On p-Adic Absolute CM-Periods II

On p-Adic Absolute CM-Periods II

By

Tomokazu Kashio^∗and HiroyukiYoshida^∗∗

Abstract

In part I of this paper, we studied thep-adic absolute CM-period symbol in the completely split case. We presented a conjecture which predicts the exact value of this symbol with supporting evidences. In this part II, we study the properties of this symbol in the general case and clarify its relation top-adic periods.

Introduction

LetF be a totally real algebraic number field andK be a CM-field which is abelian overF. We fix an embedding F ⊂Cp and letp be the prime ideal of F induced from this embedding. For τ ∈ Gal(K/F), the p-adic absolute CM-period symbol is defined by the formula

lg_p,K/F(id, τ) =−μ(τ)

2h₀ log_pα₀+ 1

|G|

χ∈Gˆ₋

χ(τ) L(0, χ)

c∈Cf(χ)p

χ(c)Xp(c).

Here we put p^h0 = (α₀) with a totally positiveα₀,h₀ being the class number ofF in the narrow sense; μ(τ) = 1 if τ = id, μ(τ) =−1 ifτ =ρ andμ(τ) = 0 otherwise, where ρ denotes the complex conjugation; G = Gal(K/F) and G₋ denotes the set of all odd characters of G; f(χ) denotes the finite part of the conductor of χ; C_f(χ)p denotes the ideal class group of F modulo f(χ)p times the product of all archimedean primes;Xp(c) is a certain class invariant which is constructed using the division values of thep-adic logarithmic multiple gamma functions. For details we refer the reader to §4 of part I. When p

Communicated by A. Tamagawa. Received June 28, 2007. Revised September 4, 2008, October 15, 2008.

2000 Mathematics Subject Classification(s): 11G15, 11G99, 11S80, 14F30.

∗Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan.

∗∗Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan.

c 2009 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

splits completely inK, Conjecture A of part I gives an explicit prediction on lgp,K/F(id, τ) as follows. We extend the embeddingF ⊂Cp to an embedding K ⊂ Cp and let P be the prime ideal of K lying above p induced from this embedding. Put P^hK = (α) where hK is the class number of K. Then the conjecture states that

lgp,K(id, τ) = 1 2hK

log_p(α^τ−1ρ/α^τ−1) +

n−1

i=1

ailog_pi

withai∈Q. Heren= [F :Q] and₁,. . .,n−1 are generators of the group of all totally positive units ofF. The “correction term” n−1

i=1 a_ilog_pi was also predicted explicitly. The purpose of this part II is to clarify the nature of the p-adic absolute CM-period symbol in the general case.

LetM be a motive over Q. In the complex case, the periods (Deligne’s periods [D]) ofM are defined using the comparison isomorphism

I^±:H_B^±(M)⊗QC∼=H_DR^± (M)⊗QC

in the usual notation. Shimura’s period symbol can be interpreted in terms of motives (cf. [Bl1]). In thep-adic case, the period is defined by the comparison isomorphism

IDR:Hp(M)⊗QpB_DR∼=H_DR(M)⊗QB_DR.

(For the notation, see the text.) We will clarify the relation of our symbol to thep-adic period in this sense.

In§1 ∼ §3, we study thep-adic period of the motive attached to an al- gebraic Hecke character of a CM-field. In §4, we introduce a certain period invariant Q which is constructed by the action of the Frobenius map on the p-adic period. In §5, we state our main conjecture which relates lg_p,K(id, τ) to aQ-invariant. In§6∼ §7, we investigate how far we can recover the p-adic period itself from the Q-invariant. In §8, we present some more supporting evidences for our main conjecture.

This paper has three appendices in which we give proofs of relevant results as promised in part I.

The authors would like to thank Professor Don Blasius for useful discus- sions. The authors would like to thank the referee for his careful reading of the manuscript and useful suggestions.

Notation. For a prime numberp,Q_pdenotes an algebraic closure ofQ_p; Cpdenotes the completion ofQp. We fix an algebraic closureQofQinC. By

an algebraic number field, we understand an algebraic extension ofQof finite degree contained inQ. We denote by ρthe complex conjugation.

LetF be an algebraic number field. The ring of integers and the group of units ofF are denoted byOF andEF respectively. We denote byE_F⁺the group of all totally positive units ofF. The class number of F is denoted byh_F. We denote byJ_F (resp. J_p,F) the set of all isomorphisms ofF intoC (resp. C_p).

For a prime idealpofF,F_p denotes the completion ofF atp. IfF is a totally real algebraic number field of degreen, ∞1, . . .,∞n denote all archimedean primes of F. For an integral ideal f of F, C_f denotes the ideal class group modulof∞1· · · ∞n. Fora∈F, a0 means that ais totally positive. By a CM-field, we understand a totally imaginary quadratic extension of a totally real algebraic number field. Throughout the paper, except in§8, we exclusively use the left action of Galois groups.

We refer the formulas and statements of Part I by prefixing I to the num- bers and labels of them. For example, (I.1.6) means the formula (1.6) of Part I, Conjecture I.A means Conjecture A of Part I, etc.

§1. Basic Comparison Isomorphisms

The comparison isomorphisms recalled in this section were conjectured by Fontaine ([Fo1]). They were established by the work of many mathematicians.

See Faltings [Fa], Tsuji [T] for example. The latter paper gives a very nice exposition on the topic.

Let K be a finite extension ofQp contained inCp. Put G= Gal(K/K).

Let B_DR and B_cris be the rings introduced by Fontaine [Fo1], [Fo2]; B_DR is a complete discrete valuation field and B_cris is a subring of B_DR; by v_DR, we denote the discrete valuation of B_DR. We have actions of Gon B_DR and on B_cris. For i ∈ Z, put B_DRⁱ = {x ∈ B_DR | v_DR(x) i}. Then we have B_DRⁱ /B_DRⁱ⁺¹∼=Cp(i) asG-modules. HereCp(i) denotes theith Tate twist.

Let X be a proper smooth algebraic variety (not necessarily connected) defined overK. It is known that there exists a canonical isomorphism

(1.1) I_DR:H_etⁿ(X×KK,Qp)⊗QpB_DR∼=H_DRⁿ (X/K)⊗KB_DR.

The isomorphism isB_DR-linear and compatible with the action ofGand filtra- tions. Taking gr₀ of the filtrations, we obtain the Hodge-Tate decomposition (1.2) I_HT:H_etⁿ(X×KK,Qp)⊗QpCp∼=⊕ⁿa=0H^n−a(X,Ω^a_X/K)⊗KCp(−a).

Suppose thatX has good reduction. Letk be the residue field ofK and W(k) be the ring of Witt vectors overk. LetK₀ be the quotient field ofW(k).

Then we have a canonical isomorphism¹

(1.3) I_cris:H_etⁿ(X×KK,Qp)⊗QpB_cris∼=H_crisⁿ (X/W(k))⊗W(k)B_cris. The isomorphism is B_cris-linear and compatible with the action of Gand the Frobenius map. More preciselyσ∈Gacts asσ⊗σand 1⊗σ on the left and the right-hand sides respectively. Concerning the Frobenius map, let Φ_cris be the Frobenius endomorphism ofB_cris and Φ be the Frobenius map acting on H_crisⁿ (X/W(k)). Then the action of the Frobenius map on the left-hand side is 1⊗Φ_cris and Φ⊗Φ_cris on the right-hand side respectively. In good reduction case,H_DRⁿ (X/K) has a K₀-structure so that H_DRⁿ (X/K)∼=H_DRⁿ (X/K₀)⊗K₀

K.² We have another comparison isomorphism

(1.4) I₀:H_crisⁿ (X/W(k))⊗W(k)K₀∼=H_DRⁿ (X/K₀).

We have the relation

(1.5) I_DR= (I₀)_DR◦(I_cris)_DR.

Here (I_cris)_DRdenotes the isomorphism obtained from (1.3) by taking⊗B_crisB_DR and (I₀)_DRdenotes the isomorphism obtained from (1.4) by taking⊗K₀B_DR.

§2. The Motive Attached to an Algebraic Hecke Character It is possible to develop the theory of p-adic CM-periods using abelian varieties with complex multiplication. This approach is especially feasible in the ordinary case. However, in general, the problem of the field of definition will make it very difficult to derive precise results. Therefore it is convenient to consider motives attached to algebraic Hecke characters. In fact, even in the complex case, the precise version of Conjecture I.1.3 (cf. [Y4], p. 83, Conjecture 3.10) is formulated in relation to such Hecke characters. By a result of Blasius [Bl2] on ap-adic property of Hodge classes on abelian variety, it is legitimate to transfer the results in§1 to CM-motives.

1We defineH_crisⁿ (X/W(k)) =H_crisⁿ (Y/W(k)), whereY is a proper smooth algebraic vari- ety overkobtained fromXby reduction. By Gillet and Messing [GM], Corollary B.3.6, H_crisⁿ (X/W(k)) together with the action of the Frobenius map does not depend on the choice ofY.

2TheK0-structure is defined by the isomorphism of Berthelot-Ogus [BO]:H_DRⁿ (X/K0)∼= H_crisⁿ (Y/W(k))⊗W(k)K0, whereY is the same as in footnote 1.

LetK be a CM-field of degree 2n. Let fbe an integral ideal ofK. Letχ be a Gr¨ossencharacter ofK of conductorfsuch that

(2.1) χ((α)) =

σ∈J_K

σ(α)^lσ, α≡1 mod^×f.

Here lσ are integers such that lσ+lσρ is independent ofσ and ρdenotes the complex conjugation as before. Put

E=Q(χ(a)|a is an integral ideal relatively prime tof), E₀=Q(χ((α))|α≡1 mod^×f).

ThenE and E₀ are algebraic number fields of finite degree; we haveE₀⊂E.

Forg∈Gal(Q/Q), we putl(g) =l_g|K. Then l defines aZ-valued function on Gal(Q/Q) which is left Gal(Q/E₀)-invariant and right Gal(Q/K)-invariant.

There exists a motive M = M(χ) over K with coefficients in E; M is characterized by the property³

L(M, s) = (L(s, σ(χ)))_σ∈JE.

The motiveM has the de Rham realizationH_DR(M) which is a freeE⊗_QK- module of rank 1;H_DR(M) has the Hodge filtration{F^m}which is a decreasing filtration byE⊗_QK-submodules. Letψ_mbe the structure function ofF^m(cf.

[Y1], §2.3); ψ_m is a Z-valued function on Gal(Q/Q) which is left invariant under Gal(Q/E) and right invariant under Gal(Q/K). We recall the following characterization of the structure function. Let σ ∈ JE, τ ∈ JK. Take s, t∈Gal(Q/Q) so thatσ=s|E,τ =t|K. Then the multiplicity ofσin the left regular representation ofE onF^m⊗K,τ C is equal toψm(s⁻¹t). We have (cf.

[Y1],§4.2)

(2.2) ψm(g) =

1 if l(g)m, 0 if l(g)< m.

We note thatM is of pure weight lσ+lσρ,σ∈JK.

For a finite placeλofE,M has theλ-adic realizationHλ(M) which is an Eλ-vector space of dimension 1. We have anEλ-linear action of Gal(K/K) on H_λ(M). It is related to thep-adic realizationH_p(M) by

Hp(M) =⊕λ|p Hλ(M).

3It is well known thatχdetermines the motiveM(χ) up to isomorphism. See, for example, [Sc], p. 51.

Take τc ∈ JK. By τc, we regard M as a motive over C. We have the Betti realizationHτ_c,B(M) which is anE-vector space of dimension 1. We have the canonical isomorphism

(2.3) i_p:H_τc,B(M)⊗QQ_p∼=H_p(M) as freeE⊗_QQp-modules of rank 1.

Take τp ∈ Jp,K. By τp, we regardK as a subfield of Cp. LetP be the prime ideal ofKinduced by this embedding. We can transfer the isomorphism (1.1) to this situation and obtain theE⊗QB_DR-linear isomorphism⁴

(1.1) I_DR:Hp(M)⊗_QpB_DR∼=H_DR(M)⊗K,τ_pB_DR.

The isomorphism is compatible with the action of Gal(Q_p/K_P) and filtrations.

Suppose thatχis unramified atP. ThenM has the crystalline realization H_cris(M). (In fact, after tensoring with an Artin motive of rank 1 unramified atP, we may assume thatM(χ) is realized as a factor of a motive generated by the motives attached to abelian varieties with complex multiplication, which are defined overKand have good reduction atτp. We construct the category of motives overKusing absolute Hodge cycles rational overK. Then Theorem 5.3 of Blasius, [Bl2] implies that an idempotent e in Mor⁰_AH(X, X), in the notation of the article of Deligne-Milne in [DMOS], gives the desired crystalline realization of (X, e). HereXis an abelian variety defined overKand have good reduction atτ_p.) Letk be the residue field of K_P and K_P,0 be the quotient field ofW(k). We identifyK_P,0with the maximal unramified extension ofQp

contained inK_P. H_cris(M) is a freeE⊗ZW(k)∼=E⊗QK_P,0-module of rank 1. The isomorphism (1.3) transfers to theE⊗_QB_cris-linear isomorphism (1.3) I_cris:H_p(M)⊗QpB_cris∼=H_cris(M)⊗W(k)B_cris.

The isomorphism is compatible with the action of Gal(Q_p/K_P) and the Frobe- nius map as noted below (1.3). Letf be the degree ofP overQ. We have (2.4) Φ^f |H_cris(M) =χ(P)⊗1.

The relation (2.4) seems to be known to specialists. Since we cannot find a reference, let us give a sketch of the proof. If (2.4) holds forM(χ₁) andM(χ₂), then it holds forM(χ₁χ₂). Therefore we may assume that the infinity type of

4Here, strictly speaking, regardingKas a subfield ofCp byτp, an algebraic closure of K is taken insideCp. This convention will be applied hereafter, because to mention the choice of an extension ofτp to an embedding of an algebraic closure ofKinto Cp

everytime is cumbersome.

χcorresponds to a CM-type Φ ofK. Let (K,Φ) be the reflex of (K,Φ). We haveE₀ =K,K ⊂E. Let (K,Φ) be the reflex of (K,Φ). Then we have K⊂Kand Φ = Inf_K/K(Φ).⁵ There exists an algebraic Hecke characterχ and a Hecke character of finite orderω ofK such thatχ= (χ◦N_K/K)×ω.

We can chooseχso that it is unramified at the prime divisor lying belowP. If (2.4) holds forM(χ), then we see easily that it also holds forM(χ). Therefore we may assume that (K,Φ) = (K,Φ), replacing χ by χ if necessary. We apply Shimura [S2], Theorem 21.4 witha =OK. Let χbe the corresponding Hecke character of K_A^× to χ. Observe that χ restricted to K^×U satisfies the conditions (19.10a) and (19.10b) of [S2], whereU is an open subgroup ofK_A^× which containsO_K^×_P and the infinite part ofK_A^×. Then we see that there exist a finite abelian extensionLofK and an abelian varietyAdefined overLwith the following properties. (i) End(A)⊗Q∼=K, 2 dimA= [K :Q]. (ii)Ais of CM-type (K,Φ) and have good reduction at every prime lying over P. (iii) Let Z(s, A/L) be the zeta function of the motive H¹(A), which is of rank 1 and with coefficients inK. We have Z(s, A/L) =L(s, χ◦N_L/K). (iv)P is unramified inL/K.

Put ψ = χ ◦N_L/K. We have H¹(A) ∼= M(ψ). Since the crystalline cohomology calculates the zeta function ofAcorrectly, (2.4) holds forH¹(A), i.e.,

(2.5) Φ^{f fL} |H_cris¹ (A) =ψ(P)⊗1,

where P is a prime factor of P in L and fL is the relative degree ofP over L. By (2.5), we see that Φ^{f fL} acts on the crystalline realization by ψ(P) which can be identified with an element of End(A). Call it Σ_Pe. Since (K,Φ) is primitive, Σ_Pe is rational overL(cf. [S2], p. 65, Proposition 30). The Frobenius automorphismσ_Pe also acts onH¹(A) byψ(P) = Σ_Pe ∈K, for a prime=p.

Now put B = RL/K(A), where RL/K denotes the restriction of scalars functor of Weil. We regardH¹(B) as a motive over K with coefficients in E.

ThenM(χ) is realized as a direct factor of H¹(B). Lete ∈Mor⁰_AH(B, B) be the idempotent such thateH¹(B) =M(χ). As a Galois module,H¹(B) is the induced module fromH¹(A). From the endmorphism Σ_Pe, we can obtain an endomorphism Σ_P of H¹(B) so that the action of σ_P on H¹(B) is given by Σ_P for =p. By (2.5), we similarly see that the action of Φ^f onH_cris¹ (A) is given by Σ_P. (To see this, we first note that the casefL = 1 is easy. We may assume that Gal(L/K) is a cyclic extension generated byσ_P. By the form of

5The homomorphism Inf_K/K fromI_K toIK is defined in [S2], p. 197; Inf_K/K(Φ) consists of all extensions of elements in ΦtoJK.

the induced module and using the fact thatA is isogenous toσ(A) overL for everyσ∈Gal(L/K), we see that the action ofσ_Pis given by the action of an (explicitly given) endomorphism Σ_P, which is rational overL. We see that Σ_P is rational overKusing the fact thatLis abelian overK. A similar argument applies to the crystalline case with the same Σ_P sinceH_cris(B) is the induced module fromH_cris(A) with respect to the action of Φ^f.) Hence we have (2.6) I_cris(Σ_P(x)) = Φ^f(I_cris(x)), x∈H_p¹(B).

We haveeΣ_P=χ(P)eonH¹(B), which implies

(2.7) eΣ_P=χ(P)e

onH¹(B). Now by (2.6) and (2.7), we have

Φ^fe(I_cris(x)) =eΦ^f(I_cris(x)) =eI_cris(Σ_P(x)) =eΣ_PI_cris(x) =χ(P)eI_cris(x) forx∈H_p¹(B). Hence (2.4) follows.

There is a K_P,0-structure on H_DR(M) ⊗K K_P which we denote by H_DR(M/K_P,0). The isomorphism (1.4) transfers to

(1.4) I₀:H_cris(M)∼=H_DR(M/K_P,0).

§3. Definition of thep-Adic Period

LetK,χ and M =M(χ) be the same as in §2. We chooseτc ∈JK and τp∈Jp,K. LetP be the prime ideal ofK determined byτp.

Now take 0=c_B ∈ H_τc,B(M). Then i_p(c_B) is a generator ofH_p(M) as anE⊗QQ_p-module. Takec_DR∈ H_DR(M) which is a generator ofH_DR(M) as anE⊗QK-module. Then we define the periodP(χ;τc, τp) by

(3.1) I_DR(i_p(c_B)⊗1) =P(χ;τ_c, τ_p)(c_DR⊗1).

We haveP(χ;τ_c, τ_p)∈(E⊗_QB_DR)^×; it is determined up to multiplication of elements of (E⊗QK)^×. Here we regard E⊗QK as a subring ofE⊗QB_DR bya⊗b→a⊗τp(b). When no confusion is likely, we abbreviateP(χ;τc, τp) to P(χ).

We assume thatχ is unramified atP. We assume that (A) c_DRbelongs toH_DR(M/K_P,0).

This condition is satisfied ifK_P=K_P,0. Then we can choose a generatorc_cris ofH_cris(M) as anE⊗ZW(k)-module so that

(3.2) I₀(c_cris⊗1) =c_DR⊗1.

We defineP(χ; τc, τp)∈(E⊗QB_cris)^× by

(3.3) I_cris(ip(cB)⊗1) =P(χ;τc, τp)(c_cris⊗1).

SinceB_cris⊂B_DR, we can regardP(χ;τc, τp)∈(E⊗QB_DR)^×. Then, by (1.5) and (3.2), we have

(3.4) P(χ;τc, τp) =P(χ;τc, τp).

In view of this relation, we writeP(χ; τc, τp) asP(χ;τc, τp), or for simplicity as P(χ). (Thep-adic period in the standard sense isP(χ). We can makeB_DRde- scend toB_cris, which is a crucial observation since the Frobenius endomorphism Φ_cris can act onB_cris.)

Now we apply the Frobenius map on the both sides of (3.3). We have I_cris(ip(cB)⊗1) = Φ_cris(P(χ))(Φ(c_cris)⊗1).

Repeating this operationf times and using (2.4), we get I_cris(i_p(c_B)⊗1) = Φ^f_cris(P(χ))(χ(P)c_cris⊗1).

Comparing with (3.3), we obtain

(3.5) Φ^f_cris(P(χ))(χ(P)⊗1) =P(χ).

We state two formal consequences of the existence of the motiveM(χ). Let χ₁ andχ₂ be Gr¨ossencharacters ofK of the type (2.1). LetEbe an algebraic number field which contains the values ofχ₁ and χ₂. We regardM(χi) as a motive overKwith coefficients inE. Then sinceM(χ₁)⊗EM(χ₂) =M(χ₁χ₂), we obtain

(3.6) P(χ₁χ₂)≡P(χ₁)P(χ₂) mod (E⊗_QK)^×

Next letL be a CM-field which containsK. Since the extension of scalars of M(χ) toLisM(χ◦NL/K), we have

(3.7) P(χ;τc, τp)≡P(χ◦N_L/K; ˜τc,τ˜p) mod (E⊗QL)^×.

Here ˜τc∈JL, ˜τp∈Jp,L denote (any) extensions ofτc and τp respectively.

§4. The Invariant Q

In this section, we will examine thep-adic periodP(χ) in more detail. We fix an embedding Q ⊂C_p. Recall our convention thatQ is a subfield of C.

Takeτc = id,τp= id in (3.1). Since Q_p⊂B_DR, we have an isomorphism E⊗_QB_DR∼=⊕σ∈J_p,EB_DR.

We write

P(χ) = (P(χ)(σ))_σ∈Jp,E

according to this decomposition. Takeσ∈Jp,E ands∈Gal(Q/Q) such that s|E=σ. By (2.2) and [Y1], Lemma 2.1, (1), we see that the σ-component of F^m⊗K,idQ_p is nonzero for m l(s⁻¹) and zero for m > l(s⁻¹). Since I_DR preserves the filtrations, we have

(4.1) v_DR(P(χ)(σ)) =−l(s⁻¹).

Since H_cris(M, W(k))⊗W(k)K_P,0 is a free E⊗QK_P,0-module of rank 1, we can write

(4.2) Φⁱ(c_cris⊗1) =Q⁽ⁱ⁾(c_cris⊗1), 1i∈Z

withQ⁽ⁱ⁾∈(E⊗QK_P,0)^×. Applying the Frobenius mapi-times on (3.3), we obtain

Φⁱ_cris(P(χ))Φⁱ(c_cris⊗1) =P(χ)(c_cris⊗1).

Therefore we obtain

(4.3) Φⁱ_cris(P(χ))Q⁽ⁱ⁾=P(χ), 1i∈Z.

We putQ=Q⁽¹⁾. Then we have

(4.4) Φ_cris(P(χ))Q=P(χ).

LetQ^ur_p be the maximal unramified extension ofQ_p and letϕ∈Gal(Q^ur_p /Q_p) be the absolute Frobenius map. Applying Φ_cris on both sides of (4.4) noting that Φ_cris acts onK_P,0 by the absolute Frobenius mapϕ, we get

Φ²_cris(P(χ))ϕ(Q) = Φ_cris(P(χ)) =P(χ)Q⁻¹. Hence we have

Q⁽²⁾=ϕ(Q)Q.

Repeating this process, we obtain the relation

(4.5) Q⁽ⁱ⁾=ϕⁱ⁻¹(Q)· · ·ϕ(Q)Q, 1i∈Z, which can also be derived directly from (4.2). We note that (4.6) Q^(f)=χ(P)⊗1∈(E⊗_QK_P,0)^× which follows from (3.5) (or directly from (2.4)).

§5. The Main Conjecture

Let K, an embedding K ⊂ C_p, P and χ be the same as before. We assume thatχis unramified atP. Suppose that the CM-fieldKis abelian over a totally real number fieldF as in our construction oflgp,K/F. UsingQ⁽ⁱ⁾, we can predict the nature oflg_p,K/F in the general case. Takeτ∈Gal(K/F). We have

Q⁽ⁱ⁾∈(E⊗_QK_P,0)^×⊂(E⊗_QQp)^× ∼=

σ∈J_p,E

Qp×

.

LetQ⁽ⁱ⁾(σ)∈Qp×

denote theσ-component ofQ⁽ⁱ⁾with respect to this decom- position. Letpbe the prime ideal ofF lying belowPand letf₀be the degree ofpoverQ. We take a Hecke characterχof the form

χ((α)) = (ρτ(α)/τ(α))^l, α≡1 mod^×f with 1l∈Z. We can takel andχso thatE⊂K.

Conjecture Q. We assume (A) and that the prime idealpis unramified inK. Then we have

lg_p,K/F(id, τ⁻¹) = 1

2llog_pQ^(f0)(id) +alog_pb witha∈Q,b∈K^×.

Since P(χ) is determined up to multiplication by elements in (E ⊗_Q (K∩K_P,0))^×, Q^(f0) is determined up to multiplication by elements of the form ϕ^f0(c)/c, c ∈ (E⊗_Q (K∩K_P,0))^× by (4.3). Here ϕ acts on E⊗Q(K∩K_P,0) through the second factor. Therefore the validity of Conjec- ture Q does not depend on the choices ofcB,c_cris andc_DR.

Suppose thatp splits completely inK. Then f₀=f and we have Q^(f)= χ(P)⊗1 by (4.6). Since ϕ^f acts trivially on K_P,0, Q^(f) is determined inde- pendently of the choices ofcB,c_cris andc_DR. Put P^hK = (α). Take a positive integermso thatα^m≡1 modf. Then we have

1

2llog_pQ^(f0)(id) = 1 2lmhK

log_p(ρτ(α^m)^l/τ(α^m)^l) = 1 2hK

log_p(ρτ(α)/τ(α)).

Thus, in this case, Conjecture I.A implies that Conjecture Q holds with an explicitly given termalog_pb.

Remark1. Ogus [O] defined Q⁽ⁱ⁾ using results in Berthelot-Ogus [BO]

on crystalline cohomology. According to his results in [O], we can show that Conjecture Q is true whenF=QandK=Q(ζn) withζna primitiventh root of unity, (p, n) = 1. We can apply similar arguments as in [Y4], Chapter III,§2 as follows. In this Remark, we putLQ(τ) = ¹

2llog_pQ⁽¹⁾(id) forτ∈Gal(K/F) and for an algebraic Hecke character as in Conjecture Q. We define σ(a) ∈ Gal(Q(ζn)/Q) fora∈(Z/nZ)^× byζn^σ(a)=ζ_n^a. Letr, s, tbe integers satisfying 0< r, s, t < n, r+s+t =n, (r, s, t, n) = 1. Then Φr,s,t ={σ(a)|a∈Hr,s,t} withH_r,s,t ={a∈(Z/nZ)^×| ar+as+at=n}is a CM-type ofQ(ζ_n).

Here fora∈Z/nZ, we denote by aan integer which satisfies 0≤ a< n, a ≡ a modn. We consider the Fermat curve F_n : xⁿ +yⁿ = 1 and a differential form ηr,s,t = x^r−1y^s−ndx on Fn. Let Ar,s,t be the factor of the Jacobian variety ofFnwhich corresponds to differential forms{η_ar,as,at| a ∈ Hr,s,t}. Then Ar,s,t is an abelian variety defined over Q, which is of CM-type (Q(ζ_n),Φ_r,s,t). We denote the algebraic Hecke character of Q(ζ_n) associated toA_r,s,tbyχ_r,s,t. Thenχ_r,s,tsatisfies thatl_σ = 1 ifσ⁻¹∈Φ_r,s,t, = 0 otherwise. We may takec_cris=c_DR=

a∈(Z/nZ)^×ηar,as,at. Note that ηr,s,t is equal toξ₋r

n,−_n^s,−_n^t in [O]. By Proposition 2.4 in [O], fora∈(Z/nZ)^× we have

log_pQ⁽¹⁾(χ_r,s,t)(σ(a))≡log_p

Γ_p(^ar_n)Γ_p(^as_n) Γp(^ar+_n ^as)

modQlog_pQ(ζ_n)^×. It gives ap-adic counterpart of (2.3) in [Y4], Chapter III. It also gives

log_pQ⁽¹⁾(χr,s,t)(σ(a)) + log_pQ⁽¹⁾(χr,s,t)(σ(−a))≡0 modQlog_pQ(ζn)^×. Note that we haveQ(χ)⁽ⁱ⁾(τ σ)≡Q(τ◦χ)⁽ⁱ⁾(σ) for anyχ,i,τ, σsince we may identify theτ σ-component of H_cris(M(χ))⊗K_P,0Qp and the σ-component of H_cris(M(τ◦χ))⊗K_P,0Q_p (cf. (8.2)). Therefore we can write

log_pQ⁽¹⁾(χ_r,s,t)(id)≡

σ∈Φr,s,t

LQ(σ⁻¹) modQlog_pQ(ζ_n)^×.

By the same arguments as for (2.5) in [Y4], Chapter III, we have

[(n−1)/2]

t=1,(t,n)=1

atLQ(σ(t)⁻¹)≡log_p

Γp(^a_n)Γp(¹_n) Γp(^a+1_n )

modQlog_pQ(ζn)^×,

wherea∈(Z/nZ)^×,T_a ={t∈(Z/nZ)^× | ^at_n+_n^t <1}, _at= 1 if t∈T_a,

=−1 otherwise. Other arguments are the same as for the proof of Theorem 2.6 in [Y4], Chapter III if we replace Γ, L(s, η) by Γp, Lp(s, ωη) respectively.

LetK a CM-subfield ofQ(ζn). For σ∈G= Gal(K/Q), we have LQ(σ⁻¹)≡

η∈Gˆ₋

η(σ) [K:Q]

L_p(0, ωη)

L(0, η) modQlog_pQ(ζn)^×.

Although we do not give the details here, we can replace modQlog_pQ(ζn)^× by modQlog_pK^× by further computations.

Remark 2. In thep-adic case, the nature oflg_p,K/F(id, τ) would depend heavily on the choice ofF. This is the main difference from the complex case.

Remark 3. Using (3.6), it is easy to see that the left-hand side of Con- jecture Q, taken moduloQlog_pK^×, is independent of the choice ofl andχ.

§6. Toward the Understanding of thep-Adic Period

Conjecture Q in the previous section gives information on theQ-invariant.

We will investigate thep-adic period itself in relation to this conjecture. We will see that, in the ordinary case, thep-adic period can be recovered from the Q-invariant modulo Q^×_p in the next section.

Let v be the additive valuation of Cp normalized by v(p) = 1. We fix an embedding Q ⊂ C_p. Let K be a CM-field and P be the prime ideal of K obtained from the embedding K ⊂ Q ⊂ C_p as before. Let e and f be the ramification index and the degree ofPoverQrespectively. We recall our notations. We take a Hecke characterχofK satisfying (2.1). We assume that χis unramified at P. The p-adic periodP(χ) =P(χ; id,id)∈(E⊗_QB_cris)^× is defined by (3.3). TheQ-invariantQ=Q⁽¹⁾ ∈(E⊗_QK_P,0)^× is defined by (4.2).

Lemma 6.1. PutG= Gal(Q/Q),H= Gal(Q/K). LetP be the prime divisor ofQinduced by the embedding Q⊂Cp and let Z be the decomposition

group ofP. For σ∈Jp,E, we take s∈Gal(Q/Q) so that s|E =σ. Then we have

(6.1) v(σ(χ(P))) = 1

e

τ∈ZH/H

l(s⁻¹τ).

Proof. Take 0< m∈Zso thatP^m= (α),α∈ OK,α≡1 modf. Since χ(P)^m=

τ∈J_K

τ(α)^lτ ∈E,

we have

v(σ(χ(P))) = 1 mv

⎛

⎝s

τ∈G/H

τ(α)^l(τ)

⎞

⎠= 1 mv

⎛

⎝

τ∈G/H

τ(α)^l(s−1τ)

⎞

⎠.

Since

v(τ(α))>0⇐⇒τ∈ZH, we have

v(σ(χ(P))) = 1 mv

⎛

⎝

τ∈ZH/H

τ(α)^l(s−1τ)

⎞

⎠.

Forτ ∈ZH/H, we havev(τ(α)) =m/e. Hence the assertion follows.

Corollary 1. IfPis of degree1 and unramified overQ, i.e.,e=f = 1, then we have

v(σ(χ(P))) =l(s⁻¹).

The assertion is obvious sinceZ ⊂H.

Corollary 2. If p splits completely in E₀, then we have v(σ(χ(P))) = f l(s⁻¹).

Proof. PutH= Gal(Q/E₀). Sincepsplits completely inE₀, we have g⁻¹Zg⊂H for allg∈G.

Since the functionl is leftH-invariant and rightH-invariant, we obtain, from (6.1),

v(σ(χ(P))) = l(s⁻¹)

e [ZH:H] = l(s⁻¹)

e [Z :H∩Z] =l(s⁻¹)

e [K_P:Q_p].

This completes the proof.

Hereafter until the end of §7, we assume that pis unramified in E. Let Q^ur_p be the maximal unramified extension ofQpandQ^ur_p be the closure ofQ^ur_p in Cp. Let ϕ ∈ Gal(Qp/Qp) be a Frobenius map which gives the absolute Frobenius automorphism of Gal(Q^un_p /Qp). We use the same letterϕ for the extension of ϕ|Gal(Q^un_p /Q_p) to the continuous automorphism of Q^ur_p . Since σ(E)⊂Q^ur_p for everyσ∈J_p,E, we have

(6.2) E⊗_QQ^ur_p ∼=⊕σ∈J_p,EQ^ur_p .

We writex∈E⊗QQ^ur_p as x= (x(σ))σ∈J_p,E according to this decomposition.

Similarly, sinceQ^ur_p ⊂B_cris, we have

(6.3) E⊗QB_cris∼=⊕σ∈J_p,EB_cris.

We writex∈E⊗_QB_cris asx= (x(σ))σ∈J_p,E according to this decomposition.

Lemma 6.2. Let x= (x(σ))σ∈J_p,E ∈ E⊗_QQ^ur_p , y = (y(σ))σ∈J_p,E = (1⊗ϕ)(x)∈ E⊗_QQ^ur_p . For σ∈ J_p,E, define σ₁ ∈J_p,E by σ₁|E =ϕ⁻¹σ|E.

Then we have y(σ) =ϕ(x(σ₁)). Similarly for x= (x(σ))_σ∈Jp,E ∈E⊗QB_cris andy= (y(σ))σ∈J_p,E = (1⊗Φ_cris)(x)∈E⊗QB_cris, we havey(σ) = Φ_cris(x(σ₁)).

Proof. Here we prove only the first assertion. The proof of the second as- sertion is similar. We writeE=Q(θ) and letf(X) be the minimal polynomial ofθoverQ. InQ^ur_p [X], we can decomposef(X) as

f(X) =

σ∈J_p,E

(X−σ(θ)).

For h ∈ Q[X] and a ∈ Q^ur_p , h(θ)⊗a is mapped to (h(σ(θ))a)_σ∈Jp,E and (1⊗ϕ)(h(θ)⊗a) is mapped to (h(σ(θ))ϕ(a))_σ∈Jp,E by the isomorphism (6.2).

It suffices to prove the relation when x(σ) = h(σ(θ))a, y(σ) = h(σ(θ))ϕ(a), σ∈Jp,E. Now we have

ϕ(x(σ₁)) =h(ϕσ₁(θ))ϕ(a) =h(σ(θ))ϕ(a) =y(σ) and the proof is complete.

Lemma 6.3. If x ∈Q^ur_p satisfies v(x) = 0, then there exists y ∈ Q^ur_p such thatϕ(y)y⁻¹=x. Moreover y is unique up to multiplication by elements inQ^×_p.