( Q p ) (see [BF], Proposition 1.17). Re-mark that the induced homomorphisms R i Γ f ( K, V ) −→ H i ( K, V ) ( i = 0 ,
1) coincide with thecomposition of r iV,p with natural embeddings H if ( K, V ) −→ H i ( K, V ) . ( ϕ, Γ) -modules. In this subsection we define an analog of theexponential map for crystalline ( ϕ, Γ)-modules. See [Na] for a more general setting. Let K/ Q p be an unramified extension. If D is a ( ϕ, Γ)-module over R ( K ) define D cris ( D ) = ( D [1 /t ]) Γ . It can be shown that D cris ( D ) is a finite dimensional K -vector space equipped with a naturaldecreasing filtration Fil i D cris ( D ) and a semilinear action of ϕ . One says that D is crystalline ifdim K ( D cris ( D )) = rg( D ) . From [Ber4], Th´eor`eme A it follows that the functor D
7→ D cris ( D ) is an equivalence between thecategory of crystalline ( ϕ, Γ)-modules and MF ϕK . Remark that if V is a p -adic representation of G K then D cris ( V ) = D cris ( D † rig ( V )) and V is crystalline if and only if D † rig ( V ) is.Let D be a ( ϕ, Γ)-module. To any cocycle α = ( a, b ) ∈ Z ( C ϕ,γ ( D )) one can associate theextension 0 −→ D −→ D α −→ R ( K ) −→ D α = D ⊕ R ( K ) e, ( ϕ − e = a, ( γ − e = b. As usual, this gives rise to an isomorphism H ( D ) ≃ Ext R ( R ( K ) , D ) . We say that cl( α ) iscrystalline if dim K ( D cris ( D α )) = dim K ( D cris ( D )) + 1 and define H f ( D ) = { cl( α ) ∈ H ( D ) | cl( α ) is crystalline } (see [Ben2], section 1.4.1). If D is crystalline (or more generally potentially semistable ) one hasa natural isomorphism H ( K, D cris ( D )) −→ H f ( D ) . Set t D = D cris ( D ) / Fil D cris ( D ) and denote by exp D : t D ⊕ D cris ( D ) −→ H ( D ) the compositionof this isomorphism with the projection t D ⊕ D cris ( D ) −→ H ( K, D cris ( D )) and the embedding H f ( D ) ֒ → H ( D ) . Assume that K = Q p . To simplify notation we will write D m for R ( | x | x m ) and e m for itscanonical basis. Then D cris ( D m ) is the one dimensional Q p -vector space generated by t − m e m .As in [Ben2], we normalise the basis (cl( α m ) , cl( β m )) of H ( D m ) putting α ∗ m = (1 − /p ) cl( α m )and β ∗ m = (1 − /p ) log( χ ( γ )) cl( β m ) . Proposition 2.1.5. i) H f ( D m ) is the one-dimensional Q p -vector space generated by α ∗ m .ii) The exponential map exp D m : t D m −→ H ( D m ) sends t − m w m to − α ∗ m . Proof.
This is a reformulation of [Ben2], Proposition 1.5.8 ii).
In this section p is an odd prime number, K is a finite unramified extensionof Q p and σ the absolute Frobenius acting on K. Recall that K n = K ( ζ p n ) and K ∞ = ∪ ∞ n =1 K n . We set Γ = Gal( K ∞ /K ) , Γ n = Gal( K ∞ /K n ) and ∆ = Gal( K /K ) . Let Λ = Z p [[Γ ]] andΛ(Γ) = Z p [∆] ⊗ Z p Λ . We will consider the following operators acting on the ring K [[ X ]] of formalpower series with coefficients in K : • The ring homomorphism σ : K [[ X ]] −→ K [[ X ]] defined by σ ∞ X i =0 a i X i ! = ∞ X i =0 σ ( a i ) X i ; • The ring homomorphism ϕ : K [[ X ]] −→ K [[ X ]] defined by ϕ ∞ X i =0 a i X i ! = ∞ X i =0 σ ( a i ) ϕ ( X ) i , ϕ ( X ) = (1 + X ) p − . • The differential operator ∂ = (1 + X ) ddX . One has ∂ ◦ ϕ = pϕ ◦ ∂. • The operator ψ : K [[ X ]] −→ K [[ X ]] defined by ψ ( f ( X )) = 1 p ϕ − X ζ p =1 f ((1 + X ) ζ − . It is easy to see that ψ is a left inverse to ϕ, i.e. that ψ ◦ ϕ = id . • An action of Γ given by γ ∞ X i =0 a i X i ! = ∞ X i =0 a i γ ( X ) i , γ ( X ) = (1 + X ) χ ( γ ) − . Remark that these formulas are compatible with the definitions from sections 1.1.1 and 1.1.6.Fix a generator γ ∈ Γ and define H = { f ( γ − | f ∈ Q p [[ X ]] is holomorphic on B (0 , } , H (Γ) = Z p [∆] ⊗ Z p H . εV,n . It is well known that Z p [[ X ]] ψ =0 is a free Λ-module generated by(1 + X ) and the operator ∂ is bijective on Z p [[ X ]] ψ =0 . If V is a crystalline representation of G K put D ( V ) = D cris ( V ) ⊗ Z p Z p [[ X ]] ψ =0 . Let Ξ εV,n : D ( V ) Γ n [ − −→ R Γ f ( K n , V ) be the mapdefined by Ξ εV,n ( α ) = (cid:26) p − n ( P nk =1 ( σ ⊗ ϕ ) − k α ( ζ p k − , − α (0)) if n > K /K (cid:0) Ξ εV, ( α ) (cid:1) if n = 0 . XTRA ZEROS 15
An easy computation shows that Ξ εV, : D cris ( V )[ − −→ R Γ f ( K, V ) is given by the formula Ξ εV, ( a ) = 1 p ( − ϕ − ( a ) , − ( p − a ) . In particular, it is homotopic to the map a
7→ − (0 , (1 − p − ϕ − ) a ) . WriteΞ εV,n : D ( V ) −→ R Γ( K n , V ) = t V ( K n ) ⊕ D cris ( V ) D cris ( V ) /V G K denote the homomorphism induced by Ξ εV,n . ThenΞ εV, ( a ) = − (0 , (1 − p − ϕ − ) a ) (mod D cris ( V ) /V G K ) . If D cris ( V ) ϕ =1 = 0 the operator 1 − ϕ is invertible on D cris ( V ) and we can writeΞ εV, ( a ) = (cid:18) − p − ϕ − − ϕ a, (cid:19) (mod D cris ( V ) /V G K ) . (2.1)For any i ∈ Z let ∆ i : D ( V ) −→ D cris ( V )(1 − p i ϕ ) D cris ( V ) ⊗ Q p ( i ) be the map given by∆ i ( α ( X )) = ∂ i α (0) ⊗ ε ⊗ i (mod (1 − p i ϕ ) D cris ( V )) . Set ∆ = ⊕ i ∈ Z ∆ i . If α ∈ D ( V ) ∆=0 , then by [PR1], Proposition 2.2.1 there exists F ∈ D cris ( V ) ⊗ Q p Q p [[ X ]] which converges on the open unit disk and such that (1 − ϕ ) F = α. A short computationshows thatΞ εV,n ( α ) = p − n (( σ ⊗ ϕ ) − n ( F )( ζ p n − ,
0) (mod D cris ( V ) /V G K ) , if n > As Z p [[ X ]] [1 /p ] is a principal idealdomain and H is Z p [[ X ]] [1 /p ]-torsion free, H is flat. Thus C † Iw ,ψ ( V ) ⊗ L Λ Q p H (Γ) = C † Iw ,ψ ( V ) ⊗ Λ Q p H (Γ) = h H (Γ) ⊗ Λ Q p D † ( V ) ψ − −−−→ H (Γ) ⊗ Λ Q p D † ( V ) i . By proposition 1.1.7 on has an isomorphism in D ( H (Γ)) R Γ Iw ( K, V ) ⊗ L Λ Q p H (Γ) ≃ C † Iw ,ψ ( V ) ⊗ Λ Q p H (Γ) . The action of H (Γ) on D † ( V ) ψ =1 induces an injection H (Γ) ⊗ Λ Q p D † ( V ) ψ =1 ֒ → D † rig ( V ) ψ =1 . Composing this map with the canonical isomorphism H ( K, V ) ≃ D † ( V ) ψ =1 we obtain a map H (Γ) ⊗ Λ Q p H ( K, V ) ֒ → D † rig ( V ) ψ =1 . For any k ∈ Z set ∇ k = t∂ − k = t ddt − k. An easy inductionshows that ∇ k − ◦ ∇ k − ◦ · · · ◦ ∇ = t k ∂ k . Fix h > − h D cris ( V ) = D cris ( V ) and V ( − h ) G K = 0 . For any α ∈ D ( V ) ∆=0 define Ω εV,h ( α ) = ( − h − log χ ( γ ) p ∇ h − ◦ ∇ h − ◦ · · · ∇ ( F ( π )) , where F ∈ H ( V ) is such that (1 − ϕ ) F = α. It is easy to see that Ω εV,h ( α ) ∈ D +rig ( V ) ψ =1 . In[Ber3] Berger shows that Ω εV,h ( α ) ∈ H (Γ) ⊗ Λ Q p D † ( V ) ψ =1 and therefore gives rise to a map E xp εV,h : D ( V ) ∆=0 [ − −→ R Γ Iw ( K, V ) ⊗ L Λ Q p H (Γ)Let Exp εV,h : D ( V ) ∆=0 −→ H (Γ) ⊗ Λ Q p H ( K, V )denote the map induced by E xp εV,h in degree 1. The following theorem is a reformulation of theconstruction of the large exponential map given by Berger in [Ber3]. Theorem 2.2.4.
Let E xp εV,h,n : D ( V ) ∆=0 Γ n [ − −→ R Γ Iw ( K, V ) ⊗ L Λ Q p Q p [ G n ] . denote the map induced by E xp εV,h . Then for any n > the following diagram in D ( Q p [ G n ]) iscommutative: D ( V ) ∆=0Γ n [ − E xp εV,h,n / / Ξ εV,n (cid:15) (cid:15) R Γ Iw ( K, V ) ⊗ L Λ Q p Q p [ G n ] ≃ (cid:15) (cid:15) R Γ f ( K n , V ) ( h − / / R Γ( K n , V ) . In particular,
Exp εV,h coincides with the large exponential map of Perrin-Riou.Proof.
Passing to cohomology in the previous diagram one obtains the diagram D ( V ) ∆=0 Exp εV,h −−−−→ H ( Γ ) ⊗ Λ Q p H ( K, V ) Ξ εV,n y y pr V,n D dR /K n ( V ) ⊕ D cris ( V ) ( h − V,Kn −−−−−−−−−−→ H ( K n , V )which is exactly the definition of the large exponential map. Its commutativity is proved in[Ber3], Theorem II.13. Now, the theorem is an immediate consequence of the following remark.Let D be a free A -module and let f , f : D [ − −→ K • be two maps from D [ −
1] to a complexof A -modules such that the induced maps h ( f ) and h ( f ) : D −→ H ( K • ) coincide. Then f and f are homotopic. Remark.
The large exponential map was first constructed in [PR1]. See [C1] and [Ben1] foralternative constructions and [PR4], [Na1] and [Ri] for generalizations. §
3. The L -invariant3.1.Definition of the L -invariant.3.1.1. Preliminaries. Let S be a finite set of primes of Q containing p and G S the Galoisgroup of the maximal algebraic extension of Q unramified outside S ∪ {∞} . For each place v wedenote by G v the decomposition at v group and by I v and f v the inertia subgroup and Frobeniusautomorphism respectively. Let V be a pseudo-geometric p -adic representation of G S . Thismeans that the restriction of V on the decomposition group at p is a de Rham representation.Following Greenberg, for any v / ∈ { p, ∞} set R Γ f ( Q v , V ) = h V I v − f v −−−→ V I v i , where the terms are placed in degrees 0 and 1 (see [F3], [BF]). Note that there is a naturalquasi-isomorphism R Γ f ( Q v , V ) ≃ C • c ( G v /I v , V I v ) . Note that R Γ( Q v , V ) = H ( Q v , V ) and R Γ f ( Q v , V ) = H f ( Q v , V ) where H f ( Q v , V ) = ker( H ( Q v , V ) −→ H ( Q ur v , V )) . XTRA ZEROS 17
For v = p the complex R Γ f ( Q v , V ) was defined in §
2. To simplify notation write H iS ( V ) = H i ( G S , V ) for the continuous Galois cohomology of G S with coefficients in V . The Bloch-Kato’sSelmer group of V is defined as H f ( V ) = ker H S ( V ) −→ M v ∈ S H ( Q v , V ) H f ( Q v , V ) ! . We also set H f, { p } ( V ) = ker H S ( V ) −→ M v ∈ S −{ p } H ( Q v , V ) H f ( Q v , V ) . From the Poitou-Tate exact sequence one obtains the following exact sequence relating thesegroups (see for example [PR2], Lemme 3.3.6)0 −→ H f ( V ) −→ H f, { p } ( V ) −→ H ( Q p , V ) H f ( Q p , V ) −→ H f ( V ∗ (1)) . We also have the following formula relating dimensions of Selmer groups (see [FP], II, 2.2.2)dim Q p H f ( V ) − dim Q p H f ( V ∗ (1)) − dim Q p H S ( V ) + dim Q p H S ( V ∗ (1)) =dim Q p t V ( Q p ) − dim Q p H ( R , V ) . Set d ± ( V ) = dim Q p ( V c = ± ) , where c denotes the complex conjugation. Assume that V satisfies the following conditions C1) H f ( V ∗ (1)) = 0. C2) H S ( V ) = H S ( V ∗ (1)) = 0. C3) V is crystalline at p and ϕ : D cris ( V ) −→ D cris ( V ) is semisimple at 1 and p − . C4) D cris ( V ) ϕ =1 = 0 . C5)
The localisation map loc p : : H f ( V ) −→ H f ( Q p , V )is injective.These conditions appear naturally in the following situation. Let X be a proper smooth varietyover Q . Let H ip ( X ) denote the p -adic etale cohomology of X . Consider the Galois representations V = H ip ( X )( m ) . By Poincar´e duality together with the hard Lefschetz theorem we have H ip ( X ) ∗ ≃ H ip ( X ) ( i )and thus V ∗ (1) ≃ V ( i + 1 − m ) . The Beilinson conjecture (in the formulation of Bloch and Kato)predict that H f ( V ∗ (1)) = 0 if w − . This corresponds to the hope that there are no nontrivial extensions of Q (0) by motives of weight > . If X has a good reduction at p , then V is crystalline [Fa] and the semisimplicity of ϕ is awell known (and difficult) conjecture. By a result of Katz and Messing [KM] D cris ( V ) ϕ =1 = 0 can occur only if i = 2 m . Therefore up to eventually replace V by V ∗ (1) the conditions C1,C3-4) conjecturally hold with except the weight − i = 2 m − . The condition D cris ( V ) ϕ =1 = 0 imples that the exponential map t V ( Q p ) −→ H f ( Q p , V ) is anisomorphism and we denote by log V its inverse. The composition of the localisation map loc p with the Bloch-Kato logarithm r V : H f ( V ) −→ t V ( Q p )coincides conjecturally with the p -adic (syntomic) regulator. We remark that if H ( Q v , V ) = 0 forall v = p (and therefore H f ( Q v , V ) = 0 for all v = p ) then loc p is injective for all m = i/ , i/ H S ( V ) = 0, then V contains a trivial subextension V = Q p (0) k . For Q p (0) our theorydescribes the behavior of the Kubota-Leopoldt p -adic L -function and is well known. Thereforewe can assume that H S ( V ) = 0 . Applying the same argument to V ∗ (1) we can also assume that H S ( V ∗ (1)) = 0 . From our assumptions we obtain an exact sequence0 −→ H f ( V ) −→ H f, { p } ( V ) −→ H ( Q p , V ) H f ( Q p , V ) −→ . (3.1)Moreover dim Q p H f ( V ) = dim Q p t V ( Q p ) − d + ( V ) , dim Q p H f, { p } ( V ) = d − ( V ) + dim Q p H ( Q p , V ∗ (1)) . (3.2) In the remainder of this § we assume that V satisfies C1-5) . Definition (Perrin-Riou) . 1) A ϕ -submodule D of D cris ( V ) is regular if D ∩ Fil D cris ( V ) = 0 and the map r V,D : H f ( V ) −→ D cris ( V ) / (Fil D cris ( V ) + D ) induced by r V is an isomorphism.2) Dually, a ( ϕ, N ) -submodule D of D cris ( V ∗ (1)) is regular if D +Fil D cris ( V ∗ (1)) = D cris ( V ∗ (1)) and the map D ∩ Fil D cris ( V ∗ (1)) −→ H ( V ) ∗ induced by the dual map r ∗ V : Fil D cris ( V ∗ (1)) −→ H ( V ) ∗ is an isomorphism. It is easy to see that if D is a regular submodule of D cris ( V ), then D ⊥ = Hom( D cris ( V ) /D, D cris ( Q p (1))is a regular submodule of D cris ( V ∗ (1)) . From (3.2) we also obtain thatdim D = d + ( V ) , dim D ⊥ = d − ( V ) = d + ( V ∗ (1)) . Let D ⊂ D cris ( V ) be a regular subspace. As in [Ben2] we use the semisimplicity of ϕ todecompose D into the direct sum D = D − ⊕ D ϕ = p − . which gives a four step filtration { } ⊂ D − ⊂ D ⊂ D cris ( V ) . XTRA ZEROS 19
Let D and D − denote the ( ϕ, Γ)-submodules associated to D and D − by Berger’s theory, thus D = D cris ( D ) , D − = D cris ( D − ) . Set W = gr D † rig ( V ) . Thus we have two tautological exact sequences0 −→ D −→ D † rig ( V ) −→ D ′ −→ , −→ D − −→ D −→ W −→ . Note the following properties of cohomology of these modules:a) The natural maps H ( D − ) −→ H ( D ) and H ( D ) −→ H ( D † rig ( V )) = H ( Q p , V ) areinjective. This follows from the observation that D cris ( D ′ ) ϕ =1 = 0 by C4) . Since H ( D ′ ) =Fil D cris ( D ′ ) ϕ =1 ([Ben2], Proposition 1.4.4) we have H ( D ′ ) = 0. The same argument works for W .b) H f ( D − ) = H ( D − ). In particular the exponential map exp D − : D − −→ H ( D − ) isan isomorphism. This follows from the computation of dimensions of H ( D − ) and H f ( D − ).Namely, since D ϕ =1 − = D ϕ = p − − = { } the Euler-Poincar´e characteristic formula [Li] togetherwith Poincar´e duality givedim Q p H ( D − ) = rg( D − ) − dim Q p H ( D − ) − dim Q p H ( D ∗− ( χ )) = dim Q p ( D − ) . On the other hand since Fil D − = D − ∩ Fil D cris ( V ) = { } one has dim Q p H f ( D − ) = dim Q p ( D − ) by [Ben2], Corollary 1.4.5.c) The exponential map exp D : D −→ H f ( D ) is an isomorphism. This follows from Fil D = { } and D ϕ =1 = { } . The regularity of D is equivalent to the decomposition H f ( Q p , V ) = H f ( V ) ⊕ H f ( D ) . (3.3)Since loc p is injective by C5) , the localisation map H f, { p } ( V ) −→ H ( Q p , V ) is also injective.Let κ D : H f, { p } ( V ) −→ H ( Q p , V ) H f ( D )denote the composition of this map with the canonical projection. Lemma 3.1.4. i) One has H f ( Q p , V ) ∩ H ( D ) = H f ( D ) . ii) κ D is an isomorphism.Proof. i) Since H ( D ′ ) = 0 we have a commutative diagram with exact rows and injective colomns0 −−−−→ H f ( D ) −−−−→ H f ( Q p , V ) −−−−→ H f ( D ′ ) y y y −−−−→ H ( D ) −−−−→ H ( Q p , V ) −−−−→ H ( D ′ ) . This gives i).ii) Since H f ( D ) ⊂ H f ( Q p , V ) one has ker( κ D ) ⊂ H f ( Q p , V ) . One the other hand (3.3) showsthat κ D is injective on H f ( V ) . Thus ker( κ D ) = { } . On the other hand, because dim Q p H f ( D ) =dim Q p ( D ) we have dim Q p H ( Q p , V ) H f ( D ) ! = d − ( V ) + dim Q p H ( Q p , V ∗ (1)) . Comparing this with (3.2) we obtain that κ D is an isomorphism. Set e = dim Q p ( D ϕ = p − ). The ( ϕ, Γ)-module W satisfiesFil D cris ( W ) = 0 , D cris ( W ) ϕ = p − = D cris ( W ) . (Recall that D cris ( W ) = D ϕ = p − .) The cohomology of such modules was studied in detail in[Ben2], Proposition 1.5.9 and section 1.5.10. Namely, H ( W ) = 0 , dim Q p H ( W ) = 2 e anddim Q p ( W ) = e. There exists a canonical decomposition H ( W ) = H f ( W ) ⊕ H c ( W )of H ( W ) into the direct sum of H f ( W ) and some canonical space H c ( W ) . Moreover there existcanonical isomorphisms i D,f : D cris ( W ) ≃ H f ( W ) , i D,c : D cris ( W ) ≃ H c ( W ) . These isomorphisms can be described explicitly. By Proposition 1.5.9 of [Ben2] W ≃ e ⊕ i =1 D m i , where D m i = R ( | x | x m i ) , m i > . By Proposition 2.1.5 H f ( D m ) is generated by α ∗ m and H c ( D m )is the subspace generated by β ∗ m (see also Proposition 1.1.9). Then i D m ,f ( x ) = xα ∗ m , i D m ,c ( x ) = xβ ∗ m . Since H ( W ) = 0 and H ( D − ) = 0 we have exact sequences0 −→ H ( D − ) −→ H ( D ) −→ H ( W ) −→ , −→ H f ( D − ) −→ H f ( D ) −→ H f ( W ) −→ . Since H f ( D − ) = H ( D ) we obtain that H ( D ) H f ( D ) ≃ H ( W ) H f ( W ) . Let H ( D, V ) denote the inverse image of H ( D ) /H f ( D ) by κ D . Then κ D induces an isomorphism H ( D, V ) ≃ H ( D ) H f ( D ) . XTRA ZEROS 21
By Lemma 3.1.4 the localisation map H ( D, V ) −→ H ( W ) is well defined and injective. Hence,we have a diagram D cris ( W ) iD,f ∼ / / H f ( W ) H ( D, V ) ρ D,f O O / / ρ D,c (cid:15) (cid:15) H ( W ) p D,f O O p D,c (cid:15) (cid:15) D cris ( W ) iD,c ∼ / / H c ( W ) , where ρ D,f and ρ D,c are defined as the unique maps making this diagram commute. FromLemma 3.1.4 iii) it follows that ρ D,c is an isomorphism. The following definition generalise(in the crystalline case) the main construction of [Ben2] where we assumed in addition that H f ( V ) = 0 . Definition.
The determinant L ( V, D ) = det (cid:16) ρ D,f ◦ ρ − D,c | D cris ( W ) (cid:17) will be called the L -invariant associated to V and D . L -invariant and the large exponential map.3.2.1. Derivation of the large exponential map. In this section we interpret L ( V, D ) interms of the derivative of the large exponential map. This interpretation is crucial for the proofof the main theorem of this paper. Recall that H ( Q p , H (Γ) ⊗ Q p V ) = H (Γ) ⊗ Λ(Γ) H ( Q p , V )injects into D † rig ( V ) . Set F H ( Q p , H (Γ) ⊗ Q p V ) = D ∩ H ( Q p , H (Γ) ⊗ Q p V ) ,F − H ( Q p , H (Γ) ⊗ Q p V ) = D − ∩ H ( Q p , H (Γ) ⊗ Q p V ) . As in section 2.2 we fix a generator γ ∈ Γ. The following result is a strightforward generalisationof [Ben3], Proposition 2.2.2. For the convenience of the reader we give here the proof which isthe same as in op. cit. modulo obvious modifications.
Proposition 3.2.2.
Let D be an admissible subspace of D cris ( V ) . For any a ∈ D ϕ = p − let α ∈ D ( V ) be such that α (0) = a. Theni) There exists a unique β ∈ F H ( Q p , H (Γ) ⊗ V ) such that ( γ − β = Exp εV,h ( α ) . ii) The composition map δ D,h : D ϕ = p − −→ F H ( Q p , H (Γ) ⊗ V ) −→ H ( W ) δ D,h ( a ) = β (mod H ( D − )) is given explicitly by the following formula: δ D,h ( α ) = − ( h − (cid:18) − p (cid:19) − (log χ ( γ )) − i D,c ( α ) . Proof.
Since D cris ( V ) ϕ =1 = 0 , the operator 1 − ϕ is invertible on D cris ( V ) and we have a diagram D ( V ) ∆=0 Exp εV,h / / Ξ εV, (cid:15) (cid:15) H ( Q p , H (Γ) ⊗ V ) pr V (cid:15) (cid:15) D cris ( V ) ( h − V / / H ( Q p , V ) . where Ξ εV, ( α ) = 1 − p − ϕ − − ϕ α (0) (see (2.1)). If α ∈ D ϕ = p − ⊗ Z p [[ X ]] ψ =0 , then Ξ εV, ( α ) = 0 andpr V (cid:16) Exp εV,h ( α ) (cid:17) = 0 . On the other hand, as V G K = 0 the map (cid:16) H (Γ) ⊗ Λ Q p H ( Q p , V ) (cid:17) Γ −→ H ( Q p , V ) is injective. Thus there exists a unique β ∈ H (Γ) ⊗ Λ H ( Q p , T ) such that Exp εV,h ( α ) =( γ − β. Now take a ∈ D ϕ = p − and set f = a ⊗ ℓ (cid:18) (1 + X ) χ ( γ ) − X (cid:19) , where ℓ ( g ) = 1 p log (cid:18) g p ϕ ( g ) (cid:19) . An easy computation shows that X ζ p =1 ℓ (cid:18) ζ χ ( γ ) (1 + X ) χ ( γ ) − ζ (1 + X ) − (cid:19) = 0 . Thus f ∈ D ϕ = p − ⊗ Z p [[ X ]] ψ =0 . Write α in the form α = (1 − ϕ ) (1 − γ ) ( a ⊗ log( X )) . ThenΩ
V,h ( α ) = ( − h − log χ ( γ ) p t h ∂ h (( γ −
1) ( a log( π )) = log χ ( γ ) p ( γ − β where β = ( − h − t h ∂ h ( a log( π )) = ( − h − at h ∂ h − (cid:18) ππ (cid:19) . This implies immediately that β ∈ D . On the other hand D ϕ = p − = D cris ( W ) = ( W [1 /t ]) Γ andwe will write ˜ a for the image of a in W [1 /t ] . By [Ben2], sections 1.5.8-1.5.10 one has W ≃ e ⊕ i =1 D m i where D m = R ( | x | x m ) and we denote by e m the canonical base of D m . Then without lost ofgenerality we may assume that ˜ a = t − m i e m i for some i. Let ˜ β be denote the image of β in W ψ =1 and let h : W ψ =1 −→ H ( W ) be the canonical map furnished by Proposition 1.1.7. Recall that h ( ˜ β ) = cl( c, ˜ β ) where (1 − γ ) c = (1 − ϕ ) ˜ β. Then ˜ β = ( − h − t h − m i ∂ h log( π ). By Lemma 1.5.1of [CC1] there exists a unique b ∈ B † ,ψ =0 Q p such that ( γ − b = ℓ ( π ). This implies that(1 − γ ) ( t h − m i ∂ h b e m i ) = (1 − ϕ ) ( t h − m i ∂ h log( π ) e m i ) = ( − h − (1 − ϕ ) ˜ β. Thus c = ( − h − t h − m i ∂ h b e m i and res( ct m i − dt ) = ( − h − res( t h − ∂ h b dt ) e m i = 0 . Nextfrom the congruence ˜ β ≡ ( h − t − m i e m i (mod Q p [[ π ]] e m i ) . it follows that res( ˜ βt m i − dt ) =( h − e m i . Therefore by [Ben2], Corollary 1.5.6 we havecl( c, ˜ β ) = ( h − β m ) = ( h − p log χ ( γ ) i W,c ( a ) . XTRA ZEROS 23
On the other hand α (0) = a ⊗ ℓ (cid:18) (1 + X ) χ ( γ ) − X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 = a (cid:18) − p (cid:19) log( χ ( γ )) . These formulas imply that δ D,h ( α ) = ( h − (cid:18) − p (cid:19) − (log χ ( γ )) − i W,c ( α ) . and the proposition is proved. L -invariant. From the definition of H ( D, V ) and Lemma 3.1.4we immediately obtain that H ( Q p , V ) H f, { p } ( V ) + H ( D − ) ≃ H ( D ) H ( D, V ) + H ( D − ) ≃ H ( W ) H ( D, V ) . Thus, the map δ D,h constructed in Proposition 3.2.2 induces a map D ϕ = p − −→ H ( Q p , V ) H f, { p } ( V ) + H ( D − )which we will denote again by δ D,h . On the other hand, we have isomorphisms D ϕ = p − exp V ∼ → H f ( Q p , V )exp V, Q p ( D − ) ≃ H f ( Q p , V ) H ( D − ) ≃ H ( Q p , V ) H f, { p } ( V ) + H ( D − ) . Proposition 3.2.4.
Let λ D : D ϕ = p − −→ D ϕ = p − denote the homomorphism making the dia-gram D ϕ = p − δ D,h ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ λ D / / D ϕ = p − ( h − V w w ♦♦♦♦♦♦♦♦♦♦♦♦♦ H ( Q p , V ) H f, { p } ( V ) + H ( D − ) commute. Then det (cid:16) λ D | D ϕ = p − (cid:17) = (log χ ( γ )) − e (cid:18) − p (cid:19) − e L ( V, D ) . Proof.
The proposition follows from Proposition 3.2.2 and the following elementary fact. Let U = U ⊕ U be the decomposition of a vector space U of dimension 2 e into the direct sum oftwo subspaces of dimension e . Let X ⊂ U be a subspace of dimension e such that X ∩ U = { } . Consider the diagrams X p / / p (cid:15) (cid:15) U U/X U i o o U f > > ⑥⑥⑥⑥⑥⑥⑥⑥ U i O O g = = ③③③③③③③③ where p k and i k are induced by natural projections and inclusions. Then f = − g. Applying thisremark to U = H ( W ) , X = H ( D, V ), U = H f ( W ) , U = H c ( W ) and taking determinants weobtain the proposition. §
4. Special values of p -adic L -functions4.1. The Bloch-Kato conjecture.4.1.1. The Euler-Poincar´e line (see [F3], [FP],[BF]). Let V be a p -adic pseudo-geometricrepresentation of Gal( Q / Q ) . Thus V is a finite-dimensional Q p -vector space equipped with acontinuous action of the Galois group G S for a suitable finite set of places S containing p . Write R Γ S ( V ) = C • c ( G S , V ) and define R Γ S,c ( V ) = cone (cid:18) R Γ S ( V ) −→ ⊕ v ∈ S ∪{∞} R Γ( Q v , V ) (cid:19) [ − . Fix a Z p -lattice T of V stable under the action of G S and set ∆ S ( V ) = det − Q p R Γ S,c ( V ) and∆ S ( T ) = det − Z p R Γ S,c ( T ) . Then ∆ S ( T ) is a Z p -lattice of the one-dimensional Q p -vector space∆ S ( V ) which does not depend on the choice of T . Therefore it defines a p -adic norm on ∆ S ( V )which we denote by k · k S . Moreother, (∆ S ( V ) , k · k S ) does not depend on the choice of S. Moreprecisely, if Σ is a finite set of places which contains S , then there exists a natural isomorphism∆ S ( V ) −→ ∆ Σ ( V ) such that k · k Σ = k · k S . This allows to define the Euler-Poincar´e line ∆ EP ( V )as (∆ S ( V ) , k · k S ) where S is sufficiently large. Recall that for any finite place v ∈ S we defined R Γ f ( Q v , V ) = h V I v − f v −−−→ V I v i if v = p (cid:20) D cris ( V ) (pr , − ϕ ) −−−−−→ t V ( Q p ) ⊕ D cris ( V ) (cid:21) if v = p. At v = ∞ we set R Γ f ( R , V ) = [ V + −→ , where the first term is placed in degree 0 . Thus R Γ f ( R , V ) ∼ → R Γ( R , V ). For any v we have a canonical morphism loc p : R Γ f ( Q v , V ) −→ R Γ( Q v , V ) which can be viewed as a local condition in the sense of [Ne2]. Consider the diagram R Γ S ( V ) / / ⊕ v ∈ S ∪{∞} R Γ( Q v , V ) ⊕ v ∈ S ∪{∞} R Γ f ( Q v , V ) O O and define R Γ f ( V ) = cone (cid:18) R Γ S ( V ) ⊕ (cid:18) ⊕ v ∈ S ∪{∞} R Γ f ( Q v , V ) (cid:19) −→ ⊕ v ∈ S ∪{∞} R Γ( Q v , V ) (cid:19) [ − . Thus, we have a distinguished triangle R Γ f ( V ) −→ R Γ S ( V ) ⊕ (cid:18) ⊕ v ∈ S ∪{∞} R Γ f ( Q v , V ) (cid:19) −→ ⊕ v ∈ S ∪{∞} R Γ( Q v , V ) . (4.1)Set ∆ f ( V ) = det − Q p R Γ f ( V ) ⊗ det − Q p t V ( Q p ) ⊗ det Q p V + . It is easy to see that R Γ f ( V ) and ∆ f ( V ) do not depend on the choice of S . Consider thedistinguished triangle R Γ S,c ( V ) −→ R Γ f ( V ) −→ ⊕ v ∈ S ∪{∞} R Γ f ( Q v , V ) . XTRA ZEROS 25
Since det Q p R Γ f ( Q p , V ) ≃ det − Q p t V ( Q p ) and det Q p R Γ f ( R , V ) = det Q p V + tautologically, we ob-tain canonical isomorphisms ∆ f ( V ) ≃ det − Q p R Γ S,c ( V ) ≃ ∆ EP ( V ) . The cohomology of R Γ f ( V ) is as follows: R Γ f ( V ) = H S ( V ) , R Γ f ( V ) = H f ( V ) , R Γ f ( V ) ≃ H f ( V ∗ (1)) ∗ , R Γ f ( V ) = coker (cid:18) H S ( V ) −→ ⊕ v ∈ S H ( Q v , V ) (cid:19) ≃ H S ( V ∗ (1)) ∗ . (4.2)These groups seat in the following exact sequence:0 −→ R Γ( V ) −→ H S ( V ) −→ M v ∈ S H ( Q v , V ) H f ( Q v , V ) −→ R Γ f ( V ) −→ H S ( V ) −→ ⊕ v ∈ S H ( Q v , V ) −→ R Γ f ( V ) −→ . The L -function of V is defined as the Euler product L ( V, s ) = Y v E v ( V, ( N v ) − s ) − where E v ( V, t ) = (cid:26) det (cid:0) − f v t | V I v (cid:1) , if v = p det (1 − ϕt | D cris ( V )) if v = p. In this paper we treat motives in the formal sense andassume all conjectures about the category of mixed motives MM over Q which are necessary tostate the Bloch-Kato conjecture (see [F3], [FP]). Let M be a pure motive over Q and let M B and M dR denote its Betti and de Rham realisations respectively. Fix an odd prime p and denote by V = M p the p -adic realisation of M . Then one has comparision isomorphisms M B ⊗ Q C ∼ → M dR ⊗ Q C , (4.3) M B ⊗ Q Q p ∼ → V (4.4)which induce trivialisations Ω ( H, ∞ ) M : det Q M B ⊗ det − Q M dR −→ C , (4.5)Ω (´ et,p ) M : det Q p V ⊗ det − Q M B −→ Q p . (4.6)The complex conjugation acts on M B and V and decomposes the last isomorphism into ± partswhich we denote again by Ω (´ et,p ) M to simplify notationΩ (´ et,p ) M : det Q p V ± ⊗ det − Q M ± B −→ Q p . (4.7)The restriction of V on the decomposition group at p is a de Rham representation and D dR ( V ) ≃ M dR ⊗ Q Q p . The comparision isomorphism V ⊗ B dR ∼ → D dR ( V ) ⊗ B dR (4.8) induces a map e Ω ( H,p ) M : det Q p V ⊗ det − Q p D dR ( V ) −→ B dR . It is not difficult to see that there exists a finite extension L of d Q urp such that Im( ˜Ω H,pM ) ⊂ Lt t H ( V ) and we define Ω ( H,p ) M : det Q p V ⊗ det − Q p D dR ( V ) −→ L (4.9)by Ω ( H,p ) M = t − t H ( V ) e Ω ( H,p ) M . We remark that if V is crystalline at p then one can take L = d Q urp (see [PR2], Appendice C.2).Assume that the groups H i ( M ) = Ext i MM ( Q (0) , M ) are well defined and vanish for i = 0 , . Itshould be possible to define a Q -subspace H f ( M ) of H ( M ) consisting of ”integral” classes ofextensions which is expected to be finite dimensional. It is convenient to set H f ( M ) = H ( M ) . The conjectures of Tate and Jannsen predict that the regulator map induces isomorphisms H if ( M ) ⊗ Q Q p ≃ H if ( V ) , i = 0 , . (4.10)In this paper M will always denote a motive satisfying the following conditions M1) M is pure of weight w − M2)
The p -adic realisation V of M is crystalline at p . M3) M has no subquotients isomorphic to Q (1) . These conditions imply that H ( M ) = H ( M ∗ (1)) = 0 and H ( M ∗ (1)) = 0 by the weightargument. and by (4.10) the representation V should satisfy the conditions C1,2,4) of section3.1.2. In particular, from (4.2) it follows thatdet Q p R Γ f ( V ) ∼ → det − Q p H f ( V ) . (4.11)The semisimplicity of ϕ is a well known conjecture which is actually known for abelian varieties.Finally C5) should follow from the injectivity of the syntomic regulator.The comparision isomorphism (4.3) induces an injective map α M : M +B ⊗ Q R −→ t M ( R )and the six-term exact sequence of Fontaine and Perrin-Riou ([F3], section 6.10) degenerates intoan isomorphism (the regulator map) r M, ∞ : H f ( M ) ⊗ Q R ∼ → coker( α M ) . The maps α M and r M, ∞ define a map R M, ∞ : det − Q t M ( Q ) ⊗ det Q M +B ⊗ det Q H f ( M ) −→ R Fix bases ω f ∈ det Q H f ( M ), ω t M ∈ det Q t M ( Q ) and ω + M B ∈ det Q M +B . Set ω M = ( ω f , ω t M , ω + M B )and define R M, ∞ ( ω M ) = R M, ∞ ( ω − t M ⊗ ω + M B ⊗ ω f ) . XTRA ZEROS 27
Using (4.11) and the isomorphisms (4.10) define i ω M ,p : ∆ EP ( V ) ∼ → det − Q p t V ( Q p ) ⊗ det Q p V + ⊗ det Q p H f ( V ) −→ Q p (4.12)by x = i ω M ,p ( x ) ( ω − t M ⊗ ω + M B ⊗ ω f ) . Consider now the case of the dual motive M ∗ (1) . Again one has ∆ EP ( V ∗ (1)) ≃ ∆ f ( V ∗ (1)) where∆ f ( V ∗ (1)) ≃ det − Q p t V ∗ (1) ( Q p ) ⊗ det Q p V ∗ (1) + ⊗ det Q p H f ( V ) . The map α M ∗ (1) : M ∗ (1) +B ⊗ Q R −→ t M ∗ (1) ( R ) is surjective and it is related to α M by thecanonical duality coker( α M ) × ker( α M ∗ (1) ) −→ R (see [F3], section 5.4). The six-term exactsequence degenerates into an isomorphism r M ∗ (1) , ∞ : H f ( M ) ∗ ⊗ Q R ≃ ker( α M ∗ (1) ) . This allows to define a map R M ∗ (1) , ∞ : det − Q t M ∗ (1) ( Q ) ⊗ det Q M ∗ (1) +B ⊗ det Q H f ( M ) −→ R . Fix bases ω t M ∗ (1) ∈ det Q t M ∗ (1) ( Q ) and ω + M ∗ (1) B ∈ det Q M ∗ (1) + B . Set ω M ∗ (1) = ( ω t M ∗ (1) , ω + M ∗ (1) B , ω f )and R M ∗ (1) , ∞ ( ω M ∗ (1) ) = R M ∗ (1) , ∞ ( ω − t M ∗ (1) ⊗ ω + M ∗ (1) B ⊗ ω f ). Then again this data defines a triv-ialisation i ω M ∗ (1) ,p : ∆ EP ( V ∗ (1))) −→ Q p . (4.13)It is conjectured that the L -functions L ( V, s ) and L ( V ∗ (1) , s ) are well defined complex functionshave meromorphic continuation to the whole C and satisfy some explicit functional equation([FP] chapitre III). One expects that they do not depend on the choice of the prime p and wewill denote them by L ( M, s ) and L ( M ∗ (1) , s ) respectively. The conjectures about special valuesof these functions state as follows. Conjecture (Beilinson-Deligne) . The L -function L ( M, s ) does not vanish at s = 0 and L ( V, R M, ∞ ( ω M ) ∈ Q ∗ . The L -function L ( M ∗ (1) , s ) has a zero of order r = dim Q p H f ( M ) at s = 0 . Let L ( M ∗ (1) ,
0) =lim s → s − r L ( M ∗ (1) , s ) . Then L ( M ∗ (1) , R M ∗ (1) , ∞ ( ω M ∗ (1) ) ∈ Q ∗ . Conjecture (Bloch-Kato) . Let T be a Z p -lattice of V stable under the action of G S . Then i ω M ,p (∆ EP ( T )) = L ( M, R M, ∞ ( ω M ) Z p ,i ω M ∗ (1) ,p (∆ EP ( T ∗ (1))) = L ( M ∗ (1) , R M ∗ (1) , ∞ ( ω M ∗ (1) ) Z p . The compatibility of the Bloch-Kato conjec-ture with the functional equation follows from the conjecture C EP ( V ) of Fontaine and Perrin-Riouabout local Tamagawa numbers ([FP], chapitre III, section 4.5.4). More precisely, defineΓ ∗ ( V ) = Y i ∈ Z Γ ∗ ( − i ) h i ( V ) (4.14)where h i ( V ) = dim Q p (gr i ( D dR ( V ))) andΓ( i ) = ( ( i − i > ( − i ( − i )! if i . The exact sequence 0 −→ t V ∗ (1) ( Q p ) ∗ −→ D dR ( V ) −→ t V ( Q p ) −→ ω M dR = ω t M ⊗ ω − t M ∗ (1) ∈ det Q p D dR ( V ). Choose bases ω + T ∈ det Z p T + and ω − T ∈ det Z p T − and set ω T = ω + T ⊗ ω − T ∈ det Z p T and ω + T ∗ (1) = ( ω − T ) ∗ ∈ det Z p T ∗ (1) + . Then theconjecture C EP ( V ) implies that i ω M ∗ (1) ,p (∆ EP ( T ∗ (1)))Ω (´ et,p ) M ∗ (1) ( ω + T ∗ (1) , ω + M ∗ (1) B ) = Γ ∗ ( V ) Ω ( H,p ) M ( ω T , ω M dR ) i ω M ,p (∆ EP ( T ))Ω (´ et,p ) M ( ω + T , ω + M B ) (4.15)(see [PR2], Appendice C). We remark that for crystalline representations C EP ( V ) is proved in[BB08]. p -adic L -functions.4.2.1. p -adic Beilinson’s conjecture. We keep previous notation and conventions. Let M be a motive which satisfies the conditions M1-3) of section 4.1.2 and let V denote the p -adicrealisation of M . We fix bases ω + M B ∈ det Q M +B , ω t M ∈ det Q t M ( Q ) and ω f ∈ det Q H f ( M ) . We also fix a lattice T in V stable under the action of G S and a base ω + T ∈ det Z p T + . Tosimplify notation we will assume that the choices of ω + M B and ω + T are compatible, namely thatΩ (´ et,p ) M ( ω + T , ω + M B ) = 1 . Let D be a regular subspace of D cris ( V ) . We fix a Z p -lattice N of D anda basis ω N ∈ det Z p N. By the analogy with the archimedian case we can consider the p -adicregulator as a map r V,D : H f ( V ) −→ coker( α V,D ) where α V,D : D −→ t V ( Q p )is the natural projection. Set ω V,N = ( ω t M , ω N , ω f ) and denote by R V,D ( ω V,N ) the determinant of r V,D computed in the bases ω f and ω t M ⊗ ω − N . Namely, R V,D ( ω V,N ) is the image of ω − t M ⊗ ω N ⊗ ω f under the induced isomorphism R V,D : det − Q p t V ( Q p ) ⊗ det Q p D ⊗ det Q p H f ( V ) −→ Q p . Now, consider the projection α V ∗ (1) ,D ⊥ : D ⊥ −→ t V ∗ (1) ( Q p ) . XTRA ZEROS 29
A standard argument from the linear algebra shows that α V ∗ (1) ,D ⊥ is surjective and is related to α V,D by the canonical duality coker( α V,D ) × ker( α V ∗ (1) ,D ) −→ Q p . This defines isomorphismsdet − Q p t V ∗ (1) ( Q p ) ⊗ det Q p D ⊥ ≃ det Q p (ker( α V ∗ (1) ,D ⊥ )) ≃ det − Q p (coker( α V,D ))and composing this map with the determinant of r V,D we have again a trivialisation R V ∗ (1) ,D ⊥ : det − Q p t V ∗ (1) ( Q p ) ⊗ det Q p D ⊥ ⊗ det Q p H f ( V ) −→ Q p . Choose a lattice N ⊥ ⊂ D ⊥ , fix bases ω t M ∗ (1) and ω N ⊥ of det Q p t V ∗ (1) ( Q p ) and det Z p N ⊥ respec-tively and set ω V,N ⊥ = ( ω t M ∗ (1) , ω N ⊥ , ω f ).Perrin-Riou conjectured [PR2] that there exists an analytic p -adic L -function L p ( T, ω N , s )which interpolates special values of the complex L -function L ( M, s ) . In particular one expectsthat if p − is not an eigenvalue of ϕ acting on D then L p ( T, ω N , s ) does not vanish at s = 0 and L p ( T, N, R V,D ( ω V,N ) = E ( V, D ) L ( M, R M, ∞ ( ω M ) . where E ( V, D ) = det(1 − p − ϕ − | D ) det(1 − p − ϕ − | D ⊥ ) == det(1 − p − ϕ − | D ) det(1 − ϕ | D cris ( V ) /D ) . Dually it is conjectured that there exists a p -adic L -function L p ( T ∗ (1) , ω N ⊥ , s ) which interpolatesspecial values of L ( M ∗ (1) , s ). One expects that if 1 is not an eigenvalue of ϕ acting on the quotent D cris ( V ∗ (1)) /D ⊥ then L p ( T ∗ (1) , ω N ⊥ , s ) has a zero of order r = dim Q H f ( M ) at s = 0 and L ∗ p ( T ∗ (1) , N ⊥ , R V ∗ (1) ,D ⊥ ( ω V ∗ (1) ,N ⊥ ) = E ( V ∗ (1) , D ⊥ ) L ∗ ( M ∗ (1) , R M ∗ (1) , ∞ ( ω M ) . These properties of p -adic L -functions can be viewed as p -adic analogues of Beilinson’s conjecturesand we refer the reader to [PR2], chapitre 4 and [C2], section 2.8 for more detail. Note that fromthe definition it is clear that E ( V, D ) = E ( V ∗ (1) , D ⊥ ) . One can also write E ( V, D ) in the form E ( V, D ) = E p ( V,
1) det (cid:18) − p − ϕ − − ϕ | D (cid:19) . Assume now that D ϕ = p − = 0 . Since M is crystalline at p ,this can occur only if M is of weight − . Set e = dim Q p D ϕ = p − = dim Q p ( D ⊥ + D cris ( V ∗ (1)) ϕ =1 ) /D ⊥ ) . Assume that the p -adic realisation V of M satisfies the conditions C1-5) of section 3.1.2. De-compose D into the direct sum D = D − ⊕ D ϕ = p − and define E + ( V, D ) = E + ( V ∗ (1) , D ⊥ ) = det(1 − p − ϕ − | D − ) det(1 − p − ϕ − | D ⊥ ) . (4.15) We propose the following conjecture about the behavior of p -adic L -functions at s = 0 . Trivial zero conjecture.
Let D be a regular subspace of D cris ( V ) . Then1) The p -adic L -function L p ( T, N, s ) has a zero of order e at s = 0 and L ∗ p ( T, N, R V,D ( ω V,N ) = − L ( V, D ) E + ( V, D ) L ( M, R M, ∞ ( ω M ) .
2) The p -adic L -function L p ( T ∗ (1) , N ⊥ , s ) has a zero of order e + r where r = dim Q H f ( M )at s = 0 and L ∗ p ( T ∗ (1) , N ⊥ , R V ∗ (1) ,D ⊥ ( ω V ∗ (1) ,N ⊥ ) = L ( V, D ) E + ( V ∗ (1) , D ⊥ ) L ∗ ( M ∗ (1) , R M ∗ (1) , ∞ ( ω M ∗ (1) ) . Remarks.
1) If H f ( M ) = 0 the p -adic regulator vanishes and we recover the conjecture formu-lated in [Ben2], section 2.3.2.2) The regulators R M, ∞ ( ω M ) and R V,D ( ω V,N ⊥ ) are well defined up to the sign and in order toobtain equalities in the formulation of our conjecture one should make the same choice of signsin the definitions of R M, ∞ ( ω M ) and R V,D ( ω V,N ⊥ ). See [PR2], section 4.2 for more detail.3) Our conjecture is compatible with the expected functional equation for p -adic L -functions.See section 2.5 of [PR2] and section 5.2.7 below. §
5. The module of p -adic L -functions5.1. The Selmer complex.5.1.1. Iwasawa cohomology. Let Γ denote the Galois group of Q ( ζ p ∞ ) / Q and Γ n = Gal( Q ( ζ p ∞ ) / Q ( ζ p n )) . Set Λ = Z p [[Γ ]] and Λ(Γ) = Z p [∆] ⊗ Z p Λ. For any character η ∈ X (∆) put e η = 1 | ∆ | X g ∈ ∆ η − ( g ) g. Then Λ(Γ) = ⊕ η ∈ X (∆) Λ(Γ) ( η ) where Λ(Γ) ( η ) = Λ e η and for any Λ(Γ)-module M one has a canonicaldecomposition M ≃ ⊕ η ∈ X (∆) M ( η ) , M ( η ) = e η ( M ) . We write η for the trivial character of ∆ and identify Λ with Λ(Γ) e η . Let V be a p-adic pseudo-geometric representation unramified outside S. Set d ( V ) = dim( V ) and d ± ( V ) = dim( V c = ± ) . Fix a Z p -lattice T of V stable under the action of G S . Let ι : Λ(Γ) −→ Λ(Γ)denote the canonical involution g g − . Recall that the induced module Ind Q ( ζ p ∞ ) / Q ( T ) isisomorphic to (Λ(Γ) ⊗ Z p T ) ι ([Ne2], section 8.1). Define H i Iw ,S ( T ) = H iS ((Λ(Γ) ⊗ Z p T ) ι ) ,H i Iw ( Q v , T ) = H i ( Q v , (Λ(Γ) ⊗ Z p T ) ι ) for any finite place v .From Shapiro’s lemma it follows immediately that H i Iw ,S ( T ) = lim ←− cores H iS ( Q ( ζ p n ) , T ) , H i Iw ( Q p , T ) = lim ←− cores H i ( Q p ( ζ p n ) , T ) . XTRA ZEROS 31
Set H i Iw ,S ( V ) = H i Iw ,S ( T ) ⊗ Z p Q p and H i Iw ( Q v , V ) = H i Iw ( Q v , T ) ⊗ Z p Q p . In [PR2] Perrin-Riouproved the following results about the structure of these modules.i) H i Iw ,S ( V ) = 0 and H i Iw ( Q v , T ) = 0 if i = 1 , v = p , then for each η ∈ X (∆) the η -component H i Iw ( Q v , T ) ( η ) is a finitely generatedtorsion Λ-module. In particular, H ( Q v , T ) ≃ H ( Q ur v / Q v , (Λ(Γ) ⊗ Z p T I v ) ι ).iii) If v = p then H ( Q p , T ) ( η ) are finitely generated torsion Λ-modules. Moreover, for each η ∈ X (∆) rg Λ (cid:16) H ( Q p , T ) ( η ) (cid:17) = d, H ( Q p , T ) ( η )tor ≃ H ( Q p ( ζ p ∞ ) , T ) ( η ) . Remark that by local duality H ( Q p , T ) ≃ H ( Q p ( ζ p ∞ ) , V ∗ (1) /T ∗ (1)).iv) If the weak Leopoldt conjecture holds for the pair ( V, η ) i.e. if H S ( Q ( ζ p ∞ ) , V /T ) ( η ) = 0then H ,S ( T ) ( η ) is Λ-torsion andrank Λ (cid:16) H ,S ( T ) ( η ) (cid:17) = (cid:26) d − ( V ) , if η ( c ) = 1 d + ( V ) , if η ( c ) = − −→ H S ( Q ( ζ p ∞ ) , V ∗ (1) /T ∗ (1)) ∧ −→ H ,S ( T ) −→ ⊕ v ∈ S H ( Q v , T ) −→ H S ( Q ( ζ p ∞ ) , V ∗ (1) /T ∗ (1)) ∧ −→ H ,S ( T ) −→ ⊕ v ∈ S H ( Q v , T ) −→ H S ( Q ( ζ p ∞ ) , V ∗ (1) /T ∗ (1)) ∧ −→ . (5.1)Define R Γ Iw ,S ( T ) = C • c ( G S , (Λ(Γ) ⊗ Z p T ) ι ) , R Γ Iw ( Q v , T ) = C • c ( G v , (Λ(Γ) ⊗ Z p T ) ι ) , R Γ S ( Q ( ζ p ∞ ) , V ∗ (1) /T ∗ (1)) = C • c ( G S , Hom Z p (Λ(Γ) , V ∗ (1) /T ∗ (1))) . Then the sequence (5.1) is induced by the distinguished triangle R Γ Iw ,S ( T ) −→ ⊕ v ∈ S R Γ Iw ( Q v , T ) −→ ( R Γ S ( Q ( ζ p ∞ ) , V ∗ (1) /T ∗ (1)) ι ) ∧ [ − R Γ Iw ,S ( T ) ⊗ L Λ Z p ≃ R Γ S ( T ) , R Γ Iw ( Q v , T ) ⊗ L Λ Z p ≃ R Γ( Q v , T )( [Ne2], Proposition 8.4.21). Γ ( η )Iw ,h ( D, V ) . For the remainder of this chapter we assume that V satisfies the conditions C1-5) of section 3.1.2 and that the weak Leopoldt conjecture holds for(
V, η ) and ( V ∗ (1) , η ) . We remark that these assumptions are not independent. Namely, by[PR2], Proposition B.5
C4) and
C5) imply the weak Leopoldt conjecture for ( V ∗ (1) , η ). Fromthe same result it follows that the vanishing of H f ( V ∗ (1)) implies the weak Leopoldt conjecture for ( V, η ) if in addition we assume that H ( Q p , V ∗ (1)) = 0 . To simplify notations we write H for H (Γ ). Fix a regular subspace D of D cris ( V ) and a Z p -lattice N of D . Set D p ( N, T ) ( η ) = N ⊗ Z p Λ, R Γ ( η )Iw ,f ( Q p , N, T ) = D p ( N, T ) ( η ) [ −
1] and R Γ ( η )Iw ,f ( Q p , D, V ) = R Γ ( η )Iw ,f ( Q p , N, T ) ⊗ Z p Q p . Consider the map E xp εV,h : R Γ ( η )Iw ,f ( Q p , T ) ⊗ Λ H −→ R Γ ( η )Iw ( Q p , T ) ⊗ L Λ H which will be viewed as a local condition at p . If v = p the inertia group I v acts trivially on Λset R Γ ( η )Iw ,f ( Q v , N, T ) = h T I v ⊗ Λ ι − f v −−−→ T I v ⊗ Λ ι i where the first term is placed in degree 0. We have a commutative diagram R Γ ( η )Iw ,S ( T ) ⊗ Λ H / / ⊕ v ∈ S R Γ ( η )Iw ( Q v , T ) ⊗ Λ H (cid:18) ⊕ v ∈ S R Γ ( η )Iw ,f ( Q v , N, T ) (cid:19) ⊗ Λ H . O O (5.2)Consider the associated Selmer complex R Γ ( η )Iw ,h ( D, V ) =cone (cid:20)(cid:18) R Γ ( η )Iw ,S ( T ) ⊕ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ,f ( Q v , N, T ) (cid:19)(cid:19) ⊗ Λ H −→ ⊕ v ∈ S R Γ ( η )Iw ( Q v , T ) ⊗ Λ H (cid:21) [ − S . Our main result about this complexis the following theorem. Theorem 5.1.3.
Assume that V satisfies the conditions C1-5) and that the weak Leopoldt con-jecture holds for ( V, η ) and ( V ∗ (1) , η ) . Let D be a regular subspace of D cris ( V ) . Assume that L ( V, D ) = 0 . Theni) R i Γ ( η )Iw ,h ( D, V ) are H -torsion modules for all i .ii) R i Γ ( η )Iw ,h ( D, V ) = 0 for i = 2 , and R Γ ( η )Iw ,h ( D, V ) ≃ (cid:0) H ( Q ( ζ p ∞ ) , V ∗ (1)) ∗ (cid:1) ( η ) ⊗ Λ H . iii) The complex R Γ ( η )Iw ,h ( D, V ) is semisimple i.e. for each i the natural map R i Γ ( η )Iw ,h ( D, V ) Γ −→ R i Γ ( η )Iw ,h ( D, V ) Γ is an isomorphism. We leave the proof of the following lemma as an easy exercise.
XTRA ZEROS 33
Lemma 5.1.4.1.
Let A and B be two submodules of a finitely generated free H -module M .Assume that the natural maps A Γ −→ M Γ and B Γ −→ M Γ are both injective. Then A Γ ∩ B Γ = { } implies that A ∩ B = { } . Since H ,S ( V ) and H ( Q v , V ) are zero, we have R Γ ( η )Iw ,h ( D, V ) = 0 . Next, by defini-tion R Γ ( η )Iw ,h ( D, V ) = ker( f ) where f : (cid:18) H ,S ( T ) ( η ) ⊕ D p ( N, T ) ( η ) ⊕ v ∈ S −{ p } H ,f ( Q v , T ) ( η ) (cid:19) ⊗ H −→ ⊕ v ∈ S H ( Q v , T ) ( η ) ⊗ H is the map induced by (5.2). If v ∈ S − { p } one has H ,f ( Q v , T ) ( η ) = H ( Q v , T ) ( η ) = H ( Q ur v / Q v , (Λ ⊗ T I v ) ι ) . Thus R Γ ( η )Iw ,h ( D, V ) = (cid:16) H ,S ( T ) ( η ) ⊗ Λ H (cid:17) ∩ (cid:16) Exp εV,h (cid:16) D p ( D, T ) ( η ) (cid:17) ⊗ Λ H (cid:17) in H ( Q p , T ) ( η ) ⊗ Λ H . Put A = Exp εV,h ( D − ⊗ H ) ⊕ X − Exp εV,h ( D ϕ = p − ⊗ H ) ⊂ H ( Q p , T ) ( η ) ⊗ Λ H . By Theorem 2.2.4 and Proposition 3.2.2 A Γ injects into H ( Q p , V ) . The H -module M = (cid:18) H ( Q p , T ) T H Q p (cid:19) ( η ) ⊗ Λ H is free and A ֒ → M. Since T G Q p = 0 one has M Γ = H ( Q p , V ) Γ ⊂ H ( Q p , V ) and we obtain that A Γ injects into M Γ . Set B = H ,S ( T ) T H Q ! ( η ) ⊗ Λ H . The weak Leopoldt conjecture for ( V ∗ (1) , η ) together withthe fact that H ( Q v , T ) are Λ-torsion for v ∈ S − { p } imply that B ֒ → M. Since the image of H ( Q v , V ) Γ in H ( Q v , V ) is contained in H f ( Q v , V ) , the image of H ,S ( V ) Γ in H S ( V ) is infact contained in H f, { p } ( V ) . From
C5) it follows that H f, { p } ( V ) injects into H ( Q p , V ) and wehave H ,S ( V ) ( η )Γ = H ,S ( V ) Γ ֒ → H f, { p } ( V ) ֒ → H ( Q p , V ) . Thus B Γ ⊂ M Γ . We shall prove that R Γ ( η )Iw ,h ( D, V ) = 0 . By Lemma 5.1.4.1 it suffices to showthat A Γ ∩ B Γ = { } . Now we claim that A Γ ∩ H f, { p } ( V ) = { } . First note that by Lemma3.1.4 H f, { p } ( V ) ֒ → H ( Q p , V ) H ( D − ) . On the other hand, from Theorem 2.2.4 it follows thatExp εV,h ( D − ⊗ H ) Γ = exp V, Q p ( D − ) ⊂ H ( D − ) . Now Proposition 3.2.2 implies that the image of A Γ in H ( Q p , V ) H ( D − ) coincides with H c ( W ) . But L ( V, D ) = 0 if and only if H D ( V ) ∩ H c ( W ) = 0 where H D ( V ) denotes the inverse image of H ( W )in H f, { p } ( V ) (see Lemma 3.1.4 iii)). This proves the claim and implies that R Γ ( η )Iw ,h ( D, V ) = 0 . We shall show that R Γ ( η )Iw ,h ( D, V ) is H -torsion. By definition, we have an exact sequence0 −→ coker( f ) −→ R Γ ( η )Iw ,h ( D, V ) −→ III ,S ( V ) ( η ) ⊗ Λ Q p H −→ , (5.3)where III ,S ( V ) = ker (cid:18) H ,S ( V ) −→ ⊕ v ∈ S H ( Q v , V ) (cid:19) . It follows from the weak Leopoldt conjecture that
III ,S ( V ) is Λ Q p -torsion. On the other hand,as H is a Bezout ring [La], the formulasrank Λ H ,S ( T ) ( η ) = d − ( V ) , rank Λ H ( Q p , T ) ( η ) = d ( V ) , rank Λ D p ( N, T ) = d + ( V )together with the fact that R Γ ( η )Iw ,h ( D, V ) = 0 imply that coker( f ) is H -torsion. We havetherefore proved that R Γ Iw ,h ( D, V ) is H -torsion. Finally, the Poitou-Tate exact sequence givesthat R Γ ( η )Iw ,h ( D, V ) = (cid:0) H ( Q ( ζ p ∞ ) , V ∗ (1)) ∗ (cid:1) ( η ) ⊗ Λ Q p H is also H -torsion. Now we prove the semisimplicity of R Γ ( η )Iw ,h ( D, V ) . First write H ,S ( V ) ( η ) ≃ Λ d − ( V ) Q p ⊕ H ,S ( V ) ( η )tor . Since H ,S ( V ) tor ⊂ H ( Q p , V ) tor = V H Q p , we have ( H ,S ( V ) tor ) Γ = 0 bythe snake lemma. Thus dim Q p H ,S ( V ) ( η )Γ = d − ( V ) . On the other hand dim Q p H f, { p } ( V ) = d − ( V ) + dim Q p H ( Q p , V ∗ (1)) by (3.2) and the dimension argument shows that in the commu-tative diagram0 −−−−→ H ,S ( V ) ( η )Γ −−−−→ H f, { p } ( V ) −−−−→ H ( Q p , V ∗ (1)) ∗ −−−−→ y y y −−−−→ H ( Q p , V ) ( η )Γ −−−−→ H ( Q p , V ) −−−−→ H ( Q p , V ∗ (1)) ∗ −−−−→ H ( Q p , V ) ( η )Γ H ,S ( V ) ( η )Γ + H ( D − ) −→ H ( Q p , V ) H f, { p } ( V ) + H ( D − )is an isomorphism.Consider the exact sequence0 −→ (cid:16) H ,S ( T ) ( η ) ⊕ D p ( N, T ) ( η ) (cid:17) ⊗ H −→ H ( Q p , T ) ( η ) ⊗ H −→ coker( f ) −→ . Recall that Exp εV,h, : D −→ H ( Q p , V ) Γ denotes the homomorphism induced by the largeexponential map. Applying the snake lemma, and taking into account that Im(Exp εV,h, ) =exp V, Q p ( D − ) = H ( D − ) and ker(Exp εV,h, ) = D ϕ = p − (see for example [BB], Propositions 4.17and 4.18 or the proof of Proposition 3.3.2) we obtaincoker( f ) Γ = ker (cid:18) H ,S ( V ) ( η )Γ ⊕ D Exp εV,h, −−−−−→ H ( Q p , V ) (cid:19) = D ϕ = p − , (by the regularity of D ) , coker( f ) Γ = H ( Q p , V ) ( η )Γ H ,S ( V ) ( η )Γ + H ( D − ) = H ( Q p , V ) H f, { p } ( V ) + H ( D − ) . XTRA ZEROS 35
Thus on has a commutative diagramcoker( f ) Γ −−−−→ D ϕ = p − y y δ D,h coker( f ) Γ −−−−→ H ( Q p , V ) H f, { p } ( V ) + H ( D − ) . (5.5)where horizontal arrows are isomorphisms, the left vertical arrow is the natural projection andthe right vertical row is the map defined in section 3.2.3. From Proposition 3.2.4 it follows thatcoker( f ) Γ −→ coker( f ) Γ is an isomorphism if and only if L ( V, D ) = 0 . On the other hand, the arguments [PR2], section 3.3.4 show that
III ,S ( V ) Γ = III ,S ( V ) Γ =0 . Remark that Perrin-Riou assumes that D cris ( V ) ϕ =1 = D cris ( V ) ϕ = p − = 0, but her proof worksin our case without modifications and we repeat it for the commodity of the reader. Considerthe commutative diagram (where we write III ( V ) instead III ,S ( V ) and H ( Q ( ζ p ∞ ) , V ∗ (1)) ∗ Γ instead ( H S ( Q ( ζ p ∞ ) , V ∗ (1)) ∗ ) Γ to abbraviate notation) (cid:15) (cid:15) (cid:15) (cid:15) / / H ( V ) Γ / / (cid:15) (cid:15) ⊕ v ∈ S H ( Q v , V ) Γ / / (cid:15) (cid:15) H ( Q ( ζ p ∞ ) , V ∗ (1)) ∗ Γ / / = (cid:15) (cid:15) III ( V ) Γ / / / / H f, { p } ( V ) / / (cid:15) (cid:15) ⊕ v ∈ Sv = p H f ( Q v , V ) ⊕ H ( Q p , V ) / / (cid:15) (cid:15) H ( V ∗ (1)) ∗ / / H ( Q p , V ∗ (1)) ∗ (cid:15) (cid:15) = / / H ( Q p , V ∗ (1)) ∗ (cid:15) (cid:15) . The top row of this diagram is obtained by taking coinvariants in the Poitou-Tate exact sequence.Thus it is exact. The middle row is obtained from the exact sequence0 −→ H S ( V ∗ (1)) −→ H ( Q p , V ∗ (1)) ⊕ M v ∈ S −{ p } H ( Q v , V ∗ (1)) H f ( Q v , V ∗ (1)) −→ H f, { p } ( V ) ∗ −→ H f ( V ∗ (1)) = 0. The exactness of the left and middlecolumns follows from the diagram (5.4). The isomorphism from the right column comes from theexact sequence0 −→ H (Γ , H S ( Q ( ζ p ∞ ) , V ∗ (1))) −→ H S ( V ∗ (1)) −→ H S ( Q ( ζ p ∞ ) , V ∗ (1)) Γ −→ H (Γ , H S ( Q ( ζ p ∞ ) , V ∗ (1))) = 0 because H (Γ , H S ( Q ( ζ p ∞ ) , V ∗ (1))) = H S ( Q , V ∗ (1))) = 0 by C2) . Now an easy diagram search shows that
III ,S ( V ) Γ = 0 . Finally, from dim Q p III ,S ( V ) Γ dim Q p III ,S ( V ) Γ it follows that III ,S ( V ) Γ = 0 . Therefore, apply-ing the snake lemma to (5.3) we obtain a commutative diagramcoker( f ) Γ / / (cid:15) (cid:15) R Γ ( η )Iw ,h ( D, V ) Γ (cid:15) (cid:15) coker( f ) Γ / / R Γ ( η )Iw ,h ( D, V ) Γ , in which the horizontal arrows are isomorphisms and the vertical arrows are natural projections.This proves that R Γ ( η )Iw ,h ( D, V ) is semisimple in degree 2. Remark that the semisimplicity indegree 3 is obvious because by ii) R Γ ( η )Iw ,h ( D, V ) Γ = R Γ ( η )Iw ,h ( D, V ) Γ = 0 . This completes theproof of Theorem 5.1.3.
Corollary 5.1.5.
The exponential map induces an isomorphism of D ϕ = p − onto coker( f ) Γ ≃ R Γ ( η )Iw ,h ( D, V ) Γ and the diagram D ϕ = p − ∼ / / λ D (cid:15) (cid:15) R Γ ( η )Iw ,h ( D, V ) Γ (cid:15) (cid:15) D ϕ = p − ( h − V / / R Γ ( η )Iw ,h ( D, V ) Γ in which the map λ D is defined in Proposition 3.2.4, commutes. p -adic L -functions.5.2.1. The canonical trivialisation. We conserve the notation and conventions of section 4.2.Let D be an admissible subspace of D cris ( V ) and assume that L ( V, D ) = 0 . We review Perrin-Riou’s definition of the module of p -adic L -functions using the formalism of Selmer complexes.Set∆ Iw ,h ( D, V ) = det − Q p (cid:18) R Γ ( η )Iw ,S ( V ) ⊕ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ,f ( Q v , D, V ) (cid:19)(cid:19) ⊗ det Λ Q p (cid:18) ⊕ v ∈ S R Γ ( η )Iw ( Q v , V ) (cid:19) . The exact triangle R Γ ( η )Iw ,S ( D, V ) −→ (cid:18) R Γ ( η )Iw ,S ( V ) ⊕ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ,f ( Q v , D, V ) (cid:19)(cid:19) ⊗ H −→ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ( Q v , V ) (cid:19) ⊗ H gives an isomorphism ∆ Iw ,h ( D, V ) ⊗ Λ Q p H ≃ det − H R Γ ( η )Iw ,S ( D, V ) . Let K denote the field offractions of H . By Theorem 5.1.3, all R i Γ ( η )Iw ,S ( D, V ) are H -torsion and we have a canonical map.det − H R Γ ( η )Iw ,S ( D, V ) ≃ ⊗ i ∈{ , } det ( − i +1 H R i Γ ( η )Iw ,S ( D, V ) ֒ → K . The composition of these maps gives a trivialization i V, Iw ,h : ∆ Iw ,h ( D, V ) −→ K . XTRA ZEROS 37
In this section we compare local conditions coming from Perrin-Riou’stheory to the Bloch-Kato’s one. Set R Γ f ( Q p , D, V ) = D [ −
1] and define S = cone (cid:18) − p − ϕ − − ϕ : R Γ f ( Q v , D, V ) −→ R Γ f ( Q p , V ) (cid:19) [ − . (5.6)Thus, explicitly S = [ D ⊕ D cris ( V ) −→ D cris ( V ) ⊕ t V ( Q p )] [ − ≃ [ D ⊕ D cris ( V ) −→ D cris ( V ) ⊕ D )] [ − , where the unique non-trivial map is given by( x, y ) (cid:18) (1 − ϕ ) y, (cid:18) − p − ϕ − − ϕ x + y (cid:19) (mod Fil D cris ( V )) (cid:19) . Thus H ( S ) = D ϕ = p − and H ( S ) = t V ( Q p )(1 − p − ϕ − ) D ≃ D cris ( V )Fil D cris ( V ) + D − . From the semi-simplicity of 1 − p − ϕ − − ϕ it follows that the natural projection H ( S ) ⊕ H f ( V ) −→ H ( S ) is anisomorphism and we have a canonical trivialization α S : det Q p S ⊗ det Q p R Γ f ( V ) ≃ det − Q p H ( S ) ⊗ det Q p H ( S ) ⊗ det − Q p H f ( V ) ≃ Q p . (5.7)Hence the distingushed triangle S −→ R Γ f ( Q p , D, V ) −→ R Γ f ( Q p , V ) −→ S [1]induces isomorphisms β S : det Q p t V ( Q p ) ⊗ det Q p R Γ f ( V ) ≃ det − Q p R Γ f ( Q p , V ) ⊗ det Q p R Γ f ( V ) ≃ det − Q p R Γ f ( Q p , D, V ) ⊗ det Q p S ⊗ det Q p R Γ f ( V ) ≃ det Q p D ⊗ det Q p S ⊗ det Q p R Γ f ( V ) (5.8)and ϑ S : det Q p t V ( Q p ) ⊗ det Q p R Γ f ( V ) β S ≃ det Q p D ⊗ det Q p S ⊗ det Q p R Γ f ( V ) id ⊗ α S ≃ det Q p D. (5.9)Fix bases ω t V ∈ det Q p t V ( Q p ) , ω D ∈ det Q p D and ω f ∈ det Q p H f ( V ) . Let R V,D ( ω V,D ) denote thedeterminant of the regulator map r V,D : H f ( V ) −→ D cris ( V ) / (Fil D cris ( V ) + D )with respect to ω f and ω t V ⊗ ω − D . Lemma 5.2.3. i) Let f : W −→ W be a semi-simple endomorphism of a finitely dimensional k -vector space W . The canonical projection ker( f ) −→ coker( f ) is an isomorphism and the tau-tological exact sequence −→ ker( f ) −→ W f −→ W −→ coker( f ) −→ induces an isomorphism det ∗ f : det k ( W ) −→ det k ( W ) ⊗ det k (ker( f )) ⊗ det − k (coker( f )) −→ det k ( W ) . Then det ∗ f ( x ) = det( f | coker( f )) . ii) The map ϑ S sends ω t V ⊗ ω − f onto det ∗ (cid:18) − p − ϕ − − ϕ | D (cid:19) − E p ( V, − R V,D ( ω V,D ) − ω D Proof.
The proof is straightforward and is omitted here. p -adic L -functions. In this subsection we interpretPerrin-Riou’s construction of the module of p -adic L -functions in terms of [Ne2]. Fix a Z p -lattice N of D and set∆ Iw ,h ( N, T ) = det − (cid:18) R Γ ( η )Iw ,S ( T ) ⊕ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ,f ( Q v , N, T ) (cid:19)(cid:19) ⊗ det Λ (cid:18) ⊕ v ∈ S R Γ ( η )Iw ( Q v , T ) (cid:19) . The module of p -adic L -functions associated to ( N, T ) is defined as L ( η )Iw ,h ( N, T ) = i V, Iw ,h (∆ Iw ,h ( N, T )) ⊂ K . Fix a generator f ( γ −
1) of L ( η )Iw ,h ( N, T ) and define a meromorphic p -adic function L Iw ,h ( T, N, s ) = f ( χ ( γ ) s − . Let now V be the p -adic realisation of a pure motive M over Q which satisfies the conditions M1-3) of section 4.1.2. As we saw in section 4.1.2 on expects that V satisfies C1-5) . We fix bases ω f ∈ det Q H f ( M ) , ω t M ∈ det Q t M ( Q ) and use the same notation for their images in det Q p H f ( V )and det Q t V ( Q p ) respectively. Choose bases ω + M B ∈ det Q M +B and ω + T ∈ det Z p T + and define the p -adic period Ω (´ et,p ) M ( ω + T , ω + M B ) ∈ Q p by ω + T = Ω (´ et,p ) M ( ω + T , ω + M B ) using the comparision isomorphism(4.4) and (4.7). Let ω N be a generator of det Z p N. Theorem 5.2.5.
Assume that V satisfies C1-5) and that the weak Leopoldt conjecture holds for ( V, η ) and ( V ∗ (1) , η ) . Let D be an admissible subspace of D cris ( V ) . Assume that L ( V, D ) = 0 . Theni) L Iw ,h ( T, N, s ) is a meromorphic p -adic function which has a zero at s = 0 of order e =dim Q p ( D ϕ = p − ) . ii) Let L ∗ Iw ,h ( T, N,
0) = lim s → s − e L Iw ,h ( T, N, s ) be the special value of L Iw ,h ( T, N, s ) at s = 0 . Then L ∗ Iw ,h ( T, N, R V,D ( ω V,N ) ∼ p Γ( h ) d + ( V ) L ( V, D ) E + ( V, D ) i ω M ,p (∆ EP ( T ))Ω (´ et,p ) M ( ω + T , ω + M B ) , where i ω M ,p and E + ( V, D ) are defined by (4.12) and (4.16) respectively and Γ( h ) = ( h − . First recall the formalism of Iwasawa descent which will be used in the proof. The
XTRA ZEROS 39 result we need is proved in [BG]. This is a particular case of Nekov´aˇr’s descent theory [Ne2]. Let C • be a perfect complex of H -modules and let C • = C • ⊗ L H Q p . We have a natural distinguishedtriangle C • X −→ C • −→ C • , where X = γ − . In each degree this triangle gives a short exact sequence0 −→ H n ( C • ) Γ −→ H n ( C • ) −→ H n +1 ( C • ) Γ −→ . One says that C • is semisimple if the natural map H n ( C • ) Γ −→ H n ( C • ) −→ H n ( C • ) Γ (5.10)is an isomorphism in all degrees. If C • is semisimple, there exists a natural trivialisation ofdet Q p C • , namely ϑ : det Q p C • ≃ ⊗ n ∈ Z det ( − n Q p H n ( C ) ≃ ⊗ n ∈ Z (cid:16) det ( − n Q p H n ( C • ) Γ ⊗ det ( − n Q p H n +1 ( C • ) Γ (cid:17) ≃ ⊗ n ∈ Z (cid:16) det ( − n Q p H n ( C • ) Γ ⊗ det ( − n − Q p H n ( C • ) Γ (cid:17) ≃ Q p where the last map is induced by (5.10). We now suppose that C ⊗ H K is acyclic and write i ∞ : det H C • −→ K for the associated morphism in P ( K ) . Then i ∞ (det H C • ) = f H , where f ∈ K . Let r be the unique integer such that X − r f is a unit of the localization H of H withrespect to the principal ideal X H . Lemma 5.2.6.2.
Assume that C • is semisimple. Then r = X n ∈ Z ( − n +1 dim Q p H n ( C • ) Γ andthere exists a commutative diagram det H C • X − r i ∞ −−−−−→ H ⊗ L Q p y y det Q p C • ϑ −−−−→ Q p in which the right vertical arrow is the augmentation map.Proof. See [BG], Lemma 8.1. Remark that Burns and Greither consider complexes over Λ ⊗ Z p Q p but since H is a B´ezout ring, all their arguments work in our case and are omitted here. By Theorem 5.1.3 the complex R Γ ( η )Iw ,h ( D, V ) is semisimple and the first assertion followsfrom Lemma 5.2.6.2 together with Corollary 5.1.5.
Now we can prove Theorem 5.2.5. Define R Γ f ( Q v , N, T ) = R Γ ( η )Iw ,f ( Q v , N, T ) ⊗ L Λ Z p , R Γ f ( Q v , D, V ) = R Γ f ( Q v , N, T ) ⊗ Z p Q p . Remark that for v = p this definition coincides with the definition given in 5.2.2. Applying ⊗ L H Q p to the map R Γ ( η )Iw ,f ( Q v , D, V ) −→ R Γ ( η )Iw ( Q v , T ) ⊗ L Λ H we obtain a morphism R Γ f ( Q v , D, V ) −→ R Γ( Q v , V ) . If v = p, then R Γ f ( Q v , D, V ) = R Γ f ( Q v , V ) and this morphism coincides with the natural map R Γ f ( Q v , V ) −→ R Γ( Q v , V ) . If v = p, then R Γ f ( Q v , D, V ) = D [ −
1] and by Theorem 2.2.4 itcoincides with the composition D − p − ϕ − − ϕ −−−−−−−→ D cris ( V ) ( h − V, Q p −−−−−−−−−→ H ( Q p , V ) . Let R Γ f,h ( D, V ) denote the Selmer complex associated to the diagram R Γ S ( V ) / / ⊕ v ∈ S R Γ( Q v , V ) ⊕ v ∈ S R Γ f ( Q v , D, V ) O O Then we have a distinguished triangle R Γ f,h ( D, V ) −→ R Γ S ( V ) ⊕ (cid:18) ⊕ v ∈ S R Γ f ( Q v , D, V ) (cid:19) −→ ⊕ v ∈ S R Γ( Q v , V ) (5.11)which induces isomorphismsdet − Q p R Γ S ( V ) ⊗ Q p (cid:18) ⊗ v ∈ S det Q p R Γ( Q v , V ) (cid:19) ⊗ det Q p D ∼ → det − Q p R Γ f,h ( D, V ) ,ξ D,h : ∆ EP ( V ) ⊗ Q p (cid:16) det Q p D ⊗ det − Q p V + (cid:17) ∼ → det − Q p R Γ f,h ( D, V ) . Next, R Γ f,h ( D, V ) = R Γ ( η )Iw ,h ( D, V ) ⊗ H Q p and for any i one has an exact sequence0 −→ R i Γ ( η )Iw ,h ( D, V ) Γ −→ R i Γ f,h ( D, V ) −→ R i +1 Γ ( η )Iw ,h ( D, V ) Γ −→ . From Theorem 5.1.3 it follows that R i Γ f,h ( D, V ) = R Γ ( η )Iw ,h ( D, V ) Γ if i = 1 R Γ ( η )Iw ,h ( D, V ) Γ if i = 20 if i = 1 , . Therefore, the isomorphism R Γ Iw ,h ( D, V ) Γ −→ R Γ Iw ,h ( D, V ) Γ induces a canonical trivializa-tion ϑ D,h : det Q p R Γ f,h ( D, V ) ∼ → Q p . By Lemma 5.2.6.2 we have a commutative diagramdet − H R Γ ( η )Iw ,h ( D, V ) X − e i V, Iw ,h −−−−−−−→ H L ⊗ H Q p y y det − Q p R Γ f,h ( D, V ) ϑ − D,h −−−−→ Q p . XTRA ZEROS 41
Since ∆ Iw ,h ( N, T ) ⊗ L Λ Z p ≃ ∆ EP ( T ) ⊗ Z p ω N ⊗ Z p ( ω + T ) − it implies that ϑ − D,h ◦ ξ D,h (∆ EP ( T ) ⊗ Z p ω N ⊗ Z p ( ω + T ) − ) = log( χ ( γ )) − e L ∗ Iw ,h ( T, N, Z p . (5.12)Consider the diagram R Γ f ( V ) / / R Γ S ( V ) ⊕ ⊕ v ∈ S ∪{∞} R Γ f ( Q v , V ) / / ⊕ v ∈ S ∪{∞} R Γ( Q v , V ) R Γ f,h ( D, V ) / / O O R Γ S ( V ) ⊕ ⊕ v ∈ S R Γ f ( Q v , D, V ) / / O O ⊕ v ∈ S R Γ( Q v , V ) O O L / / O O S ⊕ V + [ − / / O O V + [ − O O (5.13)where L = cone ( R Γ f,h ( D, V ) −→ R Γ f ( V )) [ −
1] and the upper and middle rows coincide with(4.1) and (5.11) up to the following modification: the map loc p : R Γ f ( Q p , V ) −→ R Γ( Q p , V ) isreplaced by Γ( h ) loc p . Hence S is isomorphic to L in the derived category D p ( Q p ) and we havean exact triangle S −→ R Γ f,h ( D, V ) −→ R Γ f ( V ) −→ S [1] . An easy diagram search shows that H ( S ) ≃ R Γ f,h ( D, V ) coincides with id : D ϕ = p − −→ D ϕ = p − and that 0 −→ H f ( V ) −→ H ( S ) −→ R Γ f,h ( D, V ) −→ −→ H f ( V ) −→ D cris ( V )Fil D cris ( V ) + D − h ) exp V −−−−−−→ H ( Q p , V ) H f, { p } ( V ) + H ( D − ) −→ . Therefore, we have a commutative diagramdet Q p S ⊗ det Q p R Γ f ( V ) α −−−−→ det Q p R Γ f,h ( D, V ) ϑ S y ϑ D,h y Q p κ −−−−→ Q p where ϑ S was defined in section 5.2.2 and κ is the unique map which makes this diagram commute.From Proposition 3.2.4 and Corollary 5.1.5 we obtain immediately that κ = (log χ ( γ )) e (cid:18) − p (cid:19) e L ( V, D ) − id Q p . (5.14)Passing to determinants in the diagram (5.13) we obtain a commutative diagram ∆ EP ( V ) ⊗ (cid:0) det( t V ( Q p )) ⊗ det − V + (cid:1) ⊗ det R Γ f ( V ) α S (cid:15) (cid:15) / / Q p ∆ EP ( V ) ⊗ (cid:0) det D ⊗ det S ⊗ det R Γ f ( V ) (cid:1) ⊗ det − V + ξ D,h ⊗ α / / id ⊗ β S (cid:15) (cid:15) det − R Γ f,h ( D, V ) ⊗ det R Γ f,h ( D, V ) duality / / id ⊗ ϑ D,h (cid:15) (cid:15) Q p ∆ EP ( V ) ⊗ (cid:16) det D ⊗ det − Q p V + (cid:17) ξ D,h ⊗ κ / / det − R Γ f,h ( D, V ) ϑ − D,h / / Q p where the maps α S and β S were defined in (5.7-5.9). The upper row of this diagram sends∆ EP ( T ) ⊗ ( ω t M ⊗ ( ω + T ) − ⊗ ω f ) ontoΓ( h ) d + ( V ) i ω M ,p (∆ EP ( T ))Ω (´ et,p ) M ( ω + T , ω +B ) . (5.15)From Lemma 5.2.3 it follows that the composition of left vertical maps ϑ S = (id ⊗ β S ) sends∆ EP ( T ) ⊗ ( ω t M ⊗ ( ω + T ) − ⊗ ω f ) ontodet ∗ (cid:18) − p − ϕ − − ϕ | D (cid:19) − E p ( V, − R V,D ( ω V,N ) ∆ EP ( T ) ⊗ ( ω N ⊗ ( ω + T ) − ) (5.16)Next, (5.12) and (5.13) give ϑ − D,h ◦ ( ξ D,h ⊗ κ )(∆ EP ( T ) ⊗ ω N ⊗ ( ω + T ) − ) = (cid:18) − p (cid:19) e L ( V, D ) − L ∗ Iw ,h ( T, N, Z p . (5.17)Putting together (5.15), (5.16) and (5.17) we obtain that L ∗ Iw ,h ( T, N, R V,D ( ω V,N ) ∼ p Γ( h ) d + ( V ) L ( V, D ) E ∗ p ( V,
1) det Q p (cid:18) − p − ϕ − − ϕ | D − (cid:19) i ω M ,p (∆ EP ( T ))Ω (´ et,p ) M ( ω + T , ω +B )and the theorem is proved. L ∗ Iw ,h ( T, N, s ) . Let e H f ( T ) denote the image of H f ( T ) in H f ( V ) andlet ω T,f be a base of det Z p e H f ( T ) . Let R V,D ( ω T,N ) denote the determinant of r V,D computed inthe bases ω t M , ω N and ω T,f . Corollary 5.2.8.
Under the assumptions of Theorem 5.2.5 one has L ∗ Iw ,h ( T, N, R V,D ( ω T,N ) ∼ p Γ( h ) d + ( V ) L ( V, D ) E + ( V, D )
III ( T ∗ (1)) Tam ω M ( T ) H S ( V /T ) H S ( V ∗ (1) /T ∗ (1)) , where III ( T ∗ (1)) is the Tate-Shafarevich group of Bloch-Kato [BK] and Tam ω M ( T ) is the productof local Tamagawa numbers of T taken over all primes and computed with respect to a fixed base ω t M of det Q t M ( Q ) . Proof.
The computation of the trivialisation of the Euler-Poincar´e line (see for example [FP],chapitre II, Th´eor`eme 5.6.3) together with the definition of i ω M ,p by (4.12) give i ω M ,p (∆ EP ( T )) = III ( T ∗ (1)) Tam ω M ( T ) H S ( V /T ) H S ( V ∗ (1) /T ∗ (1)) Ω (´ et,p ) M ( ω + T , ω + M B ) [ ω f : ω T,f ] . Since R V,D ( ω T,N ) = R V,D ( ω V,N ) [ ω f : ω T,f ] the corollary follows from Theorem 5.2.5.
Recall that we set h i ( V ) = dim Q p (gr i D dR ( V )) and m = P i ∈ Z ih i ( V ) . Since V is crystalline, det Q p ( V ) is a one dimensional crystalline representation anddet Z p ( T ) = T ( m ) where T is an unramified G Q p -module of rank 1 over Z p . The module ( T ⊗ XTRA ZEROS 43 W ( F p )) ϕ =1 e m where e m = ( t − ⊗ ε ) ⊗ m is a Z p -lattice in det Q p ( D cris ( V )) = D cris ( V ( m )) whichdepends only on T and which we denote by D cris ( T ( m )) . Let D ⊥ be the dual regular module. The exact sequence0 −→ D −→ D cris ( V ) −→ (cid:0) D ⊥ (cid:1) ∗ −→ Q p D ⊗ det − Q p D ⊥ ≃ det Q p D cris ( V )and we fix a lattice N ⊥ ⊂ D ⊥ such thatdet Z p N ⊗ det − Z p N ⊥ ≃ D cris ( T ( m )) . Set Γ
V,h ( s ) = Q j> − h ( j + s ) dim Fil j D dR ( V ) . The conjecture δ Z p ( V ) of [PR1] proved in [BB] impliesthat for h ≫ L Iw ,h ( T ∗ (1) , N ⊥ , − s ) ∼ Λ ∗ Γ − V,h ( s ) Y − h There are canonical and functorial isomorphisms h i : H i ( C ϕ,γ ( D ( T ))) ∼ → H i ( K, T ) which can be described explicitly by the following formulas:i) If i = 0 , then h coincides with the natural isomorphism D ( T ) ϕ =1 ,γ =1 = H ( K, T ⊗ Z p A ϕ =1 ) = H ( K, T ) . ii) Let α, β ∈ D ( T ) be such that ( γ − α = (1 − ϕ ) β. Then h sends cl( α, β ) to the class of thecocycle µ ( g ) = ( g − x + g − γ − β, where x ∈ D ( T ) ⊗ A K A is a solution of the equation (1 − ϕ ) x = α. iii) Let b γ ∈ G K be a lifting of g ∈ Γ and let x be a solution of ( ϕ − x = α. Then h sends α to the class of the 2-cocycle µ ( g , g ) = b γ k ( h − b γ k − b γ − x where g i = ˆ γ k i h i , h i ∈ H K . Proof. The isomorphisms h i were constructed in [H1], Theorem 2.1. Remark that i) followsdirectly from this construction (see [H1], p.573) and that ii) is proved in [Ben1], Proposition 1.3.2and [CC2], Proposition I.4.1. The proof of iii) follows along exactly the same lines. Namely, it isenough to prove this formula modulo p n for each n . Let α ∈ D ( T ) /p n D ( T ) . By Proposition 2.4of [H1] there exists r ≥ y ∈ D ( T ) /p n D ( T ) such that ( ϕ − α = ( γ − r β. Let N x = ( D ( T ) /p n D ( T )) ⊕ ( ⊕ ri =1 ( A K /p n A K ) t i ) , where ϕ ( t i ) = t i + ( γ − r − i ( α ) and γ ( t i ) = t i + t i − . Then N x is a ( ϕ, Γ)-module and we havea short exact sequence 0 −→ D −→ N x −→ X −→ X = N x /M ≃ ⊕ ri =1 A K /p n A K ¯ t i . An easy diagram search shows that the connectinghomomorphism δ D : H ( C ϕ,γ ( D ( X ))) −→ H ( C ϕ,γ ( D ( T ))) sends cl(0 , ¯ t r ) to − cl( α ) . The functor V ( D ) = ( D ⊗ A K A ) ϕ =1 is a quasi-inverse to D . Thus one has an exact sequence of Galois modules0 −→ T /p n T −→ T x −→ V ( X ) −→ T x = V ( N x ) . From the definition of x it follows immediately that t r − x ∈ T x . By ii), h (cl(0 , ¯ t r )) can be represented by the cocycle c ( g ) = g − γ − t r and we fix its lifting ˆ c : G K −→ N x putting ˆ c ( g ) = g − γ − t r − x ) . As g ˆ c ( g ) − ˆ c ( g g ) + ˆ c ( g ) = − µ ( g , g ) , the connectingmap δ T : H ( K, V ( X )) −→ H ( K, T /p n T ) sends cl( c ) to − cl( µ ) and iii) follows from thecommutativity of the diagram H ( C ϕ,γ ( X )) δ D −−−−→ H ( C ϕ,γ ( T /p n T )) h y h y H ( K, V ( X )) δ T −−−−→ H ( K, T /p n T ) . XTRA ZEROS 45 Proposition A.3. The complexes R Γ( K, T ) and C ϕ,γ ( T ) are isomorphic in D ( Z p ) .Proof. The proof is standard (see for example [BF], proof of Proposition 1.17). The exact se-quence 0 −→ T −→ D ( T ) ⊗ A K A ϕ − −−−→ D ( T ) ⊗ A K A −→ −→ C • c ( G K , T ) −→ C • c ( G K , D ( T ) ⊗ A K A ) ϕ − −−−→ C • c ( G K , D ( T ) ⊗ A K A ) −→ R Γ( K, T ) is quasi-isomorphic to the total complex K • ( T ) = Tot • (cid:16) C • c ( G K , D ( T ) ⊗ A K A ) ϕ − −−−→ C • c ( G K , D ( T ) ⊗ A K A ) (cid:17) . On the other hand C ϕ,γ ( T ) = Tot • (cid:16) A • ( T ) ϕ − −−−→ A • ( T ) (cid:17) , where A • ( T ) = [ D ( T ) γ − −−→ D ( T )].Consider the following commutative diagram of complexes D ( T ) β (cid:15) (cid:15) γ − / / D ( T ) β (cid:15) (cid:15) / / / / (cid:15) (cid:15) · · · C ( G K , D ( T ) ⊗ A K A ) / / C ( G K , D ( T ) ⊗ A K A ) / / C ( G K , D ( T ) ⊗ A K A ) / / · · · in which β ( x ) = x viewed as a constant function on G K and β ( x ) denotes the map G K −→ D ( T ) ⊗ A K A ) defined by ( β ( x )) ( g ) = g − γ − x. This diagram induces a map Tot • ( A • ( T ) ϕ − −−−→ A • ( T )) −→ K • ( T ) and we obtain a diagram C ϕ,γ ( T ) −→ K • ( T ) ← R Γ( K, T )where the right map is a quasi-isomorphism. Then for each i one has a map H i ( C ϕ,γ ( T )) −→ H i ( K • ( T )) ≃ H i ( K, T )and an easy diagram search shows that it coincides with h i . The proposition is proved. Corollary A.4. Let V be a p -adic representation of G K . Then the complexes R Γ( K, V ) , C ϕ,γ ( D † ( V )) and C ϕ,γ ( D † rig ( V )) are isomorphic in D ( Q p ) . Proof. This follows from Theorem 1.1 of [Li] together with Proposition A.2. A.5. Recall that K ∞ /K denotes the cyclotomic extension obtained by adjoining all p n -throots of unity. Let Γ = Gal( K ∞ /K ) and let Λ(Γ) = Z p [[Γ]] denote the Iwasawa algebra of Γ. For any Z p -adic representation T of G K the induced representation Ind K ∞ /K T is isomorphic to( T ⊗ Z p Λ(Γ)) ι and we set R Γ Iw ( K, T ) = C • c ( G K , Ind K ∞ /K T ) . Consider the complex C Iw ,ψ ( T ) = h D ( T ) ψ − −−−→ D ( T ) i in which the first term is placed in degree 1. Proposition A.6. There are canonical and functorial isomorphisms h i Iw : H i ( C Iw ,ψ ( T )) −→ H i Iw ( K, T ) which can be described explicitly by the following formulas:i) Let α ∈ D ( T ) ψ =1 . Then ( ϕ − α ∈ D ( T ) ψ =0 and for any n there exists a unique β n ∈ D ( T ) such that ( γ n − β n = ( ϕ − α. The map h sends cl( α ) to ( h n (cl( β n , α ))) n ∈ N ∈ H ( K n , T ) . ii) If α ∈ D ( T ) , then h (cl( α )) = − ( h n ( ϕ ( α ))) n ∈ N . Proof. The proposition follows from Theorem II.1.3 and Remark II.3.2 of [CC2] together withProposition A.2. Proposition A.7. The complexes R Γ Iw ( K, T ) and C Iw ,ψ ( T ) are isomorphic in the derived cat-egory D (Λ(Γ)) . Proof. We repeat the arguments used in the proof of Proposition A.1.2 with some modifications.For any n > −→ Ind K n /K T −→ ( D ( T ) ⊗ Z p Z p [ G n ] ι ) ⊗ A K A ϕ − −−−→ ( D ( T ) ⊗ Z p Z p [ G n ] ι ) ⊗ A K A −→ . Set D (Ind K ∞ /K T ) = D ( T ) ⊗ Z p Λ(Γ) ι and D (Ind K ∞ /K ( T )) ˆ ⊗ A K A = lim ←− n ( D ( T ) ⊗ Z p Z p [ G n ] ι ) ⊗ A K A . As Ind K n /K T are compact, taking projective limit one obtains an exact sequence0 −→ Ind K ∞ /K T −→ D (Ind K ∞ /K ( T )) ˆ ⊗ A K A ϕ − −−−→ D (Ind K ∞ /K ( T )) ˆ ⊗ A K A −→ . Thus R Iw ( K, T ) is quasi-isomorphic to K • Iw ( T ) = Tot • (cid:16) C • c ( G K , D (Ind K ∞ /K T ) ˆ ⊗ A K A ) ϕ − −−−→ C • c ( G K , D (Ind K ∞ /K T ) ˆ ⊗ A K A ) (cid:17) . We construct a quasi-isomorphism f • : C Iw ,ψ ( T ) −→ K • Iw ( T ) . Any x ∈ D ( T ) can be writ-ten in the form x = (1 − ϕψ ) x + ϕψ ( x ) where ψ (1 − ϕψ ) x = 0 . Then for each n > γ n − y n = ( ϕψ − x has a unique solution y n ∈ D ( T ) ψ =0 ([CC2], Propo-sition I.5.1). In particular, y n = γ n +1 − γ n − y n +1 and we have a compatible system of elements Y n = | G n |− X k =0 γ k ⊗ γ k ( y n ) ∈ D ( T ) ⊗ Z p Z p [ G n ] ι . Put Y = ( Y n ) n ≥ ∈ D (Ind K ∞ /K T ) . Then( γ n − Y n = ( γ − Y (mod D (Ind K n /K T )) . Let η x ∈ C ( G K , D (Ind K ∞ /K T ) ˆ ⊗ A K A ) be the map defined by η x ( g ) = g − γ − ⊗ x ) . Define f : D ( T ) −→ K ( T ) = C ( G K , D (Ind K ∞ /K T ) ˆ ⊗ A K A ) ⊕ C ( G K , D (Ind K ∞ /K T ) ˆ ⊗ A K A ) by f ( x ) = ( Y, η x ) and f : D ( T ) −→ C ( G K , D (Ind K ∞ /K T ) ˆ ⊗ A K A ) ⊂ K ( T ) by f ( z ) = − η ϕ ( z ) . It is easy to check that f • is a morphism of complexes. This gives a diagram C Iw ,ψ ( T ) −→ K • Iw ( T ) ← R Γ Iw ( K, T )in which the right map is a quasi-isomorphism. Using Proposition A.1.4 it is not difficult to checkthat for each i the induced map H i ( C Iw ,ψ ( T )) −→ H i ( K • Iw ( T )) ≃ H i Iw ( K, T )coincides with h i Iw . The proposition is proved. XTRA ZEROS 47 Corollary A.8. The complexes R Γ Iw ( K, T ) and C † Iw ,ψ ( T ) are isomorphic in D (Λ(Γ)) . Proof. One has D † ( T ) ψ =1 = D ( T ) ψ =1 ([CC1], Proposition 3.3.2) and D † ( T ) / ( ψ − 1) = D ( T ) / ( ψ − 1) ([Li], Lemma 3.6). This shows that the inclusion C † Iw ,ψ ( T ) −→ D ( T ) ψ =1 is a quasi-isomorphism. Remark A.9. These results can be slightly improved. Namely, set r n = ( p − p n − . The methodused in the proof of Proposition III.2.1 [CC2] allows to show that ψ ( D † ,r n ( T )) ⊂ D † ,r n − ( T ) for n ≫ . Moreover, for any a ∈ D † ,r n ( T ) the solutions of the equation ( ψ − x = a are in D † ,r n ( T ) . Thus C † ,r n Iw ( T ) = h D † ,r n ( T ) ψ − −−−→ D † ,r n ( T ) i , n ≫ C † Iw ,ψ ( T ) . Further, as ϕ ( A † ,r/p ) = A † ,r we can consider the complex C † ,r n ϕ,γ ( T ) = (cid:2) D † ,r n − ( T ) f −→ D † ,r n ( T ) ⊕ D † ,r n − ( T ) g −→ D † ,r n ( T ) (cid:3) , n ≫ f and g are defined by the same formulas as before. 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