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 unityζ
pn such thatζp
pn+1=ζpn. LetKn=Qp(ζpn)and ᐁ= lim
←OK×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
xkµ
β, (1.1)
then we haveψk(β)=(1−pk−1)·
Zpx
kµ
β, so it will have a trivial zero atk=1. Since 1−pk−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−pk−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 expQp(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×
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=OCp/pOCp. Let de-note the projective limit of the diagram
¯
O← O¯← O¯←···, (2.1)
where the transition maps are given byx→xp. The ringis a perfect ring with charac-teristicp >0 (see [5]). Forx∈,x=(xn)n∈Nsatisfiesxn∈O¯, andxnp+1=xn. For each
n, choose ˜xn∈OCpto be a representative ofxn. Then one can show that for eachm, limn→∞x˜p
n
n+mexists and the limitx(m)does not depend on the choices of the represen-tatives. Hence,xgives rise to a sequence(x(m))
m∈NinOCpsuch that(x(m+1))p=x(m).
On the other hand, if we have a sequence(x(m))
m∈NinOCpsuch 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)∈OCp,
x(m+1)p
=x(m). (2.2)
Define a functionv:→Q∪{∞}by
vx(m) m∈N
:=vx(0), (2.3)
wherevis the valuation ofCpsuch thatv(p)=1. The ringis complete with respect tov.
LetW ()denote the Witt vector ring of. Recall that the underlying set of W () is the setN = {(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
+px1
p−1
+···+pnx n
p−n
+···, (2.5)
where forx∈,[x]p−1 is the unique elementwofW ()such thatwp=[x]. Let
θ:W () →OCp (2.6)
be defined by
θx0, x1, . . .
=
∞
n=0
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
p 0, x
p 1, . . . , x
p
n, . . .). The mapϕis an isomorphism, semilinear overW (¯Fp). The ringW ()can also be endowed withp -adic topology andI-adic topology. Letε=(1, ζp, ζp2, . . .)∈. The element[ε]∈W ()
has the property θ([ε])=1. The elementϕ−1([ε])=[(ζ
p, ζp2, . . .),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 →B ϕ=1 max 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 continuousGQp-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 ))=(BdRi ⊗V )
GQp. The dimensions have the following
relation:
dimQp
Dcrys(V )
≤dimQp
DdR(V )
≤dimQp(V ). (2.10)
If dimQp(DdR(V )) = dimQp(V ), then V is called a de Rham representation. If
dimQp(Dcrys(V ))=dimQp(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 →Bmaxϕ=1⊗V →BdR/BdR+ ⊗V →0; (2.11)
taking the Galois cohomology, we have a map
BdR/BdR+ ⊗V
GQp
→H1Q p, V
. (2.12)
Then the Bloch-Kato exponential map
expV:
BdR⊗VGQp →H1Qp, V (2.13)
is defined as the composition
BdR⊗V
GQp
→BdR/B+dR⊗V
GQp
→H1Q p, V
The kernel of this map is Fil0DdR(V )+Dcrys(V )ϕ=1, and the image is He(Qp, V ):= ker{H1(Q
p, V )→H1(Qp, Bcrysϕ=1⊗V )}.
For a Galois representationV, letV (k)denote thekth cyclotomic twist ofV. That is, letχdenote the cyclotomic character,ζσ
pn=ζpχ(σ )n for alln≥1,V (k):=V (χk).
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(Q
p,Qp(1))
Q×p⊗Q
pis given by the Kummer map. To be more precise, it is generated byγ1+p,γp, where, forα∈Q×
p,
γα:τ →logε
. . . ,τ
α1/pn α1/pn , . . .
⊗e (2.15)
is the Kummer class. Hence, we haveH1(Q
p,Qp(1))Qp2, then the exponential map forQpis
expQp(1):DdR
Qp(1)
→H1Q
p,Qp(1)
. (2.16)
Lemma2.2. It follows that
expQp(1)
e
t
= γ1+p
log(1+p). (2.17)
Proof. In the exact sequence
0 →Qp →B ϕ=1
max →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 expQp(1)(e/t)is represented by
τ →(τ−1)·
log1+p t·log(1+p)⊗e
=tlog(11+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)
logεγ1+p(τ)
= γ1+p(τ) log(1+p).
(2.19)
Fork >1, it is easy to see that dimQpDdR(Qp(k))=dimQpH1(Qp,Qp(k))=1 and
expQp(k)is an isomorphism. In some sense,γpandγ1+pshould have the same positions inH1(Q
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 overis a pair(V , ρ), where (1)Vis a de Rham representation ofGQp, (2)ρ:×GQp→GlQp(V )is continuous inσ and is analytic ink.
Definition2.4. Let(V , ρ)overbe a family of Galois representations ofGQ p and
letVkdenote the Galois representation ofGQpsuch 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∈GQp,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−p1−k)/(1−p−k))(k−1)! expVk(1k)is an analytic family of cohomology classes inH1(Q
p,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 LPI = {xk· 1a+pnZp|k∈I, a∈Qp}. An algebraicI-distribution with values inM is a finitely
ad-ditive function µ:LPI→M. Let ᏰI
alg(Qp, M) denote all the algebraicI-distributions with values inM. ForX⊂Qp, a compact open subset, letLPI(X)= {xk·1(a+pnZp)∩X}, then ᏰI
alg(X, M)is defined with respect to these test functions. Especially, we have ᏰI
alg(Z×p, M), ᏰIalg(Zp, M). Let Ᏸalg+ (Qp, M) (resp., Ᏸalg− (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, xNf∈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∈ᏰI
alg(Y , A), then for eachn∈Zand every compact open subset
XofY, defineµLPI
n(X)to be the norm of the continuous linear functionµ:LP
I(X)∩
Ꮽn(X)→Aobtained by restrictingµ toLPI(X)∩Ꮽn(X). IfX⊂Qp is compact, then actually
µᏭn(X)= sup a∈X, j≥0
a+pnZp
x−a
pn
j
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 ofGQp, 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)·1Zp
x ν, (3.5)
thenf (x)(−tx)ω=f ν, hence(−tx)ω=ν.
Forµ∈ᏰI
alg(Qp, A), define an operatorϕᏰas
Qp
f (x)ϕᏰµ:=
Qp
f (px)µ. (3.6)
If Ais a Dieudonne module, thenϕ can act on it, hence bothϕand ϕᏰcan act on ᏰI
alg(Qp, A). Then we defineΦ=ϕᏰ⊗ϕ.
Lemma3.5. The twist mapT winduces a map
Ᏸcont
Qp, D(V )
Φ=1
→Ᏸcont
Qp, D
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 haveZ×
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)·1Zp
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(Zp×, 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 expansiona+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 ]a
nisr-bounded.
Lemma 3.8. A distribution µ∈Ᏸcont(Zp,Cp) has order r if and only if Ꮽµ(T )has orderr.
4. Fourier transformation and Coleman power series. Recall that we fixed ζpn
which is apnth root of unity. Letε
n:=(ζpn, ζpn+1, . . .)∈, note thatεp n
n =ε,[εn]∈
W (). For x∈Qp, x=p−n·y with some n∈Nand y∈Zp, defineεx :=εny ∈. Obviously, this is well defined, and we get an element [εx]∈ W (). For x ∈Q
p, exp(tx)=+∞k=0((tx)k/k!) converges. Defineε(x):=[εx]/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≡0modpn,
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+pmZ
p,f is constant, hence the integral equals
a
a+pmZpf (x)ε(xy)µHaar(x)=
1 pm
amodpm f (a)
x∈a+pmZp
ε(xy). (4.3)
From property (ii) ofε(x), ifyis outside ofp−mZ
p, 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−mZ
pfor somem. Sinceε(p−my)is locally constant, this implies thatᏲalg(f )is locally constant. Extend the above definition to test function{xk·1
a+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(xk·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
wheref∈LP[1−h,+∞), then we have
Ᏺ(h) alg
xk·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(−∞,h−1]
alg (Qp, D(V )), defineᏲ(h)alg(µ)such that
Qp
f (x)Ᏺ(h) alg(µ):=
Qp
Ᏺ(h)
alg(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)≡xp(modp). LetFbe the one-dimensional Lubin-Tate formal group overZpcorresponding tofπand let[+]denote the formal addition. LetWn
π:= {x∈Cp|fπ(n)(x)=0},Kn=Qp(Wπn), 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) gβϕ◦fπ(x)=
w∈Wπ1gβ(x[+]w).
Assume thatβ∈ᐁsuch thatβn≡1(modωn). Thengβ(T )≡1 mod(p, T ), hence we can define
loggβ(T ):=loggβ(T )− 1 p
w∈Wπ1
loggβ
T [+]w. (4.8)
The property (ii) of the Coleman power series implies thatloggβ(T )has integral coef-ficients. Define an algebraic distributionµβ∈Ᏸ+alg(Zp,Qpur)such that
Zp
(1+T )xµ
β(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
ur
p )Φ=1and has the following Galois
property:
σ
Qp f (x)µβ
=
Qp
fψ(σ )xµβ ∀σ (4.10)
for allf (x):Qp→Qp.
Proof. It is easy to see that
Z× p
(1+T )xµ β=
Zp
(1+T )xµ β−
pZp
(1+T )xµ
By property (ii),
gβ◦fπ(X)=
w∈Wπ1 gβ
X[+]w; (4.12)
letX=η(T ), then
gβ◦fπη(T )=
ζ∈µp
gβη(T )[+]η(ζ−1)
=
ζ∈µp gβη
ζ(1+T )−1. (4.13)
By usingfπ◦η=ηϕ◦[p], we see that
gβ◦ηϕ◦[p]=
ζ
gβηζ(1+T )−1; (4.14)
taking logarithm, and using the definition forµβ, we have
ϕ
Zp
1+[p]Txµβ
= ζ
Zp
ζx(1+T )xµβ=p
pZp
(1+T )xµβ. (4.15)
Hence,
Z× p
(1+T )xµ
β=loggβη(T )− 1 pϕ
loggβ◦ηϕ◦[p]T
=loggβ◦η− 1
ploggβ◦fπ◦η(T ) =loggβ◦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
ση(T )=η(1+T )ψ(σ )−1 ∀σ∈GQp. (4.18)
From this property, we see that
σ
Zp
(1+T )xµβ
=σloggβ◦η(T )
=loggβ◦σ
η(T )
=loggβ◦η
(1+T )ψ(σ )−1
=
Zp
(1+T )ψ(σ )xµβ,
so for generalf, we have
σ
Zp f (x)µβ
=
Zp
fψ(σ )xµβ; (4.20)
by extendingµβtoQp, we have for allf,
σ
Qp f (x)µβ
=
Qp
fψ(σ )xµβ. (4.21)
To show thatµβis 1-admissible, by definition andLemma 3.3, we only need to show that
pn(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)=ϕ
nlogg
β(0)=loggβ(0), hence, bounded. Forj=1, ifa=0, thena+pnZpxµβis bounded. Ifa=0, then
pnZpxµβ=ϕn(
Zpxµβ)= ϕn(Ω·g
β(0)/gβ(0))=αnΩ(gβ(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/pm)TrKm/Kn(x)form1. For a crystalline representation V, that is, a
finite-dimensionalQp-vector space such thatGQp 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
GK∞
dR ⊗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−∞tkdk with dk∈K∞⊗D(V ), defineδV (−k)(F )to be
tkd
k. ForI⊂Z, we have the algebraic distributionᏰIalg(Qp, D(V ))fromSection 3. For
h∈Z, we defined the algebraic Fourier transformationᏲalg(h):Ᏸ (−∞,h−1]
alg (Qp, D(V ))→ Ᏸ[1−h,+∞)
alg (Qp, BdR⊗V )as
Qp
f (x)Ᏺ(h) alg(µ):=
Qp
Ᏺ(h)
alg(f )µ, (5.2)
then Perrin-Riou and Colmez proved that the image is fixed byGQp, and the Perrin-Riou
exponential map Exph,V is defined as the composition of the following maps:
Ᏸ(−∞,h−1] alg
Qp, D(V )
→Ᏸ[1−h,+∞) alg
Qp, BdR⊗V
GQp
→Ᏸ[1−h,+∞)
alg (Z×p, BdR/BdR+ ⊗V ) GQp
→H1Q
p,Ᏸ[1alg−h,+∞)
Z×
p, V
,
(5.3)
where the last map is the connecting map of the following exact sequence:
0 →ᏰI alg
Z×
p, V
→ᏰI alg
Z×
p, B ϕ=1 max⊗V
→ᏰI alg
Z×
p, BdR/B+dR⊗V
Recall that
˜ Ᏸtemp
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 Exph,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
xkExp
h,V(µ)=expV (k)
(1−ϕ)−11−p−1ϕ−1(k+h−1)!
Z× p
µ (−tx)k
,
a+pnZ p
xkExp
h,V(µ)=(k+h−1)! expV (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(Q
p,Ᏸ[halg−1,∞)(Z×p, V )). That is, for λ ∈ Ᏸ0(Z×p,Qp) and
µ∈Ᏸ(−∞,1−h]
alg (Qp, D(V )),ξ∈H1(Qp,Ᏸ[halg−1,∞)(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 onᏰIalg(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(Q
p,ᏰIalg(Z×p, V )).
(iii)The mapExph,V is sesquilinear with respect to these actions, that is,
Exph,V(λ∗µ)=λ
√
∗Exph,V(µ), (5.9)
where√is induced byx→x−1and defined to be
Z× p
f (x)λ√=
Z× p
Proof. These follow from the definitions.
For the “negative” power, Colmez proved the following theorem.
Theorem5.4(Colmez). Assume thatV is a crystalline representation,h∈Z,k≥h, then
exp∗V (−k)
Z× p
x−kExp h,V(µ)
=(−1)h−11−p−1ϕ−1 Z
(tx)k
(k−h)!µ. (5.11)
Remark5.5. Colmez [2] provedTheorem 5.4, fork1; we will prove the statement
fork≥hin another paper [11]; this can also be found in [12]. From his proof, we can get the following theorem.
Theorem5.6. Assume thath∈Zandµ∈Ᏸ˜temp(Qp, D(V ))Φ=1, fork≥h,n≥1, then
exp∗V (−k)
1+pnZpx −kExp
h,V(µ)
=(−1)h−1ϕ−n
pn
Zp ε
x
pn
(tx)k
(k−h)!µ. (5.12)
Proof. Chooser∈Nlarge enough such thatFr=
Qp[ε
x](µ/(−tx)r)exists.
Theo-rem IV.1.1 in [2] implies that
δV (−k)◦Tn
Fr
=exp∗V (−k)
(−1)h+r−1(k−h)!
(k+r )!
1+pnZpx −kExp
h,V(µ)
; (5.13)
by using [2, the formula in II.2.1], we get
exp∗V (−k)
1+pnZpx −kExp
h,V(µ)
=(−1)h−1·p−n
p−nZp
ε(x) (tx)
k
(k−h)!µ, (5.14)
and the theorem follows from the conditionΦ(µ)=µ.
The significance of these two theorems is that fork1,(exp∗V (−k))−1gives rise to an analytic family of cohomology. Theorems5.4and5.6are called explicit reciprocity law.
To get the symmetric form of the explicit reciprocity law, one defines the following pairing:
[·,·]D(V ): ˜Ᏸtemp
Qp, D(V )
Φ=1
×Ᏸ˜tempQp, DV∗(1)Φ=1
→Ᏸtemp
Z×
p,Qp
(5.15)
as
Z× p
f (x)µ, µD(V )=
Z× p×Z×p
fx−1yµ⊗µ. (5.16)
The pairing in the cohomology side is defined as
(·,·)V:H1
Qp,ᏰtempZ×p, V
×H1Qp,ᏰtempZ×p, V∗(1)
→H2Q
p,ᏰtempZ×p×Z×p, V⊗V∗(1)
Ᏸtemp
Z×
p, H2
Zp,Qp(1)
Ᏸtemp
Z×
p,Qp
.
From Theorems5.4and5.6, we have the following theorem.
Theorem5.7(Perrin-Riou and Colmez). Assume thatVis crystalline representation ofGQp,µ∈Ᏸ˜temp(Qp, D(V ))Φ=1,µ∈Ᏸ˜temp(Qp, D(V∗(1)))Φ=1, then
Exph,V(µ),Exp1−h,V∗(1)
µ=(−1)hδ
−1∗µ, µ
D(V ), (5.18)
whereδ−1is defined by
Qp
f (x)δ−1∗µ=
Qp
f (−x)µ. (5.19)
Perrin-Riou proved this theorem forV=Qp(1)and Colmez proved it for general
crys-talline representation.
Moreover, as Iwasawa modules, those pairings have the following properties.
Proposition5.8. (i)Forµ∈Ᏸ˜temp(Qp, D(V ))Φ=1andµ∈Ᏸ˜temp(Qp, D(V∗(1)))Φ=1, the integral
Z× p
xiµ, µ
D(V )=
Z× p
x−iµ,
Z× p
xiµ
D(V )
, (5.20)
where the last pairing is defined inSection 3.
(ii)Forξ∈H1(Q
p,Ᏸtemp(Z×p, V ))andξ∈H1(Qp,Ᏸtemp(Z×p, V∗(1))), the integral
Z× p
xiξ, ξ
V=
Z× p
xiξ∪
Z× p
x−iξ, (5.21)
where the cup product is given by
H1Qp, V (i)
∪H1Qp, V∗(1−i)
→H2Qp,Qp(1)
Qp. (5.22)
(iii)[·,·]D(V ) is sesquilinear for the first variable and linear for the second variable,
that is,
δ∗µ, µD(V )=δ
√
∗µ, µD(V ),
µ, δ∗µD(V )=δ∗
µ, µD(V ).
(5.23)
(iv)(·,·)V is linear for the first variable and sesquilinear for the second variable, that
is,
δ∗ξ, ξV=δ∗
ξ, ξV,
ξ, δ∗ξV=δ
√
∗ξ, ξV.
Proof. (i) and (ii) are just from definitions, which can also be found in Colmez [2].
For (iii), we have
Z× p
xiδ∗µ, µ
D(V )=
Z× p
x−i(δ∗µ),
Z× p
xiµ
D(V )
=
Z×p
Z×p
x−iy−iδ(y)µ(x),
Z×p xiµ
D(V )
=
Z×p
y−iδ(y)
Z×p x−iµ,
Z×p xiµ
D(V )
=
Z×px
iδ√∗µ, µ
D(V ),
(5.25)
and (iv) is similar to (iii).
6. Iwasawa’s explicit reciprocity law. Recall thatKn=Qp(ζpn)and letD=(1+ T )(d/dT ). Forβ∈lim←O×Kn, letgβ(T )∈Zp[[T ]]denote the Coleman power series.
Theorem6.1(Iwasawa). Letαn, βn∈OK
n such thatαn≡1(ωn)andβn∈Kn×sits
in a norm coherence sequenceβ=(βn)n, letgβdenote the Coleman power series
corre-sponding toβ, and define
αn, βn
n=
α1/pn n
σβn−1
,
αn, βnn=p−nTrKn/Qp
logαnDloggβωnmodpn.
(6.1)
Then
αn, βn
n=ζ [αn,βn]n
pn , (6.2)
whereωn=ζpn−1.
In the following, we will show that Perrin-Riou-Colmez explicit reciprocity law, The-orem 5.2, implies Iwasawa’s explicit reciprocity law.
Recall that we have the Bloch-Kato exponential map expKn,V:(BdR⊗V)GKn→H1(Kn, V ). LetV=Qp(1)and letUndenote the principal units ofOKn. To an element of lim←Un,
we will associate an element in ˜Ᏸ1(Qp, D(V ))Φ=1. To an element in lim←K×n, we will as-sociate an element inH1(Q
p,Ᏸ0(Z×p, V )). Forβ∈lim←Un, defineµβ∈Ᏸ+alg(Zp, D(V ))as
Zp
(1+T )xµβ=loggβ(T )⊗
e
t, (6.3)
and extend it toQpby defining
p−nZp
f (x)µβ=pn
Zp
fp−nxµβ=ϕ−n
Zp
Then
Qp
f (x)Φµβ=ϕ
Qp
f (px)µβ
=1p·
Qp
f (px)µβ
=1p·p
Qp f (x)µβ
=
Qp
f (x)µβ,
(6.5)
henceΦµβ=µβ. ByProposition 4.2,µβis 1-admissible. Note that Coleman power series has the propertygβ1β2=gβ1·gβ2. So we get a map
lim
←Un →Ᏸ1Qp, DQp(1)Φ=1. (6.6) On the other hand, forβ∈lim←K×n,β=(βn), thenβngivesγβn∈H
1(K
n,Zp(1))defined by Kummer map. By using Colmez’s theorem inSection 5, we get an element ˆβ(τ):= lim
←γβn(τ)∈lim←H1(Kn,Zp(1))H1(Qp,Ᏸ0(Z×p,Zp(1))). Hence, we have a map
lim
←K×n →H1
Qp,Ᏸ0
Z×
p,Zp(1)
,
β→β,ˆ (6.7)
which has the propertyβ1·β2=β1+β2.
We can also state this map by using integral, namely, forβ∈lim
←Kn×, ˆβ∈H1(Qp,Ᏸ0(Z×p,
Zp(1)))is the element such that
1+pnZp
ˆ β=γβn,
a+pnZp
ˆ
β=γσa(βn). (6.8)
Especially, forβ∈lim←Un, we have
µβ∈Ᏸ1Qp, DQp(1)Φ=1, βˆ∈H1Qp,Ᏸ0Z×p,Zp(1). (6.9)
The element(p,1−ζp,1−ζp2, . . .)∈lim
←Kn×gives an element inH1(Qp,Ᏸ0(Z×p,Zp(1))), we denote it by ˆp. Fixa, b∈Z×
p such thata≡b(modp),a=b. For example, we can takea=1 andb=1+p. Then the element(. . . , (ζpan−1)/(ζpbn−1), . . .)∈lim←Un, hence gives a distribution, which we denote byµab. Recall thatδa∈Ᏸ0(Z×p,Zp)is defined to be the Dirac measure
Z×p
f (x)δa=f (a). (6.10)
Letδab=δa−δb∈Ᏸ0(Z×p,Zp). The following lemma describes the relationship between
µβand ˆβ,µab and ˆp.
Lemma6.2. (i)There is a homomorphismµ: lim
←Un→Ᏸ1(Qp, D(Qp(1)))Φ=1, which
(ii)There is a mapξ: lim
←K×n→H1(Qp,Ᏸ0(Z×p,Zp(1))), which sendsβtoβˆ. (iii)Forβ∈lim←Un,Exp1,Qp(1)(µβ)=βˆ.
(iv)Fora, b∈Z×
p,a≡b(modp),a=b,
Exp1,Qp(1)
µab
=δ √
ab∗p.ˆ (6.11)
Proof. We have already proved (i) and (ii). For (iii), byTheorem 5.2,
a+pnZpExp1,Qp(1)
µβ
=expQp(1)
ϕ−n
pn
Zp ε
ax
pn
µβ
=expQp(1)
ϕ−n
pn
loggβσaωn
t
=expQp(1)
logσ
aβn
t
=γσa(βn)
=
a+pnZp
ˆ β.
(6.12)
(iv) Letβ=(. . . , (ζpan−1)/(ζpbn−1), . . .). From (iii), we have Exp1,Qp(1)(µab)=βˆ. So we only need to look at the relation between ˆβand ˆp. Letpa=(. . . , ζpan−1, . . .), then by (ii)
we have ˆβ=pa−pb. The integral
1+pnZpδa−1∗pˆ=
Z×p
11+pnZp(xy)δa−1(x)p(y)ˆ
=
Z× p
11+pnZpa−1yp(y)ˆ
=
a+pnZpp(y)ˆ
=
1+pnZp σa(p)
=
1+pnZppa,
(6.13)
hencepa=δa−1∗pˆ. And we have ˆβ=(δa−1−δb−1)∗pˆ=δ
√
ab∗pˆ. Hence, the lemma follows.
The relation Exp1,Qp(1)(µab)=δ √
ab∗pˆcan make us extend Perrin-Riou exponential map to some elements with denominatorδaband we can define
Exp1,Qp(1)
µ
ab
δab
=p.ˆ (6.14)
a≡bmodp. This can be seen as if we havec≡dmodp, then the Amice transformation ofδab∗µcdis given by
Zp(1+T )
xδ
ab∗µcd=log(((1+T )ac−1)((1+T )bd−1)/((1+
T )bc−1)((1+T )ad−1))which is the same as the Amice transformation ofδ
cd∗µab. Hence, δab∗µcd =δcd∗µab. We denote this pseudomeasure µab/δab by µp. For a crystalline representationW such that there is a Galois inclusionQp(1)⊂W, we have the following theorem.
Theorem6.3. ForW above, the mapExp1,V(µ)can be extended to the set including µpby using the inclusionH1(Qp,Ᏸ0(Z×p,Qp(1)))H1(Qp,Ᏸ0(Z×p, W )). Forµ∈ µp ⊕ ˜
Ᏸtemp(Qp, D(W ))Φ=1,µ∈Ᏸ˜(Qp, D(W∗(1)))Φ=1,
Exp1,W(µ),Exp0,W∗(1)
µW= −
δ−1∗µ, µD(W ). (6.15)
Proof. We only need to show that forµ=µp, this will follow from the definition,
the sesquilinear property of the exponential map, and the pairings
Exp1,W
µab,ExpW∗(1),0
µ= −δ−1∗µab, µ,
δ√ab∗Exp1,W
µp
,Exp0,W∗(1)
µ= −δ−1∗δab∗µp, µ
,
δ √
ab∗
Exp1,W
µp
,Exp0,W∗(1)
µ= −δ √
ab∗
δ−1∗µp, µ
,
(6.16)
since the convolution inᏰ0(Z×p,Qp)has cancellation law. This implies that
Exp1,W
µp,Exp0,W∗(1)
µ= −δ−1∗µp, µ. (6.17)
(This can also be seen from that ifδab∗µ=0, then
ykδ
ab∗µ=0, hence
ykµ=0, henceµ=0.)
Now, we use Perrin-Riou and Colmez explicit reciprocity law to prove Iwasawa’s ex-plicit reciprocity law. Assume thatαn, βn ∈OKn\ {0}, βn sits in a norm coherence sequence. Then βn =un·βn·ω
j
n forun a (p−1)th root of unity, βn≡1(modωn),
j≥0. We know that theunwill give(αn, un)n=1 andσun=1. So we will consider the
caseβn≡1(modωn)and the caseβn=ωnseparately. Forβwithβn≡1(modωn), we haveµβ∈Ᏸ1(Qp, D(Qp(1)))Φ=1. Then(−tx)µβ∈Ᏸ1(Qp,Qp)Φ=1.
Lemma6.4. (i)Forαn∈OK
n\[0],expKn,Qp(1)((log(αn)/t)⊗e)=γαn.
(ii)For allµ∈Ᏸ˜temp(Qp, D(Qp(1)))Φ=1,Exp1,Qp(1)(µ)=xExp0,Qp(−txµ)
(iii)Ifβhas the propertyβn≡1(modωn),Exp0,Qp(−txµβ)=β/xˆ .
(iv)The integralZpε(x/pn)xµ
p=ζpn/ωn.
Proof. (i) Since(log[αn]/t)⊗e∈Bcrysϕ=1(1) is a lifting of (log(αn)/t)⊗e, by the
definition of exponential map and the Kummer class, we see (i). From [2] we know that
1+pnZpx
kExp
1,Qp(1)(µ)=k! expQp(k+1)
ϕ−n
pn
Zp µ (−tx)k
,
1+pnZpx
k+1Exp
0,Qp(−txµ)=k! expQp(k+1)
ϕ−n
pn
Zp µ (−tx)k
,
(6.18)
(iii) follows from (ii) andLemma 6.2(iii).
(iv) Sinceµab=δab∗µp and this relation does not depend on the choices ofaand
b, we can takea=1+pn,b=1,
Zp ε
x
pn
xµab=
Zp ε
ax
pn
axµp−
Zp ε
bx
pn
bxµp
=
Zp ε
x
pn
1+pnxµp−xµp
=pn
Zp ε
x
pn
xµp
(6.19)
sinceµab corresponds to the power series((1+T )1+p
n
−1)/T, hence
Zp ε
x
pn
xµab=(1+T )
T (1+T )1+pn−
1 d dT
(1+T )1+pn
−1 T
T
=ωn
=ζpn·ωn ωn
pn·ω1 n
=pnζpn
ωn
,
(6.20)
and this completes the proof of this lemma.
Now, we come to the proof of Iwasawa’s explicit reciprocity law in two cases.
Case1. Assume thatβn≡1(modωn)sits in the norm coherence sequenceβ. Take V=Qp(1),h=1,k=1, andµ=µβ∈Ᏸ˜temp(Qp, D(V ))Φ=1, and usingTheorem 5.6, we have
exp∗Qp
1+pnZpx −1βˆ
=ϕ−n pn
Zp ε
x
pn
(tx)µβ
, (6.21)
that is,
exp∗Qp
1+pnZpx
−1βˆ= 1
pnDloggβ
ωn
. (6.22)
We already know that
expQp(1)
logα
n
t ⊗e
=γαn. (6.23)
From the definition of the Hilbert symbol, we have
αn, βn
From the definition of the dual exponential map, we have
γαn, γβn
=
γαn,
1+pnZp
ˆ β
≡
γαn,
1+pnZpx −1βˆ
=αn, βn
.
(6.25)
Hence Iwasawa’s explicit reciprocity law follows.
Case2. Forβn=ωn,gβ(T )=T. Using the sesquilinear property of Exph,V, we see
that
exp∗Qp
1+pnZpx
−1pˆ=ϕ−n
pn
Zp ε
x
pn
(tx)µp
, (6.26)
that is,
exp∗Qp
1+pnZppˆ
= 1
pnDlog(T )
ωn byLemma 6.4(iv), (6.27)
combined with
expQp(1)
logα
n
t ⊗e
=γαn, (6.28)
then Iwasawa’s explicit reciprocity law follows as inCase 1.
Remark6.5. Lemma 6.4(iv) can be interpreted as a completion of the theory of
Cole-man power series. Namely,µpis the distribution whose Amice transformation is log(T ) in the sense thatxµpcorresponds toDlog(T ).
7. A trivial zero problem. Recall thatKn=Qp(ζpn)andᐁ=lim
←OK×n. Forβ∈ᐁ, we
have a 1-admissible distributionµβ∈Ᏸ1(Qp,Qp[−1])Φ=1. Consider the integral
ψk(β)=
Z×p xkµ
β, (7.1)
then
ψk(β)=
Zp xkµ
β−
pZp xkµ
β
=
Zp xkµ
β−
Qp
1pZpx
kµ β
=
Zp xkµ
β−
Qp
1pZpx
kΦµ β
=
Zp
xkµβ−ϕ
Qp
1pZp(px)(px)
kµ β
=1−pk−1
Zp xkµβ.
The Euler factor 1−pk−1forcesψ
1=0 atk=1. Since 1−pk−1is not an analytic function ofk, hence we cannot take the derivative directly. Butψkis an analytic function ofk, thendψk/dk|k=1must exist. Glenn Stevens predicted that the derivative ofψkat 1 will give the Kummer classγp. Based on the previous sections, now we can prove that this is true.
Lemma7.1. The integralZ×
pµab=(1−1/p)log(a/b).
Proof. Since we have
Z× p
(1+T )xµ
ab=logg(T ), (7.3)
withg(T )=((1+T )a−1)/((1+T )b−1), henceg(0)=a/b, and
Z×p
µab=logg(T )|T=0=
1−p1
loga
b. (7.4)
Lemma7.2. Assume thatµ∈Ᏸtemp(Z×p,Qp)such that
Z×pµ=0, then
Z×p µ δab =
1
log(a/b)·lims→0 1 s
Z×p
xsµ. (7.5)
Proof. Letν=µ/δabandµ=δab∗ν,
Z× p
xsµ=
Z× p
xsδ ab∗ν
=
Z×p
xysδab(y)ν(x)
=as−bs Z×
p
xsν,
(7.6)
hence
Z×p ν=lim
s→0
Z×p
xsν
=lim s→0
1 as−bs
Z× p
xsµ
= 1
log(a/b)·lims→0 1 s
Z× p
xsµ.
(7.7)
Letκr:ᐁ→H1(Qp,Qp(1−r ))be given byκr(β)=
Z×px−
rβˆ.
Lemma7.3. The following diagram is commutative forr≥1:
H1Q
p,Qp(r )×H1Qp,Qp(1−r )
κr
Qp
Homᐁ,Qp(r )
Γ×ᐁ Q
p.