IJMMS 2004:12, 607–635
PII. S0161171204301249
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
SHAOWEI ZHANG
Received 16 January 2003
Let F be a one-dimensional Lubin-Tate formal group over Zp . Colmez (1998) and Perrin-Riou
(1994) proved an explicit reciprocity law for tempered distributions over the formal group
Gm . In this paper, the general explicit reciprocity law over the formal group F is proved.
2000 Mathematics Subject Classification: 11S40.
1. Formal groups. In this section, we review some basic facts on formal groups.
De Shalit [3, Chapter I] is the source for this section. We just give the description on
Coleman power series and the associated measures, and then study such measures
extensively. These measures are the basic examples for our theory. For admissible distributions, see [8].
Let p be an odd prime. For α ∈ Z×
p , let π = pα. Let fπ (x) ∈ Zp [[x]] be a Frobenius
corresponding to π , so fπ (x) ≡ π x(mod deg 2), fπ (x) ≡ x p (mod p). Let F be the onedimensional Lubin-Tate formal group over Zp corresponding to fπ , let [+] denote the
(n)
formal addition. Let Wπn := {x ∈ Cp | fπ (x) = 0}, Kn = Qp (Wπn ), K∞ = ∪n≥1 Kn . Hence,
K∞ /Qp is a totally ramified extension with Galois group Z×
p . We call this tower the LubinTate tower corresponding to the formal group F. Let R = Zp [[T ]], ᐁ = lim Kn× , where
←
the map is with respect to the norm map. Let η : Gm → Ᏺπ be an isomorphism, then
ur
ϕ−1
= α, where ϕ = Frobp is a generator
η∈Q
p [[T ]] and η(T ) = ΩT +· · · , such that Ω
ur
ϕ
of Gal(Qp /Qp ). We have f ◦ η = η ◦ [p], where [p] = (1 + T )p − 1 is the Frobenius of
−n
the formal group Gm . Let ωn = ηϕ (ζpn − 1). Then it is easy to see that
(n)
(i) ωn ∈ Wπ ,
(ii) fπ (ωn ) = ωn−1 ,
−n
−n+1
−n+1
◦ [p](ζpn − 1) = ηϕ
(ζpn−1 − 1) = ωn−1 .
as fπ (ωn ) = fπ ηϕ (ζpn − 1) = ηϕ
(n)
Let Tπ = lim Wπ be the Tate module, where the inverse limit is taken with respect
←
to fπ . Let κ : Gal(K∞ /Qp ) → Z×
p be the character given by the action of GQp on Tπ . If
the formal group is Gm , this character is just χ, the cyclotomic character. We know that
κ = χψ, where ψ is an unramified character.
Assume β ∈ ᐁ, then Coleman’s theorem tells us that there is a unique (Coleman)
power series gβ ∈ Zp [[T ]] such that
(i) gβ (ωi ) = βi , for all i ≥ 1,
ϕ
(ii) gβ ◦ fπ (x) = w∈Wπ1 gβ (x[+]w).
To get a rough idea for what Coleman power series is, we look at some examples.
Consider the Gm case, let βn = 2
+ ω−1 (2)(ζpn − 1), where ω is the Teichmüller
character. Then we have gβ (T ) = 2
+ ω−1 (2)T . If we take βn = (ζpan − 1)/(ζpbn − 1),
then gβ (T ) = ((1 + T )a − 1)/((1 + T )b − 1).
608
SHAOWEI ZHANG
Assume β ∈ ᐁ such that βn ≡ 1(mod ωn ). Then gβ (T ) ≡ 1 mod(p, T ), where p is the
maximal ideal in Zp , hence we can define
β (T ) := log gβ (T ) − 1
log gβ T [+]w ,
logg
p
1
(1.1)
w∈Wπ
β (T ) has integral coefficients.
property (ii) of the Coleman power series implies that logg
ur
+
Define an algebraic distribution µβ ∈ Ᏸ (Zp , Qp ) such that
alg
Zp
(1 + T )x µβ (x) = log gβ ◦ η(T ).
(1.2)
Proposition 1.1. (i) The restriction µβ |Z×p is a measure and its Amice transformation
β ◦ η(T ).
is logg
ur Φ=1
(ii) The distribution µβ can be extended to a distribution in Ᏸ1 (Qp , Q
and has
p )
the following Galois property:
σ
Qp
f (x)µβ =
Qp
f ψ(σ )x µβ ,
∀σ ,
(1.3)
for all f (x) : Qp → Qp .
Proof. It is easy to see that
Z×
p
(1 + T )x µβ =
Zp
(1 + T )x µβ −
pZp
(1 + T )x µβ .
(1.4)
By property (ii),
gβ ◦ fπ (X) =
gβ X[+]w .
(1.5)
1
w∈Wπ
Put X = η(T ), then
gβ ◦ fπ η(T ) =
gβ η(T )[+]η(ζ − 1) =
gβ η ζ(1 + T ) − 1 .
ζ∈µp
(1.6)
ζ∈µp
By using fπ ◦ η = ηϕ ◦ [p], we see
ϕ
gβ ◦ η ◦ [p] =
gβ η ζ(1 + T ) − 1 .
(1.7)
ζ
Taking logarithm and using the definition of µβ , we have
ϕ
Zp
x
1 + [p]T µβ =
ζ
Zp
ζ x (1 + T )x µβ = p
We write down this useful property as follows.
pZp
(1 + T )x µβ .
(1.8)
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
609
Proposition 1.2. For µβ as above,
ϕ
Zp
(1 + T )px µβ = p
pZp
(1 + T )x µβ .
(1.9)
Now continue the proof of Proposition 1.1.
The integral
Z×
p
(1 + T )x µβ = log gβ η(T ) −
= log gβ ◦ η −
1 ϕ log gβ ◦ ηϕ ◦ [p]T
p
1
log gβ ◦ fπ ◦ η(T )
p
(1.10)
β ◦ η(T )
= logg
has integral coefficients, hence µβ |Z×p is a measure. By Proposition 1.2 we extend µβ to
Qp by defining
p −n Z
f (x)µβ = p n ϕ−n
p
Zp
f p −n x µβ ,
(1.11)
for any f (x) with support in p −n Zp . It is easy to see that this definition does not depend
on n.
To prove the second property, since
η(T ) : Gm → Ᏺπ ,
(1.12)
we can show that
σ η(T ) = η (1 + T )ψ(σ ) − 1 ,
∀σ ∈ GQp .
(1.13)
To see this, define
hσ (T ) := σ η(T ) − η (1 + T )ψ(σ ) − 1 .
(1.14)
(n)
For Tn = ζpn − 1, η(Tn ) ∈ Wπ ,
(σ η) σ Tn = σ η Tn = κ(σ ) π η Tn = η κ(σ ) Tn
ψ(σ )
κ(σ )
−1 ,
= η ζp n − 1 = η 1 + σ Tn
(1.15)
hence hσ (σ Tn ) = 0 for all n ≥ 1. The function hσ (T ) has infinitely many zeros by
Weierstrass lemma, then hσ (T ) = 0.
From this property, we see that
σ
Zp
(1 + T )x µβ = σ log gβ ◦ η(T ) = log gβ ◦ σ η(T )
= log gβ ◦ η (1 + T )ψ(σ ) − 1 =
(1.16)
Zp
(1 + T )ψ(σ )x µβ ,
610
SHAOWEI ZHANG
so for general f , we have
σ
Zp
f (x)µβ =
Zp
f ψ(σ )x µβ .
(1.17)
From the extension definition (1.11) of µβ , we have, for all f ,
σ
Qp
f (x)µβ =
Qp
f ψ(σ )x µβ .
(1.18)
To show that µβ is 1-admissible, by definition and [8, Proposition 3.3], we only need to
show that p n(1−j) a+pn Zp (x − a)j µβ is r -bounded for j = 0, 1. For j = 0, if a ≠ 0, then
since µβ |Z×p is a measure, the integral p n a+pn Zp µβ is always bounded. If a = 0, then
p n pn Zp µβ = ϕn ( Zp µβ ) = ϕn log gβ (0) = log gβ (0), hence bounded.
For j = 1, if a ≠ 0, then a+pn Zp xµβ is bounded. If a = 0, then pn Zp xµβ = ϕn ( Zp xµβ ) =
ϕn (Ω · (gβ (0)/gβ (0))) = αn Ω(gβ (0)/gβ (0)), hence bounded.
Next, we will restrict to the Gm case. µβ is an example which is a distribution on Qp
but not a measure. We will show an example of distribution which is not a tempered
distribution. µβ can be extended to negative power. For k > 0, define
Zp
−1
x −k µβ = 1 − p −k−1
x −k µβ ,
Z×
p
(1.19)
define νβ ∈ Ᏸ+
alg (Zp , K∞,cyc ) such that
a+p n Zp
x k νβ := p nk k!
Zp
ε
ax µβ
,
pn x k
k ≥ 0.
(1.20)
The relation between µβ and νβ is that
νβ ⊗
e
e
= Ᏺalg µβ ⊗
.
t
t
(1.21)
Lemma 1.3. The distribution νβ is a distribution over Zp , but it is not a tempered
β (T ) ≡ logg
β (0)(mod(p, T p−1 )).
distribution if logg
Proof. The additivity of νβ follows from the relation
 
pε ax ,
a + ip n x
p n+1
=
ε

p n+1
0,
i=0
p−1
x≡0
(mod p).
(1.22)
x ≡ 0
β ≡ 0 mod(p, T p−1 ), hence there exists i, 1 ≤ i ≤
β (T ) = ai T i , since logg
Assume logg
p −1, such that ai ≡ 0(mod p). Then 1+pn Zp νβ = log βn has denominator with valuation
n − 1 − i/(p − 1), so νβ is not a measure.
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
611
If νβ is r -admissible, assume r ∈ N, then for all X,
[n(r −j)]
j
sup p
(x − a) νβ n
a+p Zp
a∈X
(1.23)
is r -bounded. Taking j = r , we would have that a+pn Zp (x − a)r νβ is bounded when
n → ∞.
We will calculate the valuation of a+pn Zp (x − a)r νβ . For a = 1,
r
r
r
(x − 1) νβ =
x k νβ =
dk .
· (−1)r −k
k
a+p n Zp
1+p n Zp
k=0
k=0
r
(1.24)
Claim 1.4. v(dk ) > i/(p − 1) − (n − 1) = v(d0 ).
To prove the claim, keeping using the relation
ε
pZp
µβ
x µβ
1
x
=
ε
pm x k
p k+1 Zp
p m−1 x k
(1.25)
and the decomposition
×
n−1 ×
Zp ∪ p n Zp ,
Z p = Z×
p ∪ pZp ∪ · · · ∪ p
(1.26)
we have
1+p n Zp
x k νβ
x µβ
n
p
xk
Zp
n−1 x µβ
x µβ
kn
= p · k!
ε
+
ε
pn x k
pn x k
p i Z×
p n Zp
p
i=0
n−1
µβ
µβ
1
x
1
kn
ε
+
= p · k!
p i(k+1) Z×p
p n−i x k p n(k+1) Zp x k
i=0
n−1
µβ
µβ
1
x
1
kn
−k−1 −1
ε
+
= p · k!
.
1−p
k
p i(k+1) Z×p
p n−i x k p n(k+1)
Z×
p x
i=0
=p
kn
· k!
ε
(1.27)
The last two terms for d0 equal
1
p n−1
Z×
p
ε
x
p −1
1 µβ = n−1 logg
µβ + n−1
β ζ1 − 1 + (p − 1)loggβ (0) .
×
p
p
p
Zp
(1.28)
β (T ) = a0 + a1 T + · · · . From the hypothesis we know that there exists a
Assume logg
minimal i with 1 ≤ i < p − 1 such that ai ≡ 0 mod p, hence the above expression equals
i
1 pa0 + a1 ζ1 − 1 + · · · + ai ζ1 − 1 + · · · ,
p n−1
(1.29)
612
SHAOWEI ZHANG
and therefore v(d0 ) = i/(p − 1) − (n − 1). For k > 0, we easily see that v(dk ) ≥ k −
(n − 1) > v(d0 ), hence the claim follows. The sequence 1+pn Zp (x − 1)r νβ could not be
bounded, hence this completes the proof of the lemma.
If r ∉ R, taking j = [r ], then p [n(r −[r ])] 1+pn Zp (x − 1)[r ] νβ cannot tend to zero.
2. Galois action on BdR . Let F be a one-dimensional height-one formal group over
Zp , let π = pα be the uniformizer with α ∈ Z×
p , and let f be a Frobenius power series
corresponding to F, then we have f (x) ≡ π x(mod deg 2), f (x) ≡ x p (mod p). Let η(x) :
Gm → F which is an isomorphism, such that η(x) = Ωx + · · · , where Ω is a p-adic unit
−n
with Ωϕ−1 = α. Let ωn = ηϕ (ζpn − 1), Kn = Qp (ωn ), and K∞ = ∪n≥0 Kn , let Tπ be
the Tate module, and let κ : Gal(K∞ /Qp ) → Z×
p be the character given by the action on
Tπ , then we know that κ = χψ with ψ an unramified character such that ψ(Frobp ) = α
and σ (ωn ) = [κ(σ )]ωn , where for a ∈ Zp , [a] denotes the unique endomorphism of
F such that [a] = aX + · · · (see [4, Section 20.1]). Let Ξ be the completion of Qpur . Let
tπ = Ωt. Then tπ has the property that σ (tπ ) = κ(σ )tπ , ϕ(tπ ) = π tπ .
If x ∈ K∞ and n ∈ N, we define Tn (x) := (1/p m ) TrKm /Kn (x) for m 1. We extend
k
k
) = Tn (ak )tπ
. Note that this
Tn to a map from K∞ ((tπ )) to Kn ((tπ )) by Tn ( ak tπ
definition does not depend on the choice of tπ since the different choice only differs
a multiple in Qp and Tn is Qp -linear. We also define Tr/Kn = (1/[Km : Kn ]) TrKm /Kn (x)
for m 1.
Recall that ᏾ is the projective limit of the following diagram:
Ō ← Ō ← · · · .
(2.1)
p
n ∈ OCp such that x
n ≡ xn (mod p).
For x ∈ ᏾, x = (xn )n∈N , xn+1 = xn . Choose x
n+m ) exists and does not depend on the choice of
Lemma 2.1. The limit limn→∞ f (m) (x
n . Denote it by x (m) and then we have f (x (m) ) = x (m−1) .
x
Proof. We first prove that f has the property, for y ∈ ᏾,
f (n) (x + py) ≡ f (n) (x) mod p n+1 y .
(2.2)
This is true for n = 1 from the definition of f . Assume it is true for n, then we have
f (n) (x + py) = f (n) (x) + p n+1 yz for some z ∈ ᏾, hence
f (n) (x) + p n+1 yz
= f (n+1) (x) mod p · p n+1 yz
= f (n+1) (x) mod p n+2 y .
f (n+1) (x + py) = f
From this we see that
n+m − f (n+1) x
n+m+1 = f (n) x
n+m − f (n+1) x
n+m + py
f (n) x
≡ 0 mod p n+1 ,
hence the lemma follows.
(2.3)
(2.4)
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
613
On the other hand, if (x (m) ) satisfies the relation f (x (m) ) = x (m−1) , then (x̄ (m) ) ∈ ᏾,
hence ᏾ is one-to-one corresponding to
(2.5)
x (m) m∈N f x (m) = x (m−1) .
Let W = {w ∈ Cp | ∃n, s.t. [π ]n w = 0}. For w ∈ W , if n ∈ N such that [π ]n w ≠ 0,
[π ]n+1 w = 0, then we say that w has order n. To w ∈ W , we can associate an element
w̄ = (w, f (−1) (w), f (−2) (w), . . . ). The association is not unique.
GK
Lemma 2.2. All elements of Cp ∞ can be written in the form x =
aw (x)w with
w∈W
aw (x) ∈ Qp tending to zero when the order of w tends to infinity.
GK
GK
Proof. Since K∞ = Q̄p ∞ , W is a base of OK∞ /Zp and Cp ∞ is a separated completion of K∞ with respect to p-adic topology, hence the proof follows from Tate-Sen-Ax
theorem.
+ GK∞
Proposition 2.3. (i) The field K∞ ((tπ )) is dense in (BdR
)
and Tn can be extended
+ GK∞
to Kn ((tπ )).
to a continuous Qp -linear map from (BdR )
+ GK∞
(ii) If F ∈ (BdR
)
, then limn→∞ p n Tn (F ) = F .
GK
+ GK∞
)
, θ(x) ∈ Cp ∞ , hence θ(x) = w∈W aw (x)w for some
Proof. (i) For all x ∈ (BdR
aw (x) ∈ Qp from Lemma 2.2. Let
+ GK∞
−1
R(x) = tπ
aw (x)[w̄] ∈ BdR
,
(2.6)
x−
w∈W
so we can repeat the above process and get
k+1 k+1
R
(x) −
x = tπ
k
i
tπ
aw R (i) (x) [w̄] ,
(2.7)
w
i=0
+ GK∞
and this shows that K∞ ((tπ )) is dense in (BdR
)
.
(ii) For F ∈ K∞ , F ∈ Km0 for some m0 , hence for n ≥ m0 , Tn (F ) = (1/p m ) TrKm /Kn (F ) =
(1/p n )F . Hence, limn→+∞ p n Tn (F ) = F . By the definition of Tn on K∞ ((tπ )) we have for
F ∈ K∞ ((tπ )), limn→+∞ p n Tn (F ) = F .
+ GK∞
For F ∈ (BdR
)
, (i) shows that we can take Fk ∈ K∞ ((tπ )) such that limk→+∞ Fk = F .
Hence
lim p n Tn (F ) = lim p n Tn lim Fk = lim lim p n Tn Fk = lim Fk = F .
(2.8)
n→+∞
n→+∞
k→+∞
k→+∞ n→+∞
k→+∞
We can change the order of the limit since p n Tn is continuous.
Recall that LA denotes the space of locally analytic functions with compact support in Qp taking values in Qp . For a p-adic Banach space A, define Ᏸcont (Qp , A) =
Homcont (LA, A) with respect to Morita topology. Define
ψ
Ᏸcont Qp , Ξ := µ ∈ Ᏸcont Qp , Ξ | σ
f (x)µ =
f ψ(σ )x µ, ∀σ ∈ GQp .
Qp
Qp
(2.9)
614
SHAOWEI ZHANG
For µ ∈ Ᏸcont (Qp , Ξ)ψ with compact support, we can define an element in BdR as
Ᏺcont (µ) =
Qp
x
ε µ.
(2.10)
Colmez called this element as the continuous Fourier transformation of µ and we continue using his notation. Recall from [8, Section 4] that µ ∈ Ᏸcont (Qp , Ξ) is said to be of
order r ∈ R̄+ if for all open compact set X ⊂ Qp and all j ≥ 0, the following sequence
is r -bounded:
E(n(r −j))
j (x − a) µ .
sup p
(2.11)
a+p n Zp
a∈X
+
n
+
Define B+
cont := ∩n≥0 ϕ (Bmax ). An element F ∈ Bcont is said to be of order r if the
[nr ] −n
sequence p
ϕ F is r -bounded. The power series A(T )= an T n ∈ Ξ[[T ]] is said
to be of order r if n−r |an | is r -bounded. Define Ξ[[T ]]ψ := {h(T ) ∈ Ξ[[T ]] such that
for all σ ∈ GQp , σ (h(T )) = h((1 + T )ψ(σ ) − 1)}.
+
)GK∞ .
Proposition 2.4. (i) For µ ∈ Ᏸcont (Qp , Ξ)ψ , Ᏺcont (µ) ∈ (Bcont
ψ
(ii) For µ ∈ Ᏸcont (Zp , Ξ) , the Amice transformation Ꮽµ (T ) is in Ξ[[T ]]ψ , µ has order
r if and only if Ꮽµ has.
(iii) For µ ∈ Ᏸcont (Zp , Ξ)ψ , if µ has order r , then Ᏺcont (µ) has order r .
(iv) For a crystalline representation V , for µ ∈ Ᏸcont (Zp , Ξ ⊗ D(V ))ψ , if µ has order
r , then Ᏺcont (µ) has order r + r (V ), where r (V ) = min{k ∈ Z | ϕ(d) = p k d for some
d ∈ D(V )}.
Proof. (i) For σ ∈ GQp and µ ∈ Ᏸcont (Qp , Ξ)ψ ,
σ Ᏺcont (µ) = σ
Qp
x
ε µ =
Qp
ψ(σ )xχ(σ ) ε
µ=
Qp
κ(σ )x ε
µ.
(2.12)
+
If κ(σ ) = 1, then σ (Ᏺcont (µ)) = Ᏺcont (µ), so Ᏺcont (µ) ∈ (Bmax
)GK∞ . On the other hand,
α−n x
]µ; we have
let Fn = Qp [εn
ϕ Fn = ϕ n
Qp
−n α x
εn
µ =
Qp
x
ε µ
(2.13)
GK
and this shows that Ᏺcont (µ) ∈ (B+
cont ) ∞ .
(ii) The Galois action gives
σ Ꮽµ (T ) = σ
Zp
(1 + T )x µ =
Zp
(1 + T )ψ(σ )x µ = Ꮽµ (1 + T )ψ(σ ) − 1 ,
(2.14)
hence Ꮽµ (T ) ∈ Ξ[[T ]]ψ .
The second statement can be found in [1].
(iii) Assume µ has order r , hence Ꮽµ (T ) has order r by (ii).
Case 1. Assume r ∈ N, then there exists a constant C > 0 such that |ak | ≤ Ckr , that
is, −vp (ak ) ≤ (log C + r log k)/log p (here the log is the logarithmic function on real
variable). The real function f (x, a, r ) = ax − r (log x/log p) has a minimum g(a, r ) at
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
615
x = r /a log p. For x ∈ R>0 , we have
vp ak + r n +
k
log k log C
k
−
≥ rn+
−r
(p − 1)p n−1
(p − 1)p n−1
log p log p
k
p
log C
=f
,
,r −
pn p − 1
log p
p
log C
≥g
,r −
.
p −1
log p
(2.15)
Take m ≤ −1 + g(p/(p − 1), r ) − log C/log p and recall that
Fn =
Qp
α
εn
−n x
k
−n α
µ = ak εn
−1 ,
−n
Fn = sup p −(vp (ak )+kvp ([εnα ]−1)) = sup p −(vp (ak )+k/(p−1)pn−1 ) ;
k
(2.16)
k
we get
nr p Fn = sup p −(nr +vp (ak )+k/(p−1)pn−1 ) ≤ p −m ,
(2.17)
k
hence Ᏺcont (µ) is r -bounded.
Case 2. For r ∈ R̄+ \ R+ , then we have limn→+∞ n−r |an | = 0. Hence, we can choose a
sequence ck → 0 such that |ak | < ck kr , then the above proof shows that we can take
mk = − 1 + g
!
log ck
p
,r −
,
p −1
log p
(2.18)
then p [nr ] Fn ≤ p −mn hence tends to zero.
(iv) Assume d1 , . . . , dk are a base of D(V ) and µ ∈ Ᏸ(Qp , Ξ ⊗ D(V ))ψ , then Ᏺcont (µ) =
−nr
|ϕ−n (bi ) ≤ cn for some cn (cn is
bi ⊗ di for some bi ∈ B+
cont with order r , p
−n(r +r (V ))
|ϕ−n (bi ⊗ di )| ≤ cn |di |, hence
bounded if r ∈ R, otherwise cn → 0), so p
Ᏺcont (µ) is (r + r (v))-bounded.
Remark 2.5. Properties (iii) and (iv) are even true for µ has support in Qp . To prove
this we only need to extend (ii) to this case.
Lemma 2.6. (i) For m ≥ n ≥ 1,

p m−n ε(x),
ε κ(σ )x =
0,
σ ∈Gal(Km /Kn )
if x ∈ p −n Zp ,
otherwise.
(2.19)
(ii) For m > 0,
σ ∈Gal(Km /K0 )
ε κ(σ )x = p m 1Zp − p m−1 1p−1 Zp .
(2.20)
616
SHAOWEI ZHANG
Proof. (i) If x ∈ p −n Zp , then
p m−n −1
ε κ(σ )x =
σ ∈G(Km /Kn )
ε 1 + a · p n x = p m−n ε(x).
(2.21)
a=0
Otherwise,
p m−n −1
ε κ(σ )x = ε(x)
σ ∈G(Km /Kn )
ε
a=0
ab
pk
= 0,
(2.22)
with some b ≠ 0 a p-unit, k ≥ 1.
(ii) If x ∈ Zp , then
ε κ(σ )x = p m − p m−1 .
(2.23)
σ ∈G(Km /K0 )
If x ∈ p −1 Zp \ Zp , then
σ ∈G(Km /K0
p−1 a
= p m−1 · (−1) = −p m−1 .
ε κ(σ )x = p m−1
ε
p
a=1
)
(2.24)
And it is easy to see that if x ∉ p −1 Zp , then
ε κ(σ )x = 0.
(2.25)
σ ∈G(Km /K0 )
So the sum equals 1Zp · (p m − p m−1 ) − 1p−1 Zp \Zp p m−1 = p m 1Zp − p m−1 1p−1 Zp .
Proposition 2.7. If µ ∈ Ᏸcont (Qp , Ξ)ψ , then
+∞
k
Tn Ᏺcont (µ) =
t p −n
k=0
+∞
k
t
T0 Ᏺcont (µ) =
Zp
k=0
xk
ε(x)
µ , if n ≥ 1,
k!
p −n Zp
xk
xk
−1
µ −p
µ .
k!
p −1 Zp k!
(2.26)
Proof. The Fourier transformation gives
Ᏺcont (µ) =
Qp
x
ε µ=
=
Qp
ε(x)
Qp
ε(x) exp(tx)µ
∞
∞
xk
(tx)k
k
µ=
µ.
tπ
ε(x) k
k!
Ω · k!
Qp
k=0
k=0
(2.27)
617
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
To prove the first identity, we need to show that
Tn
xk
xk
ε(x) k
ε(x) k
µ = p −n
µ
Ω · k!
Ω · k!
Qp
p −n Zp
(2.28)
(note that the right-hand side is indeed in Kn ).
Now, assume µ has compact support p −m Zp for some m, then
Tn
xk
1
ε(x) k µ = m
Ω
p
p −m Zp
=
1
pm
= p −n
σ
σ ∈G(Km /Kn )
σ
p −m Z
p −n Z
xk
µ
Ωk
xk
ε κ(σ )x
µ
Ωk
p
ε(x)
p
p −m Zp
ε(x)
(2.29)
xk
µ.
Ωk
The same proof works for the second formula.
The Galois action on BdR has the following property.
Lemma 2.8. Suppose V is a p-adic representation of GK∞ , then (BdR ⊗ V )GK∞ is a
GK
finite-dimensional vector space over BdR ∞ with dimension dimQp V and can have a base
ϕ=1
consisting of elements in (Bmax ⊗ V )GK∞ .
+
⊗V )GK∞ is
Proof. By an argument similar to [2, Corollary B.14], we can see that (BdR
+ GK∞
a finite-dimensional vector space over (BdR )
and we can have a base v1 , . . . , vd such
that v1 , . . . , vd ∈ (W (᏾)⊗V )GK∞ . Assume e1 , . . . , ed are a base of V /Qp , and (v1 , . . . , vd ) =
+
⊗ V )GK∞ ), then θ(det(A)) ≠ 0.
(e1 , . . . , ed )A with A ∈ GLd ((BdR
For k ≥ 1, 1 ≤ i ≤ d, let vi,k = m∈Z π −km ϕm−1 (η([ε] − 1)k+1 vi ), then this is a con+
vergence sum and it converges to an element in (Bcrys
⊗ V )GK∞ . Consider that v1,k /
−1
k+1
−1
tends to ϕ (vi ) when k → ∞, hence when k 1, vi,k /ϕ−1 (η([ε]
ϕ (η([ε] − 1))
− 1))k+1 , . . . , vd,k /ϕ−1 (η([ε] − 1))k+1 are linearly independent, hence v1,k , . . . , vd,k are
linearly independent.
ϕ=1
−k
−k
−k
−k
vi,k ) = (tπ
vi,k ) we see that tπ
v1,k , . . . , tπ
vd,k ∈ (Bmax ⊗ V )GK∞ are a
From ϕ(tπ
GK
base of (BdR ⊗ V )GK∞ over BdR ∞ .
ϕ=1
Lemma 2.9. H 1 (K∞ , BdR ⊗ V ) = 0 and H 1 (K∞ , Ᏸr (Z×
p , Bmax ⊗ V )) = 0.
Proof. Assume τ → cτ is a cocycle from GK∞ → V . From [2] we know that H 1 (K∞ ,
W (m᏾ ) ⊗ T ) = 0, where T ⊂ V is a Galois invariant lattice. Let ω = (ω0 , ω1 , ω2 , . . .) ∈ ᏾,
then [ω]cτ ∈ H 1 (K∞ , W (m᏾ )⊗T ) = 0, so we can find a c ∈ W (᏾)⊗V such that [ω]cτ =
(1−τ)c. From the ramification property, we can show that [ω]p−1 ≡ 0(mod p) in Acrys ,
GK
hence a = n∈Z (ϕn ([ω])/p n ) is convergent in Acrys∞ and ϕ(a) = pa. Let
c =
1 ϕn [ω]−1 ϕ[ω] c ,
pa
pn
ϕ=1
(2.30)
then it is easy to see that (1−τ)c = cτ . Since c ∈ (Bmax ⊗V ), this shows that the incluϕ=1
sion map h1 (K∞ , V ) → H 1 (K∞ , Fil−1 (Bmax ⊗V )) is the zero map. By similar arguments as
618
SHAOWEI ZHANG
ϕ=1
in [2, Lemmas B.18 and B.19] we conclude that H 1 (K∞ , BdR ⊗V ) = 0, H 1 (K∞ , Ᏸr (Z×
p , Bmax ⊗
V )) = 0.
Lemma 2.10. For a p-adic representation V of GQp , as a Ᏸ0 (Z×
p , Qp )-module the following sequence is exact:
GK∞ 0 → H 1 Γ , Ᏸ0 Z×
→ H 1 K∞ , Ᏸ0 Z×
Γ → 0.
→ H 1 Qp , Ᏸ0 Z×
p,V
p,V
p,V
(2.31)
Proof. The proof can be found in [2].
3. Perrin-Riou exponential map. For I ⊂ Z, we have defined LP I , ᏰIalg (Qp , A), Φ,
and Ᏺalg in [8, Section 4]. For a Galois module A, in this section, the Galois action on
ᏰIalg (Qp , A) is defined by
f (x)σ (µ) = σ
f κ(σ )x µ ,
(3.1)
where µ ∈ ᏰIalg (Qp , A) and σ ∈ GQp . Recall that we have the following formulas:
Ᏺalg x k 1a+pn Zp (y) = p −n
k!
ε(ay)1p−n Zp (y),
(−ty)k
1
1
Ᏺalg x k 1Z×p =
· 1Zp − p −1
1 −1 .
(−tx)k
(−tx)k p Zp
(−∞,h−1]
Proposition 3.1. For a de Rham representation V , if µ ∈ Ᏸalg
then
(h)
Ᏺalg (µ)
∈
[1−h,+∞)
Ᏸalg
(Qp , BdR ⊗ V )
(3.2)
(3.3)
(Qp , Ξ⊗D(V ))ψ ,
is fixed by GQp .
Proof. For σ ∈ GQp ,
a+p n Zp
(h)
x k σ Ᏺalg (µ) = σ
κ(σ )−1 a+p n Zp
κ(σ )x
k
(h)
Ᏺalg (µ)
−n
(k + h − 1)!κ(σ )k
× σ
ε κ(σ )−1 ax (−tx)−k µ
=p
p −n Zp
= p −n (k + h − 1)!
=
(h)
a+p n Zp
p −n Zp
ε(ax)(−tx)−k µ
(3.4)
(h)
x k Ᏺalg (µ),
(h)
hence we have σ (Ᏺalg (µ)) = Ᏺalg (µ).
Let exp denote the connecting map of the following exact sequence:
× × ϕ=1
×
+
+
+
0 → Ᏸ+
alg Zp , V → Ᏸalg Zp , Bmax ⊗ V → Ᏸalg Zp , BdR /BdR ⊗ V → 0.
(3.5)
619
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
Proposition 3.2. The following diagram is commutative:
×
GQp
+
Ᏸ+
alg Zp , BdR /BdR ⊗ V
× H 1 Qp , Ᏸ+
alg Zp , V
exp
k
1+p n Zp x µ
DKn V κ k
(3.6)
k
1+p n Zp x µ
H 1 K, V κ k .
exp
GQp
+
×
, it is easy to see that 1+pn Zp x k µ ∈
Proof. For µ ∈ Ᏸ+
alg (Zp , BdR /BdR ⊗ V )
×
k
1
k
DKn (V (κ k )) and for ξ ∈ H 1 (Qp , Ᏸ+
alg (Zp , V )), 1+p n Zp x ξ ∈ H (Kn , V (κ )), then the
proposition follows from the following commutative diagram:
k
1+p n Zp x
× Ᏸ+
alg Zp , V
0
+
BdR /BdR
⊗V
ϕ=1
Bmax ⊗ V
V
0
0
k
1+p n Zp x
k
1+p n Zp x
× ϕ=1
Ᏸ+
alg Zp , Bmax ⊗ V
+
×
Ᏸ+
alg Zp , BdR /BdR ⊗ V
(3.7)
0.
Now we define the Perrin-Riou exponential map E = Eh,V as the composition of the
following ones:
Φ=1,ψ Ᏺalg
GQp
Ᏸ−
→
Ᏸ+
alg Qp , Ξ ⊗ D(V )
alg Qp , BdR ⊗ V
GQp
×
Fil<0
+
→
Ᏸ+
alg Zp , BdR /BdR ⊗ V
× exp
→
H 1 Qp , Ᏸ+
.
alg Zp , V
(3.8)
The invariance of Φµ = µ gives the following identity:
p −n Zp
f (x)
1
µ = ϕ−n
(−tx)k
Zp
f
x
µ
p n (−tx)k
(3.9)
for all n ∈ Z, k ≥ 0, and f a local constant function with compact support p −n Zp .
Theorem 3.3. The integrals of the exponential map give
x k Eh,V (µ) = (k + h − 1)! expV (κ k ) (1 − ϕ)−1 1 − p −1 ϕ−1
µ
,
k
Z×
Z×
p
p (−tx)
−n ax
ϕ
µ
x k Eh,V (µ) = (k + h − 1)! expV (κ k )
ε
·
.
pn
pn
(−tx)k
a+p n Zp
Zp
(3.10)
620
SHAOWEI ZHANG
Proof. By formulas (3.2) and (3.3), we have
(h)
Z×
p
x k Ᏺalg (µ) =
Qp
1
1
−1
µ
1
−
p
1
−1
Z
(−tx)k p
(−tx)k p Zp
Qp
µ
= (k + h − 1)!(1 − ϕ)−1 1 − p −1 ϕ−1
,
k
Z×
p (−tx)
(h)
(h) x k Ᏺalg (µ) =
Ᏺalg x k 1a+pn Zp µ
=
a+p n Zp
(h) Ᏺalg x k 1Z×p µ
(k + h − 1)!
(3.11)
Qp
1
ε(ax)1pn Zp
(−tx)k
ϕ−n
ax
µ
= (k + h − 1)! n
ε
.
p
p n (−tx)k
Zp
=
Qp
(k + h − 1)!p −n ·
Hence, by Proposition 1.2, we have
Z×
p
x k Eh,V (µ) = expV (κ k )
Qp
= expV (κ k ) (k + h − 1)!(1 − ϕ)−1 1 − p −1 ϕ−1
a+p n Zp
Ᏺalg x k · 1Z×p µ
x k Eh,V (µ) = expV (κ k )
Qp
Ᏺalg x k 1a+pn Zp µ
Z×
p
µ
,
(−tx)k
(3.12)
ax
ϕ−n
µ
= expV (κ k ) (k + h − 1)! n
ε
.
p
p n (−tx)k
Zp
Lemma 3.4. For a de Rham representation V such that Fil−1 D(V ) = D(V ), µ ∈
j
a+p n Zp (x − a) Eh,V (µ) restricted
(−∞,0]
Ᏸalg (Qp , Ξ ⊗ D(V ))Φ=1,ψ , a ∈ Z×
p , n ≥ 1, and j ≥ 0,
[0,∞)
1
×
to H (K∞ , Ᏸalg (Zp , V )) is represented by the cocycle
a,n,j (µ) ,
τ → (τ − 1) EulB β
(3.13)
where
−n
a,n,j (µ) = j!p nj (−t)j ϕ
β
pn
Z×
p
[ε]ax
µ
.
xj
(3.14)
Proof. We first consider
ϕ−n
µ
ax
.
x k E1,V (µ) = expV (κ k ) k! n
ε
p
p n (−tx)k
a+p n Zp
Zp
(3.15)
621
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
Put
γa,n,k (µ) = k!
a,n,k (µ) = k!
γ
ϕ−n
pn
ϕ−n
pn
Zp
ε
µ
ax
,
p n (−tx)k
(3.16)
k
ax µ
(−atx)i
ε
.
k
i!
(−tx)
Zp
i=0
By the formula ε(1/p n ) = [εn ] exp(−t/p n ), we have
x
ε
pn
k
k
x x 1 −tx i
(−tx)i
−n
mod Filk+1 BdR .
≡ εn
=ϕ
ε
n
i! p
i!
i=0
i=0
(3.17)
Let
k
ax ax
(−atx)i
−n
∈ Filk+1 BdR
−ϕ
ε
δa,n,k (x) = ε
pn
i!
i=0
k
ax (−atx)i
ax
−n
− ε
=ϕ
ε
,
pn
i!
i=0
(3.18)
then
ϕ−n
pn
ϕn δa,n,k
µ
(−tx)k
−1
1
1
+
∈ BdR
⊗ D(V ) ⊂ BdR
⊗ BdR
⊗ V ⊂ BdR
⊗V.
a,n,k (µ) − γa,n,k (µ) = k!
γ
Zp
(3.19)
a,n,k (µ) ∈ Bmax ⊗ V is a lifting of γa,n,k under the projection map Bmax →
Hence, γ
+
. By [2, Lemma 0.5.1], we see that the above cohomology class expV (κ k ) (γa,n,k (µ))
BdR /BdR
a,n,k (µ)). Moreover,
is represented by the cocycle τ → (1 − τ) EulB ((1 − ϕ)γ
ϕ−n
a,n,k (µ) = (1 − ϕ)k! n
(1 − ϕ)γ
p
ϕ−n
= k! n
p
k
ax µ
(−atx)i
ε
k
i!
(−tx)
Zp
i=0
k
ax µ
(−atx)i
ε
k
i!
(−tx)
Z×
p
i=0
(3.20)
since
ϕ
k
ax µ
(−atx)i
ε
i
i!
(−tx)
Zp
i=0
=
k
ax 1
(−atx)i
Φµ,
ε
k
i!
(−tx)
Zp
i=0
(3.21)
and Φ(µ · 1Zp ) = 1pZp · µ.
k
Hence, for a ∈ Z×
p , n ≥ 1, a+p n Zp x E1,V (µ) is represented by
∗
(µ) ,
τ → (τ − 1) EulB γa,n,k
(3.22)
622
SHAOWEI ZHANG
where
∗
γa,n,k
(µ) = k!
ϕ−n
pn
k
ax ν
(−atx)i
.
ε
i!
(−tx)k
Z×
p
i=0
(3.23)
Therefore,
j
(−a)j−k x k E1,V (µ)
a+p n Zp k
k=0
j j
=
x k E1,V (µ)
(−a)j−k
k
a+p n Zp
k=0
a+p n Zp
(x − a)j E1,V (µ) =
j (3.24)
is represented by
∗
j
(−a)j−k EulB γa,n,k
τ → (τ − 1)
(µ)
k
k=0
j k
j
ax µ
(−atx)i
ϕ−n
j−k
= (τ − 1) EulB
ε
(−a) k! n
k
p
i!
(−tx)k
Z×
p
k=0
i=0
k
j (−atx)i
j
ϕ−n
atx
j−k
·
= (τ − 1) EulB
e
k!
(atx)
k
p n Z×p
i!
k=0
k=0
µ
×
(−tx)j
ax ϕ−n
µ
,
j! ε
= (τ − 1) EulB
p n Z×p
(−tx)j
j
and
j
a+p n Zp (x − a) E1,V (µ)
(3.25)
is represented by
ax µ
ϕ−n
,
τ → (τ − 1) EulB j!p nj (−t)−j n
ε
p
xj
Z×
p
(3.26)
and the lemma follows.
Lemma 3.5. The following diagram is commutative:
(−∞,h] Ᏸalg
Φ=1,ψ
Qp , Ξ ⊗ D V (κ)
µ→(−tx)µ
Eh+1,V (κ)
[−h,∞) ×
H 1 K, Ᏸalg
Zp , V (κ)
(−∞,h−1] Ᏸalg
Φ=1,ψ
Qp , Ξ ⊗ D(V )
Eh,V
ξ→x −1 ξκ −1
[1−h,∞) ×
H 1 K, Ᏸalg
Zp , V .
(3.27)
623
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
(−∞,h]
Proof. First note that all the maps are well defined. For µ ∈ Ᏸalg
(−∞,h−1]
D(V (κ)))ψ , (−tx)µ ∈ Ᏸalg
Z×
p
x k Eh,V (−tx)µ
−1
= (k + h − 1)! expV (κ k ) (1 − ϕ)
Z×
p
(Qp , Ξ ⊗
(Qp , Ξ ⊗ D(V ))ψ ,
1 − p −1 ϕ−1
Z×
p
µ
,
(−tx)k−1
(3.28)
x k · x −1 Eh+1,V (κ) (µ)
= (k + h − 1)! expV (κ k ) (1 − ϕ)−1 1 − p −1 ϕ−1
Z×
p
µ
,
(−tx)k−1
hence the lemma follows.
Theorem 3.6. Assume V is a crystalline representation and h ∈ Z such that Fil−h D(V )
temp (Qp , Ξ ⊗ D(V ))Φ=1,ψ , under the map
= D(V ). For µ ∈ Ᏸ
Φ=1,ψ
[1−h,∞) ×
→ H 1 Qp , Ᏸalg
Qp , Ξ ⊗ D(V )
Zp , V
Γ
res
[1−h,∞) ×
→
H 1 K∞ , Ᏸalg
,
Zp , V
(−∞,h−1] Ᏸalg
(3.29)
Γ
the image is in H 1 (K∞ , Ᏸtemp (Z×
p , V )) with Γ = Gal(K∞ /Qp ).
Proof. Since x 1−h E1,V (κ 1−h ) ((−tx)h−1 µ) = Eh,V (µ), by Lemma 3.5, replacing V by
V (1−h) and µ by (−tx)h−1 µ, we can assume h = 1. From Lemma 3.5 for a ∈ Z×
p , n ≥ 1,
and j ≥ 0, a+pn Zp (x − a)j E1,V (µ) is represented by the cocycle
τ → (τ − 1) EulB j!p
nj
−j
(−t)
ϕ−n
pn
ax µ
.
ε
xj
Z×
p
(3.30)
From [8, Lemma 14], we know that if µ is of order r , then µ/x j is also of order r . From
Proposition 2.4, we know that Z×p [εax ](µ/x j ) has order r + r (V ), that is,
p [n(r +r (V )+1)] ϕ−n
is r -bounded. Hence, p [n(r +r (V )+1)]
Z×
p
ax µ
ε
xj
j
a+p n Zp (x − a) E1,V (µ)
(3.31)
represented by
τ → (τ − 1) EulB j!(−t)−j p [n(r +r (V )+1)] ϕ−n
ax µ
ε
×
xj
Zp
(3.32)
Γ
is (r + r (V ))-bounded. Hence, the image is in H 1 (K∞ , Ᏸr +r (V )+1 (Z×
p , V )) .
4. The logarithmic map. The proof of Theorem 3.6 suggests that Ᏺcont (µ) has some
deep arithmetic meaning for the distribution µ. In this section, we construct a logarithmic map on the cohomology side, which is related to Ᏺcont (µ).
ϕ=1
Suppose V is a de Rham representation. The Qp -space ∪n∈N (Bmax ⊗ V (κ −i ))GKn ⊂
ϕ=1
∪n∈N (Bmax ⊗ V )GKn has dimension dimQp V , so ∪n∈N (Bmax ⊗ V (κ −i ))GKn is a closed
624
SHAOWEI ZHANG
ϕ=1
subspace of Bmax ⊗ V . Let Wi be one supplement. Suppose h ∈ Z, h ≥ 1, µ ∈ H 1 (Qp ,
Ᏸh− (Z×
, V )), and τ → µτ is a 1-cocycle representation of µ. Suppose µ has the propp
ϕ=1
erty 1+pn Zp x −i µ ∈ He1 (Kn , V (κ −i )) = ker{H 1 (Kn , V (κ −i )) → H 1 (Kn , Bmax ⊗ V (κ −i ))}.
ϕ=1
For such a µ, there is a cn,i ∈ Bmax ⊗ V (κ −i ) such that
1 − κ(σ )−i σ cn,i =
1+p n Zp
x −i µσ .
(4.1)
The choice of Wi makes it possible to choose cn,i ∈ Wi uniquely for n 1.
Theorem 4.1. Suppose V is a de Rham representation of GQp , h ≥ 1 is an integer, and
−i
1
−i
µ ∈ H 1 (Qp , Ᏸh− (Z×
p , V )) such that 1+p n Zp x µ ∈ He (Kn , V (κ )) for n ≥ 1, 0 ≤ i ≤ h−1.
Choose cn,i ∈ Wi such that
1 − κ(σ )−i σ cn,i =
1+p n Zp
x −i µ,
(4.2)
h−1 ϕ=1
then the sequence (1/(h − 1)!)p n i=0 h−1
cn,i converges to an element in (Bmax ⊗
i
V )GK∞ which is denoted by L h,V (µ).
ϕ=1
Proof. By Lemma 2.9, H 1 (K∞ , Ᏸh− (Z×
p , Bmax ⊗ V )) = 0, the inflation-restriction sequence gives the following isomorphism:
∼
ϕ=1
GK∞ ϕ=1
→
H 1 Γ , Ᏸh− Z×
.
H 1 Qp , Ᏸh− Z×
p , Bmax ⊗ V
p , Bmax ⊗ V
(4.3)
ϕ=1
GK∞
)) correspond to the image of µ under the map
Let µ ∈ H 1 (Γ , Ᏸh− (Z×
p , (Bmax ⊗ V )
ϕ=1
1
×
1
H 1 (Qp , Ᏸh− (Z×
p , V )) → H (Qp , Ᏸh− (Zp , Bmax ⊗V )). Then µ−µ is a coboundary in H (Qp ,
ϕ=1
Ᏸh− (Z×
p , Bmax ⊗ V )),
ϕ=1
that is, there exists ν ∈ Ᏸh− (Z×
p , Bmax ⊗ V ) such that µσ − µσ = (1 −
ϕ=1
−i
−i GK∞
−i
and (1−κ(σ ) σ )un,i
σ )ν. Let un,i = cn,i − 1+pn Zp x ν, then un,i ∈ (Bmax ⊗V (κ ))
= 1+pn Zp x −i µσ for σ ∈ GKn , and we have
h−1
h−1
n
i h−1
lim p
(−1)
(−1)
cn,i − p
un,i
n→∞
i
i
i=0
i=0
h−1
= lim p n
ν = 0.
1 − x −1
n
h−1
i
n→∞
(4.4)
1+p n Zp
+
)ϕ=1
Here, again, we use [8, Lemma 14]. Note that there exists a k such that un,i ∈(t −k Bmax
−i
⊗V (κ ) (which does not depend on n, i) [2]. The theorem will follow from the following
lemma.
ϕ=1
GK∞
)) and k ∈ N such that for
Lemma 4.2. Suppose µ ∈ H 1 (Γ , Ᏸh− (Z×
p , (Bmax ⊗ V )
ϕ=1
−i
n ≥ 1 and 0 ≤ i ≤ h − 1, the image 1+pn Zp x µ is zero in H 1 (Γn , (Fil−k Bmax ⊗ V )GK∞ ).
Suppose
Fil−k Bmax ⊗ V
ϕ=1
GK∞
GKn
−k ϕ=1
= ∪+∞
Bmax ⊗ V κ −i
⊕ Wi
n=0 Fil
(4.5)
625
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
is a direct summand decomposition (since the first component is closed). Suppose σ →
µσ is a cocycle representation of µ, take un,i ∈ Wi such that 1+pn Zp x −i ξσ = (1 −
h−1
κ −i (σ )σ )un,i for all σ ∈ GKn , then the sequence vn = p n i=0 (−1)i h−1
un,i converges
i
to an element in (Fil−k Bmax ⊗ V )GK∞ .
ϕ=1
Proof. The cocycles satisfy µσ −1 τ = µσ −1 +σ −1 µτ = µτ +τµσ −1 , and this means that
1+p n Zp
σ −1 µτ = µτ + (τ − 1)µσ −1 ,
−i −1
−1
x σ µτ = σ
κ i (σ )x −i µτ .
(4.6)
κ(σ )+p n Zp
This gives
κ(σ )+p n Zp
x −i µτ = κ(σ )−i σ
1+p n Zp
i
·σ
= tπ
1+p n Zp
x −i · σ −1 µτ
−i
tπ x
· σ −1 µτ
i
−i
(1 − τ)σ tπ
un,i −
= tπ
1+p n Zp
(4.7)
−i
tπ x µσ −1
by
−i
−i
(1 − τ)un−1,i = tπ
·
tπ
κ(σ )+p n Zp
σ ∈Γn−1 /Γn
= (1 − τ)
σ
x −i µσ
−i
un,i
tπ
−σ
σ ∈Γn−1 /Γn
1+p n Zp
−i
tπ x µσ −1
.
(4.8)
From the choice of Wi , we can cancel (1 − τ) and get
vn − vn−1 = p n−1
h−1
(−1)i
i=0
h−1 i
−i
(σ − 1)tπ
un,i
tπ
i
σ ∈Γn−1 /Γn
−i
−σ
tπ x µσ −1 .
(4.9)
1+p n Zp
The sum
pn
−i
h−1 i
tπ σ
tπ x µσ −1
i
1+p n Zp
i=0
h−1
µσ −1 → 0
1 − κ(σ )−1 x
= pn σ
h−1
(−1)i
(4.10)
1+p n Zp
as κ(σ ) ∈ 1 + p n−1 Zp , µσ −1 → µ1 which is of order 1− . Hence, to show that vn is a
Cauchy sequence, we only need to show that for σn ∈ Γn−1 the sequence
h − 1 i
p
(−1)
κ σn σn − 1 un,i
i
i=0
n
tends to zero.
h−1
i
(4.11)
626
SHAOWEI ZHANG
Case 1 (h = 1). This is equivalent to showing that
lim p n σn − 1 un,i = 0.
(4.12)
n→+∞
Put Tσ =
p−1
j=0
σ j , then
p
Tσn σn − 1 un,0 = σn − 1 un,0 = −
1+p n Zp
µσnp .
(4.13)
Since µσnp has order 1− , we have
lim p n
n→∞
1+p n Zp
µσnp = 0,
(4.14)
hence
lim p n Tσn σn − 1 un,0 = 0.
n→∞
(4.15)
The theorem follows from the following proposition.
Proposition 4.3. There exists a constant C > 0 such that, for all n ∈ N, σ ∈ Γn ,
F ∈ (Fil−k Bmax ⊗ V )GKn ,
Tσ (F ) ≥ CF ,
(4.16)
where · is the norm on Bmax .
In the following we will develop several lemmas to prove the above proposition.
d
Lemma 4.4. There exist a ζ ∈ Z×
p and a Bmax -morphism g : Bmax ⊗ V → Bmax such that
ϕ=ζ
ϕ=1
g(Bmax ⊗ V )GK∞ is contained in ((Bmax )GK∞ )d , where d = dimQp V .
ϕ=1
Proof. Suppose e1 , . . . , ed ∈ (Bmax ⊗V )GK∞ such that they are a base of BdR ⊗V over
d
BdR by Lemma 2.8. Assume v1 , . . . , vd are a base of V /Qp and assume ei = j=1 aij vj ,
ϕ=1
then aij ∈ Bmax , the σ action gives det(σ ) · det(σ (aij )) = det(aij ). Consider the onedimensional representation detQp (V ) which has a base e = v1 ∧· · ·∧vd , then the above
formula gives e1 ∧· · ·∧ed = (det(aij))v1 ∧· · ·∧vd , hence σ (e) = (det(aij)/ det(σ (aij )))e.
But detQp (V ) is a one-dimensional representation, so it must be of the form φ0 φκ k ,
where φ0 is a finite-order N of character, φ is an unramified character such that
ur
φ(Frobp ) = u for some u ∈ Z×
p . Take ∈ Qp such that Frobp () = u, then the
above identity means that det(aij )/σ (det(aij ))=φ0 (σ )·φ(σ )·κ k (σ ). Assume σ ∈ GK∞
and raise Nth power, we get (det(aij )/σ (det(aij )))N = φN (σ ). Take ∆ = (det(aij ))N ,
then this gives σ (∆) = ∆ for all σ ∈ GK∞ . On the other hand, ϕ(∆) = ϕ(det(aij )·)N =
det(aij )N · N · uN = uN · ∆. Now, taking ζ = uN , g : x → ∆x, then g(ei ) = ∆ei and
ϕ(∆ei ) = uN δei = ζ∆ei , and the lemma follows.
−k +
+
Corollary 4.5. For k ∈ N, there exist αk ∈ Zp and gk : tπ
Bmax ⊗ V → (Bmax
)d such
+
)ϕ=αk ,GK∞ )d .
that gk maps (Fil−k Bmax ⊗ V )GK∞ to ((Bmax
ϕ=1
627
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
s
+
+
· g(Bmax
⊗ V ) (Bmax
)d , and let gk =
Proof. Use the g in Lemma 4.4, assume tπ
k+s
αk = π
ζ.
k+s
g,
tπ
Lemma 4.6. For k ∈ N, > 0, there exists n ∈ N such that for σ ∈ Γn and F ∈
ϕ=1
(Fil−k Bmax ⊗ V )GK∞ ,
gk σ (F ) − σ gk (F ) ≤ F .
(4.17)
1 ∈ GQp a lifting of γ1 ; we can
Proof. Let γ1 be a topology generator of Γ1 and γ
1 in the p-Sylow subgroup of GQp . γ
1 induces a continuous map from Γ1 → GQp
choose γ
"1
by sending x to γ
logγ x
1
∈ GQp under such a map. The map
. So, for σ ∈ Γ1 , we can get σ
−σ
◦ gk : Bmax ⊗ V → (Bmax )d and the map gk ◦ σ − σ ◦ gk : (Fil−k (Bmax ⊗ V ))GK∞ →
gk ◦ σ
ϕ=1
d
(Bmax ) are the same if we restrict to (Bmax ⊗ V )GK∞ , and
ϕ=1
−k −k −k −k tπ vi − σ
g k tπ
− 1 tπ
gk tπ
vi = gk σ
vi + 1 − σ
vi
gk σ
(4.18)
−k
−k
−k +
+
v1 , . . . , tπ
vd are a base of tπ
Bmax ⊗ V over Bmax
, and when σ → 1 the above
since tπ
goes to zero, hence the lemma follows.
Lemma 4.7 (Coleman-Colmez exact sequence). For α ∈ Zp , α ≠ 0, r = vp (α),
∼
+
ψ
)ϕ=α,GK∞ →
Ᏸr − (Z×
(i) if α ∉ p N , then (Bmax
p , Ξ ⊗ Qp ) ,
r
(ii) if α = p , then the following sequence is exact:
+ ϕ=pr ,GK∞
ψ
→ Ᏸr − Z×
→ Qp → 0.
0 → Qp t r → Bmax
p , Ξ ⊗ Qp
(4.19)
Proof. See [2, Appendix A].
Lemma 4.8. For r ∈ R+ , there exists Cr > 0 such that for µ ∈ Ᏸr (Zp , Ξ ⊗ D(V ))ψ and
a ∈ Zp ,
p−1
δia ∗ µ ≥ Cr µ.
(4.20)
i=0
Proof. This is the same as [2, Lemma III.2.6]. Note the ∗ above is induced by the
addition in Zp .
Corollary 4.9. For r ∈ R+ , there exists Cr > 0 such that for µ ∈ Ᏸr (Zp , Ξ ⊗ D(V ))ψ
and a ∈ 1 + pZp ,
p−1
δai ∗ µ ≥ Cr µ.
(4.21)
i=0
Here the ∗ is induced by the multiplication in Z×
p.
+
Corollary 4.10. There exist C > 0 and an n0 ∈ N such that for F ∈ (Bmax
)ϕ=α,GK∞
and σ ∈ Γn0 ,
Tσ (F ) ≥ CF .
(4.22)
628
to
SHAOWEI ZHANG
Proof. Using Corollary 4.9, F corresponds to µ, F = µ, and Tσ (F ) corresponds
δai ∗ µ, hence
Tσ (F ) = δai ∗ µ ≥ Cµ = CF .
(4.23)
Corollary 4.11. There exist a C > 0 and an n ∈ N such that Tσ (gk (F )) ≥ Cgk (F )
ϕ=1
for all F ∈ (Fil−k Bmax ⊗ V )GK∞ .
+
Proof. The corollary follows since gk (F ) ∈ (Bmax
)ϕ=α,GK∞ .
Now Proposition 4.3 follows since gk is just a multiple by p k+s ∆.
Case 2 (general h). In this case we need to show that
p
n
h−1
i
(−1)
i=0
h − 1 −i
κ σn σn − 1 un,i → 0
i
(4.24)
as n → ∞.
Lemma 4.12. For l ∈ N and σ ∈ Γ , put Tl,σ =
p−1
j=0
κ(σ )−lj σ j , then the sequence
h−1 h − 1 −i
p
Tl,σn
κ σn σn − 1 un,i
i
l=0
i=0
n
k
(4.25)
tends to zero as n → ∞.
Proof. For l ≥ 0 and l ≤ k
p −i
−i
p
· σn − 1 un,i = −
Tl,σn κ σn σn − 1 un,i = κ σn
1+p n Zp
x −i µσnp ,
(4.26)
then (4.25) equals
h−1
h−1
(−1)i
−p
i
i=0
n
h−1
Tl,σn
l=0,l≠i
1+p n Zp
x −i µσnp = −p n
with Rl ∈ p n(h−1−l) Zp [[T ]] and p n Rl · Yl ∈ p n(h−l) Zp [[T ]]
[2, Lemma III.3.3], the proof of the lemma follows.
h−1
Rl · Yl
(4.27)
l=0
1+p n Zp (x
− 1)l µσnp → 0. By
Lemma 4.13. There exist a C > 0 and an n ∈ N such that
h−1
Ti,σ (F ) ≥ CF h
i=0
for all F ∈ (Fil−k Bmax ⊗ V )GK∞ and σ ∈ Γn .
ϕ=1
(4.28)
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
629
Proof. The lemma follows using Proposition 4.3 h times.
This completes the proof of Lemma 4.2.
So, this completes the proof of the existence of the logarithm.
GK∞
Corollary 4.14. For µ ∈ H 1 (K, Ᏸh− (Z×
, if µτ = 0 for all τ ∈ GK∞ and
p , V ))
ϕ=1
x −i µ ∈ He1 (Kn , V (κ−i)), then L h,V (µ) ∈ (Bmax ⊗ V )GK∞ .
Proof. Since (1−κ(τ)−i τ)cn,i = 1+pn Zp x −i µτ for τ ∈ GK∞ , τ(cn,i ) = cn,i and cn,i =
1+p n Zp
ϕ=1
0 for n 1, hence L h,V (µ) ∈ (Bmax ⊗ V )GK∞ .
5. The relationship between the exponential map and the logarithmic map. For a
crystalline representation V , assume h ∈ Z such that Fil−h D(V ) = D(V ).
temp (Qp , Ξ ⊗ D(V ))Φ=1,ψ , there exists i 1 such that Ᏺcont (µ/
Lemma 5.1. For µ ∈ Ᏸ
i
(−tx) ) exists.
Proof. See [2, Section II.2.3].
ϕ=1
−i
GKn
Let Σ = (∪+∞
).
n=0 ((⊕i∈Z t Bmax ) ⊗ V (−i))
temp (Qp , Ξ ⊗ D(V ))Φ=1,ψ , choose r as in Lemma 2.1 and asTheorem 5.2. For µ ∈ Ᏸ
−h
sume h ≥ 1 such that Fil D(V ) = D(V ), then
L (h+r ),V (r ) E(h+r ),V (r )
µ
(−tx)r
≡
Qp
x
ε
µ
mod Σ.
(−tx)r
(5.1)
ϕ=1
Proof. Let F = Qp [εx ](µ/(−tx)r ), then it is easy to see that F ∈ FV0 (Bmax ⊗V (r ))GK∞ .
By the definition of L h,V , we only need to calculate, for 0 ≤ i ≤ h + r − 1, the integral
1+p n Zp
x −i · x r Eh,V (µ) =
1+p n Zp
x r −i Eh,V (µ)
= expV (κ r −i )
1+p n Zp
= expV (κ r −i )
Qp
(h)
x r −i Ᏺalg (µ)
(h) Ᏺalg 11+pn Zp · x r −i µ
(r − i + h − 1)!
ε(x)µ
(−tx)r −i
Qp
µ
,
= expV (κ r −i ) (h + i − 1 − r )!i!δV (κ r −i ) ◦ Tn Ᏺcont
r
(−tx)
(5.2)
= expV (κ r −i )
p −n · 1p−n Zp ·
where we have used Proposition 2.7 to get δV (κ r −i ) ◦ Tn (Ᏺcont (µ/(−tx)r )). Let dn,i =
(−1)i (h + r − 1 − i)!i!δV (κ i−r ) ◦Tn (Ᏺcont (µ/(−tx)r )) and cn,i = eB ◦Fil<0 (dn,i ). Then cn,i ∈
ϕ=1
Bmax ⊗ V such that the above cohomology class is represented by the cocycle
τ → (1 − τ) ◦ cn,i = 1 − κ r −i τ cn,i .
(5.3)
630
SHAOWEI ZHANG
Γ
By Theorem 4.1, Eh+r ,V (κ r ) (µ/(−tx)r ) is tempered in H 1 (K∞ , Ᏸ(h+r +r (V ))− (Z×
p , V )) , under the inflation-restriction exact sequence
GK∞ → H 1 Qp , Ᏸ(h+r +r (V ))− Z×
0 → H 1 Γ , Ᏸ(h+r +r (V ))− Z×
p,V
p,V
Γ
→ H 1 K∞ , Ᏸ(h+r +r (V ))− Z×
→ 0;
p,V
(5.4)
we can lift Eh+r ,V (κ r ) (µ/(−tx)r ) to an element in H 1 (Qp , Ᏸ(h+r +r (V ))− (Z×
p , V )), the lifting
ϕ=1
is not unique. By Corollary 4.14, modulo (Bmax ⊗ V )GK∞ the left-hand side of (5.1) is
equal to
h+r
−1
h+r −1
1
(−1)i ·
· cn,i
· pn
i
(h + r − 1)!
i=0
h+r
−1
h+r −1
1
pn ·
=
(−1)i
i
(h + r − 1)!
i=0
· eB ◦ Fil<0 (−1)i (h + r − 1 − i)!i!δV (κ i−r ) Tn Ᏺcont
= pn
h+r
+1
µ
(−tx)r
eB ◦ Fil<0 ◦δV (κ i−r ) Tn (F )
(5.5)
i=0
= p n eB ◦ Fil<0
h+r
−1
δV (κ i−r ) Tn (F )
i=0
= p n eB ◦ Fil<0 Tn (F )
= eB ◦ Fil<0 p n Tn (F )
→ eB ◦ Fil<0 (F ) = Fil<0 (F ) = F .
Theorem 5.3. Assume V is a de Rham representation, h ≥ 1 an integer, and µ ∈
−i
1
−i
H 1 (Qp , Ᏸh− (Z×
p , V )) such that 1+p n Zp x µ ∈ He (Kn , V (κ )) for n ≥ 1 and 0 ≤ i ≤ h−1,
then
δV (κ −k ) ◦ Tm L h,V (µ) = exp∗
V (κ k )
(−1)h−1
k(k − 1) · · · (k − h + 1)
1+p m Zp
x −k µ .
(5.6)
n
Proof. We will use the isomorphism κ : Γ → Z×
p , Γn 1 + p Zp , to transfer the
to
the
measures
on
Γ
.
Assume
γ
is
a
topology
generator of Γ and
measure on Z×
0
p
[Kn :Qp ]
γn = γ0
is a generator of Γn . It follows that
i
−i
δV (κ i−k ) ◦ Tm p n tπ
un,i ,
δV (κ −i ) ◦ Tm p n un,i = tπ
(5.7)
δV (κ i−k ) ◦ Tm ◦ γ = κ(γ)k−i ◦ Tm ,
(5.8)
for γ ∈ Γm ,
631
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
hence
k−i
−i
1 − κ γn
−i
δV (κ i−k ) ◦ Tm p n tπ
un,i = δV (κ i−k ) ◦ Tm 1 − γn tπ
un,i ,
pn
−i
−i
un,i = tπ
κ(x)−i µγn .
1 − γ n tπ
(5.9)
Γn
The cocycle relation gives
µγn =
[Kn :K
m ]−1
−i l
tπ
γm
l=0
−l
γm
Γm
l −i
κ γm
x µγm ,
(5.10)
and we get
−i
tπ κ(x) µγn
Γn

[Kn :K
m ]−1
−i
= tπ δV (κ −k ) ◦ Tm 
κ(γ)l(k−i)
δV (κ i−k ) ◦ Tm
l=0

(5.11)
κ(x) µγm  .
−i
−1 Γ
γm
The element δV (κ −k ) ◦ Tm (L h,V (µ)) is the limit of


[Kn :K
h−1
m ]−1
h−1
pn
l(k−i)
(−1)i ·
κ γm
κ(x)−i µγm  .
k−i δV (κ −k ) ◦ Tm 
−l
i
γm
Γn
1 − κ γn
i=0
l=0
(5.12)
Case 1 (h = 1). The above formula becomes


[Kn :K
m ]−1
l(k−i)
pn
κ γm
κ(x)−i µγm 
k · δV (κ −k ) ◦ Tm 
−l
γm
Γn
1 − κ γn
l=0


[Kn :K
m ]−1
lk
pn

=
κ γm
µγm 
k · δV (κ −k ) ◦ Tm
−l
γm
Γn
1 − κ γn
l=0

pm

−k

δV (κ −k ) ◦ Tm

, if m ≥ 1,
κ(x)
µ
γ

m
 k log κ γm
Γm
→

p
1


−k

·
◦
T
κ(x)
µ
·
δ

0
γ0 .
V (κ −k )
p − 1 k log κ γ0
Γ0
(5.13)
Hence, we come to the formula
δV (κ −k ) ◦ Tm L 1,V (µ) =
1
δ −k ◦ Tr/Km
k log κ γm V (κ )
Γm
κ(x)−k µγm .
(5.14)
Therefore, the theorem follows from the following lemma.
Lemma 5.4 (Kato [5]). If V is a de Rham representation of GK , for any unramified
character ψ, the map which sends x ∈ D(V ) to the cocycle τ → x log ψ(τ) ∈ D(V ) ⊂
BdR ⊗ V is the zero map, and the map which sends x to τ → x log χ(τ) (hence τ →
x log κ(τ)) gives an isomorphism of D(V ) to H 1 (K, BdR ⊗V ), and exp∗
V coincides with the
632
SHAOWEI ZHANG
map H 1 (K, V ) → H 1 (K, BdR ⊗ V ) → D(V ), where the last map is the inverse map of the
above isomorphism.
For any K ⊂ K∞ and a de Rham representation V of GK , we have the following diagram:
H 1 (K, V )
(5.15)
G GK
H 1 Γ , BdR ∞ ⊗ D(V ) H 1 Γ , BdR ⊗ V K∞ H 1 K, BdR ⊗ V
∼
D(V ).
GK
If c ∈ H 1 (K, V ) and τ → cτ is the cocycle representation, we can choose c ∈ H 1 (Γ , BdR ∞ ⊗
D(V )) such that c and c have the same images in H 1 (K, BdR ⊗V ). Let τ → cτ be a cocycle
representation of c . The element δV ◦Tr/K (cγ /logp κ(γ)) ∈ D(V ). Using Kato’s lemma,
we see that it is the same as exp∗
V (c), hence
exp∗
V (c) = δV ◦ Tr/K
1
cγ .
log κ(γ)
(5.16)
Using this formula for K = Km , V = V (κ −k ) (hence γ = γm ), c =
V (κ
−k
1+p m Zp
x −k µ ∈ H 1 (Km ,
)), then
exp∗
V (κ −k )
1+p m Zp
x −k µ = δV (κ −k ) ◦ Tr/Km
1
log κ(γ)
Γm
κ −k (x)µγm .
(5.17)
This completes the proof of the theorem for Case 1; the general case follows an argument similar to [2, Section 3.3].
6. Explicit reciprocity law. By constructing the logarithmic map, now we can evaluate the integral of the analytic cohomology class at “negative” power (≤ −h).
Lemma 6.1. Assume V is a crystalline representation and h ∈ Z, h ≥ 1, such that
ϕ=1
(0)
−i GKn
, then δV (κ −k ) ◦ Tm (F ) = 0
Fil−h D(V ) = D(V ), F ∈ Σ = ⊕i∈Z ∪+∞
n=0 FV (Bmax ⊗ V (κ ))
for k ≥ h.
ϕ=1
G
Proof. We can assume F ∈ FV0 (Bmax ⊗ V (−i)) Kn0 for some n0 1 and some i.
0
Since Σ is a finite-dimensional Qp -vector space, F ∈ BdR
⊗ D(V ), hence F = bi ⊗ di .
G
∼
0
For σ ∈ GKn0 , σ (F ) = F , hence σ (bi ) = bi , and bi ∈ (BdR
) Kn0 →
Kn0 ; this shows that
Tm (F ) ∈ D(V ). Hence, δV (κ −k ) ◦ Tm (F ) = 0 for k ≥ 1. This proves the lemma.
Theorem 6.2. Assume V is a crystalline representation of GQp and suppose h ∈ Z
temp (Qp , Ξ ⊗ D(V ))Φ=1,ψ , k ≥ h, then
and µ ∈ Ᏸ
x −k Eh,V (µ) = (−1)h−1 · (1 − ϕ)−1 1 − p −1 ϕ−1
(tx)k
µ ,
Z×
Z×
p
p (k − h)!
(6.1)
−n ax (tx)k
∗
−k
h−1 ϕ
x Eh,V (µ) = (−1)
ε
µ .
expV (κ −k )
pn
p n (k − h)!
a+p n Zp
Zp
exp∗
V (κ −k )
633
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
Proof. First of all, assume h ≥ 1 and Fil−h D(V ) = D(V ). Assume r ∈ N is sufficiently large such that Fr = Ᏺcont (µ/(−tx)r ) exists. Using Theorem 5.3, h + r for h,
V (κ r ) for V , and Eh+r ,V (κ r ) (µ/(−tx)r ) for µ, we get
δV (κ −k ) ◦ Tm L h+r ,V (r ) ◦ Eh+r ,V (r )
µ
r
(−tx)
(−1)h+r −1 (k − h)!
∗
−k−r
r
x
· x Eh,V (µ) .
= expV (κ k )
(k + r )!
1+p m Zp
(6.2)
By Theorem 5.2 and Lemma 6.1, this gives
exp∗
V (κ k )
(k + r )!
µ
δV (κ −k ) ◦ Tm Ᏺcont
x −k Eh,V (µ) = (−1)h+r −1
.
(k − h)!
(−tx)r
1+p m Zp
(6.3)
Case 1 (m = 0). By Proposition 2.7, it gives
(−1)h+r −1
(k + r )!
· (−1)r · (1 − ϕ)−1 1 − p −1 ϕ−1
(k − h)!
= (−1)h−1 (1 − ϕ)−1 1 − p −1 ϕ−1
Z×
p
Z×
p
(tx)k
µ
(k + r )!
(6.4)
(tx)k
µ.
(k − h)!
Case 2 (m = n ≥ 1). By Proposition 2.7, the above formula gives
(−1)h+r −1
(k + r )!
· p −n ·
(k − h)!
p −n Zp
ε(x)
µ
(tx)k+r
= (−1)h−1
(k + r )! (−tx)r
Zp
ε
x
(tx)k
µ.
n
p
(k − h)!
(6.5)
This completes the proof of the theorem for this special case of h, the case for general
h follows from the relation Eh,V = lh−1 Eh−1,V , where lh−1 = hδ1 + δ1 .
Define the pairing
temp Qp , Ξ ⊗ D(V ) Φ=1,ψ × Ᏸ
temp Qp , Ξ ⊗ D V ∗ (1) Φ=1,ψ → Ᏸtemp Z×
[·, ·]D(V ) : Ᏸ
p , Qp
(6.6)
by the formula
Z×
p
f (x) µ, µ D(V ) :=
×
Z×
p ×Zp
f x −1 y µ ⊗ µ .
(6.7)
634
SHAOWEI ZHANG
We must check that the integral really takes values in Qp . From the definition of µ and
µ , we know that [µ, µ ] will take values in Ξ. For any Galois element σ , we have
σ
×
Z×
p ×Zp
f x −1 y µ ⊗ µ =
=
×
Z×
p ×Zp
×
Z×
p ×Zp
f
−1 ψ(σ )y µ ⊗ µ ψ(σ )x
(6.8)
f x −1 y µ ⊗ µ ,
so [µ, µ ] takes values in Qp , and the above pairing is well defined.
Define the paring (µ, µ )V to be
∗
× H 1 Qp , Ᏸtemp Z×
H 1 Qp , Ᏸtemp Z×
p,V
p , V (1)
×
→ H 2 Qp , Ᏸtemp Z×
p × Zp , Qp (1)
2
→ Ᏸtemp Z×
p , H Qp , Qp (1)
→ Ᏸtemp Z×
p , Qp ,
(6.9)
×
where for ξ ∈ H 2 (K, Ᏸtemp (Z×
p × Zp , Qp (1))), we associate a distribution µ as
Z×
p
f (x)µ =
×
Z×
p ×Zp
f xy −1 ξ(x, y).
(6.10)
We need to check that Z×p f (x)µ is a cocycle. This is equivalent to checking that
f (x)σ µ = σ ( Z×p f (x)µ) since
Z×
p
Z×
p
f (x)σ µ = σ
×
Z×
p ×Zp
−1 f κ(σ )x κ(σ )y
ξ =σ
Z×
p
f (x)µ .
(6.11)
temp (Qp , Ξ⊗D(V ∗ (1)))Φ=1,ψ ,
temp (Qp , Ξ⊗D(V ))Φ=1,ψ and µ ∈ Ᏸ
Lemma 6.3. (i) For µ ∈ Ᏸ
Z×
p
x i µ, µ D(V ) =
Z×
p
x −i µ,
Z×
p
xi µ
!
,
(6.12)
D(V )
where the last pairing is defined in [8, Section 2].
1
×
∗
(ii) For ξ ∈ H 1 (Qp , Ᏸtemp (Z×
p , V )) and ξ ∈ H (Qp , Ᏸtemp (Zp , V (1))),
Z×
p
x i ξ, ξ V =
Z×
p
xiξ ∪
Z×
p
x −i ξ ,
(6.13)
where the cup product is given by
H 1 Qp , V κ i ∪ H 1 Qp , V ∗ χκ −i → H 2 Qp , Qp (1) Qp .
(6.14)
(iii) If ξ or ξ restricted to K∞ is zero, then (ξ, ξ ) = 0.
(iv) The pairing [·, ·]D(V ) is sesquilinear for the first variable, linear for the second
variable, that is, for δ ∈ Ᏸ0 (Z×
p , Qp ),
√
δ ∗ µ, µ D(V ) = δ ∗ µ, µ D(V ) ,
µ, δ ∗ µ D(V )
= δ ∗ µ, µ D(V ) .
(6.15)
635
ON EXPLICIT RECIPROCITY LAW OVER FORMAL GROUPS
(v) The pairing (·, ·)V is linear for the first variable, sesquilinear for the second variable,
that is, for δ ∈ Ᏸ0 (Z×
p , Qp ),
δ ∗ ξ, ξ V = δ ∗ ξ, ξ V ,
√
ξ, δ ∗ ξ V = δ ∗ ξ, ξ V .
(6.16)
Proof. The only thing we need to prove is (iii) and this follows from Lemma 2.10.
Hence, (·, ·)V induces a pairing
Γ
Γ
∗
× H 1 K∞ , Ᏸtemp Z×
→ Ᏸtemp Z×
(·, ·)V : H 1 K∞ , Ᏸtemp Z×
p,V
p , V (1)
p , Qp . (6.17)
temp (Qp , Ξ
Theorem 6.4. Assume h ∈ Z and V is a crystalline representation. For µ ∈ Ᏸ
Φ=1,ψ
∗
Φ=1,ψ
temp (Qp , Ξ ⊗ D(V (1)))
and µ ∈ Ᏸ
,
⊗ D(V ))
Eh,V (µ), E1−h,V ∗ (1) µ = (−1)h δ−1 ∗µ, µ .
Proof. For i 0, by Theorem 3.3, we have
x i Eh,v (µ) = (h + i − 1)! expV (κ i ) (1 − ϕ)−1 1 − p −1 ϕ−1
Z×
p
Z×
p
(6.18)
µ
(−tx)i
(6.19)
and by Theorem 6.2, we have
exp∗
V (κ i )
Z×
p
x −i E1−h,V ∗ (1) µ = (−1)h (1 − ϕ)−1 1 − p −1 ϕ−1
Z×
p
(tx)i
.
(i − 1 + h)!
(6.20)
Then the theorem follows from the definition of Bloch-Kato dual exponential map and
the duality between 1 − ϕ and 1 − p −1 ϕ−1 in [2, Lemma IV.4.6].
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Y. Amice and J. Vélu, Distributions p-adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), Société Mathématique de France, Paris, 1975, pp. 119–131 (French).
P. Colmez, Théorie d’Iwasawa des représentations de de Rham d’un corps local [Iwasawa
theory of de Rham representations of a local field], Ann. of Math. (2) 148 (1998), no. 2,
485–571 (French).
E. de Shalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication, Perspectives in
Mathematics, vol. 3, Academic Press, Massachusetts, 1987.
M. Hazewinkel, Formal Groups and Applications, Pure and Applied Mathematics, vol. 78,
Academic Press, New York, 1978.
K. Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR .
I, Arithmetic Algebraic Geometry (Trento, 1991), Lecture Notes in Math., vol. 1553,
Springer-Verlag, Berlin, 1993, pp. 50–163.
B. Perrin-Riou, Théorie d’Iwasawa des représentations p-adiques sur un corps local, Invent.
Math. 115 (1994), no. 1, 81–161 (French).
A. Wiles, Higher explicit reciprocity laws, Ann. of Math. (2) 107 (1978), no. 2, 235–254.
S. Zhang, On a trivial zero problem, Int. J. Math. Math. Sci. 2004 (2004), no. 6, 295–318.
Shaowei Zhang: Department of Mathematics and Statistics, Boston University, MA 02215, USA
E-mail address: swz@math.bu.edu