ON A TRIVIAL ZERO PROBLEM

PII. S0161171204210419 http://ijmms.hindawi.com

© Hindawi Publishing Corp.

ON A TRIVIAL ZERO PROBLEM

SHAOWEI ZHANG Received 25 October 2002

One trivial zero phenomenon forp-adic analytic function is considered. We then prove that the first derivative of this function is essentially the Kummer class associated withp.

2000 Mathematics Subject Classification: 11S40.

1. Introduction. In this paper we always fix an odd primep >2. Forn≥1, fix a pⁿth primitive root of unityζpⁿ such thatζ_p^p_n+1=ζpⁿ. LetKn=Qp(ζpⁿ)and ᐁ= lim←O_K^×_n. Forβ∈ᐁ, we will define a 1-admissible distributionµβ∈Ᏸ1(Qp,Qp[−1])^Φ=1 (seeSection 3). Consider the integral

ψk(β)=

Z^×p

x^kµβ, (1.1)

then we haveψk(β)=(1−p^k−1)·

Zpx^kµβ, so it will have a trivial zero atk=1. Since 1−p^k−1is not an analytic function ofk, hence we cannot take the derivative directly.

Butψkis an analytic function ofk, so the derivative exists. This phenomenon in which the zero is forced by Euler factor is called trivial zero problem. Ferrero and Greenberg [4] considered the trivial zero problem for the first time in 1978 and found that the derivative has deep arithmetic meaning. The behavior of the derivative of some Kubota- Leopoldtp-adicL-function with trivial zero has a deep relation with some arithmetic Iwasawa module (see [6]). The second such trivial zero phenomenon was found by Mazur et al. in [8], and then they conjectured that the derivative has a relation withL- invariant. This conjecture was proved by Greenberg and Stevens in 1993 (see [7]). The functionψkis very close to Coates-Wileskth derivative (seeSection 7); actually, it only differs by the factor(1−p^k−1), and was calledCoates-Wiles homomorphismin de Shalit [3]. The question to find the derivative atk=1 ofψkwas proposed by Glenn Stevens in 1997. Simultaneously, we also tried to understand how the Bloch-Kato exponential map exp_Qp(1)can miss the Kummer classγp. Glenn Stevens predicted that the derivative of ψkat 1 will give the Kummer classγp. We will prove this in this paper.

LetCpdenote the completion of ¯Qp. For a fieldK⊂Cp, letOK denote the ring of integers. Choose Iwasawa’s log :C^×p→Cp such that log(p)=0. InSection 2, we will review Fontaine’s rings briefly and describe Bloch-Kato exponential map. InSection 3, we will define distributions and explain cohomology groups as Iwasawa module. In Section 4, we will introduce algebraic Fourier transformation and use Coleman power series to give some special distributions. InSection 5, we will review Perrin-Riou and Colmez theorems. In Section 6, we will show that Iwasawa’s explicit reciprocity law

is actually a special case of Perrin-Riou’s theorem. InSection 7, we use the theory we developed so far to prove our theorem.

2. Fontaine’s rings and Bloch-Kato exponential map. Let ¯O=O_Cp/pO_Cp. Let᏾^de- note the projective limit of the diagram

O¯← O¯← O¯←···, (2.1)

where the transition maps are given byx→x^p. The ring᏾is a perfect ring with charac- teristicp >0 (see [5]). Forx∈᏾^,x=(xn)n∈Nsatisfiesxn∈O, and¯ x_n+1^p =xn. For each n, choose ˜xn∈O_Cpto be a representative ofxn. Then one can show that for eachm, limn→∞x˜n^pn+mexists and the limitx^(m)does not depend on the choices of the represen- tatives. Hence,xgives rise to a sequence(x^(m))m∈NinO_Cpsuch that(x^(m+1))^p=x^(m). On the other hand, if we have a sequence(x^(m))_m∈NinO_Cpsuch that(x^(m+1))^p=x^(m), then(¯x^(m))m∈Nis an element in᏾. Hence,᏾is in one-to-one correspondence with the set

x^(m)

m∈N| ∀m∈N, x^(m)∈O_Cp,

x^(m+1)p

=x^(m)

. (2.2)

Define a functionv_᏾:᏾→Q∪{∞}by v_᏾

x^(m)

m∈N

:=v x⁽⁰⁾

, (2.3)

wherevis the valuation ofCpsuch thatv(p)=1. The ring᏾is complete with respect tov_᏾.

LetW (᏾)denote the Witt vector ring of᏾. Recall that the underlying set of W (᏾) is the set᏾^N = {(x0, x1, . . .)|xi∈᏾}. The ring structure is given in terms of Witt polynomials (see [10]). Since ¯Ois an ¯Fp-algebra,W (᏾)is aW (¯Fp)-algebra. Forx∈᏾, let

[x]:=(x,0,0, . . .)∈W (᏾) (2.4) denote the Teichmüller representative ofx. For(x0, x1, . . . , xn, . . .)∈W (᏾), we have the identity

x0, x1, . . . , xn, . . .

= x0

+p x1

p⁻¹

+···+pⁿ xn

p⁻ⁿ

+···, (2.5) where forx∈᏾^,[x]p−1 is the unique elementwofW (᏾^{)such thatwp}=[x].

Let

θ:W (᏾⁾ →O_Cp (2.6)

be defined by

θ

x0, x1, . . .

=

∞ n=0

pⁿx_n⁽ⁿ⁾. (2.7)

Then it is easy to see thatθis aZp-homomorphism and it is surjective. The Frobenius on

᏾induces a continuous Frobenius map onW (᏾)with respect to the product topology, we denote it byϕ, which sends(x0, x1, . . . , xn, . . .)to(x^p₀, x₁^p, . . . , xn^p, . . .). The mapϕis an isomorphism, semilinear overW (¯Fp). The ringW (᏾⁾can also be endowed withp- adic topology andI-adic topology. Letε=(1, ζp, ζ_p2, . . .)∈᏾. The element[ε]∈W (᏾) has the property θ([ε])=1. The elementϕ⁻¹([ε])=[(ζp, ζ_p2, . . .),0,0, . . .]. Let u= ([ε]−1)/(ϕ⁻¹[ε]−1). The kernel ofθis a principal ideal ofW (᏾), which is generated byu[5].

We will useB⁺_dR,BdR,Acrys, B⁺_crys,Bcrys,Amax, andBmaxfrom Colmez [2].

Lemma2.1. The following sequences are exact:

0 →Qp →Bmax^ϕ=1 Fil^<0

→BdR/B⁺_dR →0, (2.8)

0 →Qp →Fil⁰Bmax ϕ−1

→Bmax →0, (2.9)

whereϕis the Frobenius ofBdRwhich is induced by the one from᏾^. Proof. See Colmez [2, Appendix A].

For a continuousG_Qp-representationV, finite-dimensionalQp-vector space, define Dcrys(V ):=(Bcrys⊗V )^GQp,DdR(V ):=(BdR⊗V )^GQp. ThenDcrys(V )is a finite-dimensional Qp-vector space, with a Frobenius action (acts onVtrivially) [5]. The operatorDdRhas a filtration given by Filⁱ(DdR(V ))=(B_dRⁱ ⊗V )^GQp. The dimensions have the following relation:

dim_Qp

Dcrys(V )

≤dim_Qp

DdR(V )

≤dim_Qp(V ). (2.10) If dim_Qp(DdR(V )) = dim_Qp(V ), then V is called a de Rham representation. If dim_Qp(Dcrys(V ))=dim_Qp(V ), thenV is called a crystalline representation. Note that a crystalline representation must be a de Rham representation. In the following, all representations are assumed to be de Rham representations. Similarly, we can also de- fineDmax(V ):=(Bmax⊗V )^GQp; Colmez proved that this is the same asDcrys(V ). For a crystalline representationV, letD(V )=Dcrys(V ).

For a de Rham representationV, taking tensor product with the exact sequence (2.8), we have the following exact sequence:

0 →V →Bmax^ϕ=1⊗V →BdR/B_dR⁺ ⊗V →0; (2.11) taking the Galois cohomology, we have a map

BdR/B_dR⁺ ⊗VG_Qp

→H¹ Qp, V

. (2.12)

Then the Bloch-Kato exponential map expV:

BdR⊗VG_Qp

→H¹ Qp, V

(2.13) is defined as the composition

BdR⊗VG_Qp

→

BdR/B⁺_dR⊗VG_Qp

→H¹ Qp, V

. (2.14)

The kernel of this map is Fil⁰DdR(V )+Dcrys(V )^ϕ=1, and the image is He(Qp, V ):= ker{H¹(Qp, V )→H¹(Qp, Bcrys^ϕ=1⊗V )}.

For a Galois representationV, letV (k)denote thekth cyclotomic twist ofV. That is, letχdenote the cyclotomic character,ζ_p^σn=ζ_p^{χ(σ )}n for alln≥1,V (k):=V (χ^k).

Consider the exampleV =Qp(1)=Qp·e; in this case, DdR(V )=Qp·(e/t) is a one-dimensional vector space, wheret=log([ε]). The isomorphismH¹(Qp,Qp(1)) Q^×p⊗Qpis given by the Kummer map. To be more precise, it is generated byγ_1+p,γp, where, forα∈Qp^×,

γα:τ →log_ε

. . . ,τ α^1/pn α^1/pn , . . .

⊗e (2.15)

is the Kummer class. Hence, we haveH¹(Qp,Qp(1))Qp², then the exponential map forQpis

exp_Qp(1):DdR

Qp(1)

→H¹

Qp,Qp(1)

. (2.16)

Lemma2.2. It follows that

exp_Qp(1) e

t

= γ_1+p

log(1+p). (2.17)

Proof. In the exact sequence

0 →Qp →Bmax^ϕ=1 →BdR/B⁺_dR →0, (2.18) log[_{1+p]/t·log(1+p)}maps to 1/t, so(log[_{1+p]/t·log(1+p))⊗emaps to(1/t)⊗e∈} DdR(Qp(1)), hence the class exp_Qp(1)(e/t)is represented by

τ →(τ−1)·

log_1+p t·log(1+p)⊗e

= 1

tlog(1+p) log

. . . , τ

(1+p)^1/pn , . . .

, . . .

−log

. . . , (1+p)^1/pn, . . . , . . .

⊗e

= 1

tlog(1+p)

log

. . . ,τ

(1+p)^1/pn (1+p)^1/pn , . . .

, . . .

= 1

tlog(1+p) log

ε^γ1+p(τ)

= γ_1+p(τ) log(1+p).

(2.19)

Fork >1, it is easy to see that dim_QpDdR(Qp(k))=dim_QpH¹(Qp,Qp(k))=1 and exp_Qp(k)is an isomorphism. In some sense,γpandγ_1+pshould have the same positions inH¹(Qp,Qp(1)). Note that fork=1, the left-hand side has dimension 1 and the right- hand side has dimension 2, so the image is a one-dimensional vector space, andγpis not in the image. In this paper, we will show that the “derivative of Bloch-Kato map” is essentiallyγp. To be a little bit more precise, we need the following definitions.

Letᐄ=Homcont(Z^×p,C^×p)which is identical toB(µp−1,1)and there is an obvious in- clusionZ⊂ᐄ.

Definition2.3. Given᏿⊂ᐄ, a rigid analytic subspace overQp, an analytic family of Galois representations over᏿is a pair(V , ρ), where (1)Vis a de Rham representation ofG_Qp, (2)ρ:᏿×G_Qp→Gl_Qp(V )is continuous inσ and is analytic ink.

Definition2.4. Let(V , ρ)over᏿be a family of Galois representations ofG_Qp and letVkdenote the Galois representation ofG_Qpsuch that the underlying space isVand the action is given by

σ◦v=ρk(σ ) σ (v)

. (2.20)

A family of classesξk∈H¹(Qp, Vk)is said to be an analytic family if there is a cocycle representationσ→ξk(σ )such that for allσ∈G_Qp,ξk(σ )is an analytic function ofk.

Now, we can go back to answer the question onγp. InSection 7, we will show that ψk=((1−p^1−k)/(1−p^−k))(k−1)! exp_V

k(1k)is an analytic family of cohomology classes inH¹(Qp,Qp(k))and(d/dk)(ψk)|k=1= −(1−p⁻¹)⁻¹γp. In other words,γpappears in the first coefficient of the “Taylor expansion” of Bloch-Kato exponential map.

3. Distributions and Iwasawa module. Let I ⊂Z be a subset and let LP^I = {x^k· 1_a+pⁿ_Zp|k∈I, a∈Qp}. An algebraicI-distribution with values inM is a finitely ad- ditive function µ:LP^I→M. Let Ᏸ^Ialg(Qp, M) denote all the algebraicI-distributions with values inM. ForX⊂Qp, a compact open subset, letLP^I(X)= {x^k·1(a+pⁿZp)∩X}, then Ᏸ^I_alg(X, M)is defined with respect to these test functions. Especially, we have Ᏸ^Ialg(Z^×p, M), Ᏸ^Ialg(Zp, M). Let Ᏸ⁺alg(Qp, M) (resp., Ᏸ⁻alg(Qp, M)) denote the case I =N (resp.,I= −N). Note that when we sayNwe always meanN= {0,1,2, . . .}.

Let LA= {locally analytic compactly supported functions inQpwith values inQp}.

LetLA= {f :Qp\ {0} →Qp|f is locally analytic and compact supported such that thereexistsN∈N, x^Nf∈LA}.LAandLAhave Morita topology.

We letAn(X)denote theQp-affinoid algebra ofB[X, p⁻ⁿ]. In particular,An(X)is a Banach algebra under the Gauss norm. For ap-adic Banach spaceA, letᏰcont(Qp, A):= {µ:LA→A|µis linear and continuous with respect to Morita topology}. Note thatµis continuous if and only if it is continuous when restricted on eachᏭn(X),n∈Z,Xopen.

Definition3.1. (a) Letµ ∈Ᏸcont(Y , A). For each n∈Z and every compact open subsetX ofY, defineµᏭn(X) to be the norm of the continuous linear function µ: Ꮽn(X)→Aobtained by restrictingµtoᏭn(X).

(b) Similarly, ifµ∈Ᏸ^Ialg(Y , A), then for eachn∈Zand every compact open subset XofY, defineµLP_n^I(X)to be the norm of the continuous linear functionµ:LP^I(X)∩ Ꮽn(X)→Aobtained by restrictingµ toLP^I(X)∩Ꮽn(X). IfX⊂Qp is compact, then actually

µᏭn(X)= sup

a∈X, j≥0

a+pⁿZp

x−a pⁿ

j

µ

. (3.1)

Definition3.2. Forr ∈R¯⁺,µ∈Ᏸcont(Qp, A)is said to be tempered of orderr if for every compact open subsetX⊂Qp,p^{−[nr ]}µᏭn(X)isr-bounded. LetᏰr(Qp, A)⊂ Ᏸcont(Qp, A) denote the set of distributions of order r. For r1 < r2, Ᏸr₁(Qp, A)

⊂Ᏸr₂(Qp, A). LetᏰtemp(Qp, A)= ∪r≥0Ᏸr(Qp, A)denote all tempered distributions with values inA. From the above remark we see thatµisr-bounded if and only if

p^{−[nr ]} sup

a∈X, j≥0

a+pⁿZp

x−a pⁿ

j

µ

(3.2)

isr-bounded. A distribution with orderr is also called anr-admissible distribution.

Lemma3.3. Forµ∈Ᏸcont(Qp, A),µhas orderrif and only if for allXcompact open, allx∈X,0≤j≤r,

p^{−[nr ]} sup

a∈X,0≤j≤r

a+pⁿZp

x−a pⁿ

j

µ

(3.3)

isr-bounded.

Proof. Sincej > r, thenp^{[n(j−r )]}

a+pⁿZp(x−a)^jtends to zero whenn→ ∞. IfV is a crystalline representation ofG_Qp, we have a twist map

Ᏸcont

Qp, D(V ) T w

→Ᏸcont

Qp, D

V (−1)

(3.4) which sendsµto(−tx)µ.

Lemma3.4. The kernelker(T w)=δ0⊗D(V ),T wis surjective.

Proof. Obviously, we haveT w(δ0⊗D(V ))=0. Ifµ∈ker(T w), then supp(µ)= {0}. Letµ1=µ−(

µ)δ0, then

f (x)µ1=

f (x)µ−(

µ)·f (0)=f (0)·( µ)−(

µ)·f (0)=0, henceµ=(

µ)⊗δ0.

For the surjectivity, givenν∈Ᏸcont(Qp, D(V (−1))), defineω∈Ᏸcont(Qp, D(V ))such that

f ω=

−t⁻¹ f−f (0)·1_Zp

x ν, (3.5)

then

f (x)(−tx)ω=

f ν, hence(−tx)ω=ν.

Forµ∈Ᏸ^I_alg(Qp, A), define an operatorϕ_Ᏸas

Qp

f (x)ϕ_Ᏸµ:=

Qp

f (px)µ. (3.6)

If Ais a Dieudonne module, thenϕ can act on it, hence bothϕand ϕ_Ᏸcan act on Ᏸ^Ialg(Qp, A). Then we defineΦ=ϕ_Ᏸ⊗ϕ.

Lemma3.5. The twist mapT winduces a map Ᏸcont

Qp, D(V )_Φ=1

→Ᏸcont

Qp, D

V (−1)_Φ=1

(3.7)

with kernel=δ0⊗D(V )^ϕ=1,image= {v∈Ᏸcont(Qp, D(V (−1)))^Φ=1|

Z^×px⁻¹ν=0}, and cokernel=D(V (−1))/(ϕ−p)D(V (−1))D(V )/(ϕ−1)D(V ).

Proof. Assume thatδ0⊗dis in the kernel,Φ(δ0⊗d)=δ0⊗d. For allf, we have fΦ(δ0⊗d)=

f δ0⊗d=f (0)⊗d, that is,ϕ(f (0)⊗d)=f (0)⊗d, henced∈D(V )^ϕ=1. Now, we calculate the image. Ifν=T w(µ)=(−tx)µ, then from the Colman-Colmez exact sequence [2], we have

Z^×px⁻¹ν=

Z^×p(−t)µ=0. On the other hand, ifνsatisfies

Z^×px⁻¹ν=0,ωmaps toν fromLemma 3.4, we need to show thatΦ(ω)=ω. That is, for allf,

f·Φ(ω)=

f ω. The calculation shows that

fΦ(ω)−

f ω

=ϕ

f (px)ω

−

f ω

=ϕ

f (px)−f (0)·1_Zp (−tx) ν

−

f ω

=(−t)⁻¹

f (x)−f (0)·1_Zp(x/p)

x ν−(−t)⁻¹

f (x)−f (0)·1_Zp

x ν

=(−t)⁻¹ 1_Z×

p·f (0) x ν=0.

(3.8)

The statement about cokernel follows immediately.

Define ˜Ᏸtemp(Qp, D(V )):=lim

←Ᏸtemp(Qp, D(V ))^Φ=1, where the transition maps are given by the above twist map.

Lemma3.6. Forµ∈Ᏸcont(Z^×p, A),µhas orderrif and only ifxµhas orderr. Proof. Assume that µ has order r with r ∈ R, then there is a constant C >0 such that for allj≥0,

a+pⁿZp(x−a)^jµ ≤Cp^[n(r−j)], hence

a+pⁿZpx(x−a)^jµ =

a+pⁿZp(x−a)^j+1µ−p^[n(r−j)]

a+pⁿZpa(x−a)^jµ ≤Cp^[n(r−j)]. Ifr∉ R, then we take thatC=Cntends to zero.

Ifµhas orderr, by using the expansion

a+pⁿZp(x−a)^r(µ/x)=

a+pⁿZp(x−a)^r(1/(a+ (x−a)))µ=

a+pⁿZp(x−a)^r·1/a·

k≥0((x−a)/a)^kµ, we see that

a+pⁿZp(x−a)^r(µ/x)

≤Cp^[n(r−j)], this proves the lemma.

Forµ∈Ᏸcont(Zp,Cp), define the Amice transformation Ꮽµ(T )=

Zp

(1+T )^xµ∈C[[T ]]. (3.9)

Definition3.7. A formal power seriesf (T )=

anTⁿ∈Cp[[T ]]is said to be of orderr ifp^{[nr ]}anisr-bounded.

Lemma 3.8. A distribution µ∈Ᏸcont(Zp,Cp) has order r if and only if Ꮽµ(T )has orderr.

Proof. See [1].

4. Fourier transformation and Coleman power series. Recall that we fixed ζpⁿ

which is apⁿth root of unity. Letεn:=(ζpⁿ, ζ_pn+1, . . .)∈᏾, note thatεn^pn =ε,[εn]∈ W (᏾). For x∈Qp, x=p⁻ⁿ·y with some n∈Nand y∈Zp, defineε^x :=εn^y ∈᏾. Obviously, this is well defined, and we get an element [ε^x]∈ W (᏾^{). For x} ∈Qp, exp(tx)=_+∞

k=0((tx)^k/k!) converges. Defineε(x):=[ε^x]/exp(tx) forx∈Qp. Then ε(x)has the following properties:

(i) ifx∈p⁻ⁿZ^×p, withn≥0, thenε(x)is apⁿth root of unity,ε(x)=1 if and only ifx∈Zp. Moreover,ε(1/pⁿ)=ζpⁿ;

(ii) it follows that

pⁿ−1

x=0

ε ax

pⁿ

=





pⁿ, ifa≡0

modpⁿ ,

0, otherwise fora∈Zp\pⁿZp; (4.1) (iii) forx, y∈Q,ε(x+y)=ε(x)ε(y);

(iv) for a cyclotomic characterχ,σ (ε(x))=ε(χ(σ )x).

Iff is a locally constant function with compact support inQp, define Ᏺalg(f )(y):=

Qp

f (x)ε(xy)µHaar(x), (4.2) whereµHaar∈Ᏸnaive(Qp,Qp)such thatµHaar(a+pⁿZp)=1/pⁿ. Sincefis locally con- stant, this means that we can find anmsuch that ona+p^mZp,f is constant, hence the integral equals

a

a+p^mZp

f (x)ε(xy)µHaar(x)= 1

p^m_amodpmf (a)

x∈a+p^mZp

ε(xy). (4.3)

From property (ii) ofε(x), ifyis outside ofp^−mZp, then this sum is zero, henceᏲalg(f ) is well defined and compactly supported. On the other hand, since f is compactly supported, we can assume thatfis supported onp^−mZpfor somem. Sinceε(p^−my)is locally constant, this implies thatᏲalg(f )is locally constant. Extend the above definition to test function{x^k·1a+pⁿZp, k≥0}, define

Ᏺalg(f)(y):=(−ty)Ᏺalg(f )(y). (4.4) Proposition4.1. The Fourier transformationᏲalgenjoys the following properties:

(i) Ᏺalg(f (x+a))(y)=ε(−ay)Ᏺalg(f )(y), fora∈Qp, (ii) Ᏺalg(ε(ax)f (x))(y)=Ᏺalg(f )(y+a),

(iii) Ᏺalg(f (cx))(y)= |c|⁻¹Ᏺalg(f )(c⁻¹y),

(iv) Ᏺalg(x^k·1a+pⁿZp)(y) = p⁻ⁿ·(k!/(−ty)^k)ε(ay)1p⁻ⁿZp(y) if k ≥ 0, n ∈ Z, a∈Qp,

(v) Ᏺalg◦Ᏺalg(f )(y)=f (−y).

Proof. The properties follow easily from the definitions.

Forh∈Z, define the twist forᏲalgas Ᏺ^(h)alg(f ):=(−ty)^h−1Ᏺalg

x^h−1f (x)

(y), (4.5)

wheref∈LP^[1−h,+∞), then we have Ᏺ^(h)alg

x^k·1a+pⁿZp

(y)=p⁻ⁿ(k+h−1)!

(−ty)^k ε(ay)1p−nZp(y) (4.6) for allk≥1−h,n∈Z,a∈Qp.

Now, we define the algebraic Fourier transformation on distributions as follows. For µ∈Ᏸ⁽alg^{−∞,h−1]}(Qp, D(V )), defineᏲ^(h)alg(µ)such that

Qp

f (x)Ᏺ^(h)alg(µ):=

Qp

Ᏺalg^(h)(f )µ. (4.7)

Forα∈Z^×p, letπ=pα. Letfπ(x)∈Zp[[x]]be a Frobenius corresponding toπ, so fπ(x)≡π x(mod deg 2)andfπ(x)≡x^p(modp). LetFbe the one-dimensional Lubin- Tate formal group overZpcorresponding tofπand let[+]denote the formal addition.

LetW_πⁿ:= {x∈Cp|fπ⁽ⁿ⁾(x)=0},Kn=Qp(W_πⁿ), andK_∞= ∪n≥1Kn. Hence,K_∞/Qpis a totally ramified extension with Galois groupZ^×p. We call this tower the Lubin-Tate tower corresponding to the formal groupF. LetR=Zp[[T ]]andᐁ=lim←O^×_Kn, where the map is with respect to the norm map. Assume thatβ∈ᐁ, then Coleman’s theorem tells us that there is a unique (Coleman) power seriesgβ∈Zp[[T ]]such that

(i) gβ(ωi)=βifor alli≥1, (ii) g_β^ϕ◦fπ(x)=

w∈Wπ¹gβ(x[+]w).

Assume thatβ∈ᐁ^{such thatβ}n≡1(modωn). Thengβ(T )≡1 mod(p, T ), hence we can define

logg β(T ):=loggβ(T )−1 pw∈Wπ¹

loggβ

T [+]w

. (4.8)

The property (ii) of the Coleman power series implies thatlogg β(T )has integral coef- ficients. Define an algebraic distributionµβ∈Ᏸ⁺_alg(Zp,Qp^ur)such that

Zp

(1+T )^xµβ(x)=loggβ◦η(T ). (4.9)

Proposition4.2. (i)The restriction ofµβ toZ^×p µβ|Z^×p is a measure and its Amice transformation isloggβ◦η(T ).

(ii)The distributionµβ|Z^×pis a measure inᏰ1(Qp,Q^urp )^Φ=1and has the following Galois property:

σ

Qp

f (x)µβ

=

Qp

f

ψ(σ )x

µβ ∀σ (4.10)

for allf (x):Qp→Qp.

Proof. It is easy to see that

Z^×p

(1+T )^xµβ=

Zp

(1+T )^xµβ−

pZp

(1+T )^xµβ. (4.11)

By property (ii),

gβ◦fπ(X)=

w∈Wπ¹

gβ

X[+]w

; (4.12)

letX=η(T ), then

gβ◦fπ η(T )

=

ζ∈µp

gβ

η(T )[+]η(ζ−1)

=

ζ∈µ_p

gβη

ζ(1+T )−1

. (4.13)

By usingfπ◦η=η^ϕ◦[p], we see that gβ◦ηϕ

◦[p]=

ζ

gβη

ζ(1+T )−1

; (4.14)

taking logarithm, and using the definition forµβ, we have ϕ

Zp

1+[p]Tx

µβ

=

ζ

Zp

ζ^x(1+T )^xµβ=p

pZp

(1+T )^xµβ. (4.15)

Hence,

Z^×p

(1+T )^xµβ=loggβη(T )−1 pϕ

loggβ◦η^ϕ◦ [p]T

=loggβ◦η−1

ploggβ◦fπ◦η(T )

=logg β◦η(T )

(4.16)

has integral coefficients, henceµβ|_Z^×_pis a measure.

To prove the second property, since

η(T ):Gm →Ᏺπ, (4.17)

by comparing the values atTn=ζpⁿ−1, we can show that σ

η(T )

=η

(1+T )^{ψ(σ )}−1

∀σ∈G_Qp. (4.18)

From this property, we see that σ

Zp

(1+T )^xµβ

=σ

loggβ◦η(T )

=loggβ◦σ η(T )

=loggβ◦η

(1+T )^{ψ(σ )}−1

=

Zp

(1+T )^{ψ(σ )x}µβ,

(4.19)

so for generalf, we have σ

Zp

f (x)µβ

=

Zp

f

ψ(σ )x

µβ; (4.20)

by extendingµβtoQp, we have for allf, σ

Qp

f (x)µβ

=

Qp

f

ψ(σ )x

µβ. (4.21)

To show thatµβis 1-admissible, by definition andLemma 3.3, we only need to show that p^n(1−j)

a+pⁿZp(x−a)^jµβisr-bounded forj=0,1. Forj=0, ifa=0, then sinceµβ|_Z^×_p is a measure, the integralpⁿ

a+pⁿZpµβ is always bounded. Ifa=0, thenpⁿ

pⁿZpµβ= ϕⁿ(

Zpµβ)=ϕⁿloggβ(0)=loggβ(0), hence, bounded.

Forj=1, ifa=0, then

a+pⁿZpxµβis bounded. Ifa=0, then

pⁿZpxµβ=ϕⁿ(

Zpxµβ)= ϕⁿ(Ω·g_β(0)/gβ(0))=αⁿΩ(g_β(0))/gβ(0)), hence, bounded.

5. Perrin-Riou and Colmez theorems. LetKn=Qp(ζpⁿ)andK_∞= ∪n≥1Kn. LetΓ= Gal(K_∞/Qp), χ:Γ Z^×p be the cyclotomic character. For x∈K_∞ and n∈N, define Tn(x)=(1/p^m)TrK_m/K_n(x)form1. For a crystalline representation V, that is, a finite-dimensionalQp-vector space such thatG_Qp has a continuous action on it and V is crystalline, letD(V ):=Dcrys(V )denote the Dieudonne module of V. Then from Colmez [2], Tn can be extended to B_dR^GK∞⊗D(V ). Then it is known thatD(V ) has a Frobenius endomorphism and a filtration which we denote by FilⁱD(V ). This filtration is decreasing, separated, and exhausted. That is,

FilⁱD(V )⊇Filⁱ⁺¹D(V ), ∩iFilⁱD(V )= {0}, ∪iFilⁱD(V )=D(V ). (5.1) If F ∈K_∞((t))⊗D(V ),F =

k−∞t^kdk with dk∈K_∞⊗D(V ), defineδ_{V (−k)}(F )to be t^kdk. ForI⊂Z, we have the algebraic distributionᏰ^Ialg(Qp, D(V ))fromSection 3. For h∈Z, we defined the algebraic Fourier transformationᏲalg^(h):Ᏸ^{(−∞,h−1]}alg (Qp, D(V ))→ Ᏸ^[1_alg^−h,+∞)(Qp, BdR⊗V )as

Qp

f (x)Ᏺ^(h)alg(µ):=

Qp

Ᏺalg^(h)(f )µ, (5.2)

then Perrin-Riou and Colmez proved that the image is fixed byG_Qp, and the Perrin-Riou exponential map Exp_h,V is defined as the composition of the following maps:

Ᏸ^{(−∞,h−1]}_alg Qp, D(V )

→Ᏸ^[1−h,+∞)_alg Qp, BdR⊗VG_Qp

→Ᏸ^[1−h,+∞)alg (Z^×p, BdR/B_dR⁺ ⊗V )^GQp

→H¹

Qp,Ᏸ^[1alg^−h,+∞)

Z^×p, V ,

(5.3)

where the last map is the connecting map of the following exact sequence:

0 →Ᏸ^Ialg

Z^×p, V

→Ᏸ^Ialg

Z^×p, Bmax^ϕ=1⊗V

→Ᏸ^Ialg

Z^×p, BdR/B⁺_dR⊗V

→0. (5.4)

Recall that

Ᏸ˜temp

Qp, D(V )

=lim←Ᏸtemp

Qp, D V (k)

, (5.5)

where the projective limit map is given byµ→(−tx)µ. Then Perrin-Riou [9] first proved the following theorem.

Theorem5.1(Perrin-Riou). Assume thatVis a crystalline representation,h∈Zsuch thatFil^−hD(V )=D(V ). Ifµ∈Ᏸ˜temp(Qp, D(V ))^Φ=1, thenExp_h,V(µ)restricted toK_∞is in H¹(K_∞,Ᏸtemp(Z^×p, V )).

FromSection 4, we know that forµβ∈Ᏸ˜temp(Qp, D(Qp(1)))^Φ=1, we could have that Ᏺalg(µβ)is not tempered, so the miracle of this theorem is that Exp_h,V sends tempered distribution to tempered distribution (not only algebraic distribution). Then Perrin-Riou gets the following theorem.

Theorem5.2(Perrin-Riou). Assume thatVis a crystalline representation,h∈Zsuch thatFil^−hD(V )=D(V ), fork≥1−h,

Z^×p

x^kExp_h,V(µ)=exp_{V (k)}

(1−ϕ)⁻¹

1−p⁻¹ϕ⁻¹

(k+h−1)!

Z^×p

µ (−tx)^k

,

a+pⁿZp

x^kExp_h,V(µ)=(k+h−1)! exp_{V (k)} ϕ⁻ⁿ

pⁿ

Zp

ε ax

pⁿ µ

(−tx)^k

,

forn≥1, a∈Z^×p.

(5.6)

The significance of this theorem is that fork∈Zp, the left-hand side (hence the right- hand side) gives an analytic family of cohomology classes in the sense ofSection 3.

The ringᏰ0(Z^×p,Qp)has an action on both the distribution sideᏰ⁽alg^{−∞,1−h]}(Qp, D(V )) and the cohomology side H¹(Qp,Ᏸ^[h−1,∞)alg (Z^×p, V )). That is, for λ ∈ Ᏸ0(Z^×p,Qp) and µ∈Ᏸ^{(−∞,1−h]}alg (Qp, D(V )),ξ∈H¹(Qp,Ᏸ^[h−1,∞)alg (Z^×p, V )), then the action∗(which is es- sentially induced by the mapZ^×p×Qp→Qp,(x, y)→xy) is defined as

Qp

f (x)λ∗µ:=

Qp

Z^×p

f (xy)λ(x)µ(y), (5.7)

Z^×p

f (x)λ∗ξ:=

Z^×p

Z^×p

f (xy)λ(x)ξ(y). (5.8)

Lemma5.3. (i)The action (5.7) commutes with the actionΦ, hence induces an action onᏰ^Ialg(Qp, D(V ))^Φ=1, and it sends tempered distributions to tempered distributions.

(ii) The action (5.8) commutes with the Galois action, hence it is well defined on H¹(Qp,Ᏸ^I_alg(Z^×p, V )).

(iii)The mapExp_h,V is sesquilinear with respect to these actions, that is,

Exp_h,V(λ∗µ)=λ^√∗Exp_h,V(µ), (5.9) where√

is induced byx→x⁻¹and defined to be

Z^×p

f (x)λ^√=

Z^×p

f x⁻¹

λ(x). (5.10)