Tohoku Math. J.
66 (2014), 309–320
A GENERALIZATION OF THE THEORY OF
COLEMAN POWER SERIES
KAZUTO OTA
(Received March 30, 2012, revised May 7, 2013)
Abstract. Shinichi Kobayashi found a generalization of the Coleman power series
theory to formal groups of elliptic curves and applied it to a study of p-adic height pairings.
In this paper, we generalize his theory of Coleman power series to general formal groups.
1. Introduction. The theory of Coleman power series [1] says that every norm com-
patible system is interpolated by a power series. This theory has been generalized in various
ways by Perrin-Riou [8] and others, and they play important roles in Iwasawa theory. On
the other hand, Kobayashi [6] found a generalization to formal groups of elliptic curves. He
studied the interpolations of “admissible norm systems” (cf. [6]), which are not norm com-
patible. Furthermore, he applied it to computations of p-adic height pairings to prove the
p-adic Gross-Zagier formula for elliptic curves at supersingular primes. We expect a general-
ization of his theory will play an important role in proving more general p-adic Gross-Zagier
formulas. In this paper, we generalize his theory to general formal groups over unramified
rings.
Let ζ
p
n
be a primitive p
n
-th root of unity such that ζ
p
p
n+1
= ζ
p
n
, m
n
the maximal ideal
of Z
p
[ζ
p
n
] and U
n
= 1 + m
n
. Then the Coleman power series theory [1] says that every
(u
n
)
n
∈ lim
←−
U
n
, where the limit is taken with respect to the norm maps, is interpolated by a
power series. Namely, there exists a unique power series f(T) ∈ Z
p
[[T ]] such that
f(ζ
p
n
− 1) = u
n
for all n ≥ 1. In terms of formal groups, every element of lim
←−
ˆ
G
m
(m
n
) can be interpolated.
Let E be an elliptic curve over Q
p
with good supersingular reduction and
ˆ
E the formal
group over Z
p
(of height two) associated to E. Then it is known that lim
←−
ˆ
E(m
n
) is trivial,
where the projective limit is taken with respect to the trace maps Tr
m/n
:
ˆ
E(m
m
) →
ˆ
E(m
n
).
However, systems (c
n
)
n
∈
∞
n=1
ˆ
E(m
n
) satisfying the equations
(1.1) Tr
n+2/n
c
n+2
− a
p
Tr
n+1/n
c
n+1
+ pc
n
= 0
in
ˆ
E(m
n
) for all n ≥ 1, where a
p
= p + 1 − #E(F
p
), are also important (cf. Kobayashi
[5], Perrin-Riou [7]). For example, some systems satisfying (1.1) are constructed from certain
integral power series and applied to a construction of p-adic height pairings in [7]. In [6],
2010 Mathematics Subject Classification. Primary 14L05; Secondary 11S31, 11E95, 13F25.
Key words and phrases. Coleman power series, formal group, Dieudonné module.
310 K. OTA
Kobayashi studied the interpolations of systems (c
n
)
n
∈
∞
n=0
ˆ
E(m
n
) satisfying (1.1) and
found a generalization of the Coleman power series theory. By his theory, a system of Heegner
points, which satisfies (1.1), can be interpolated by a power series. Thus, he applied the
system to computations of the p-adic height of Heegner points. This computation played an
important role in his proof of the p-adic Gross-Zagier formula, which relates the p-adic height
of a Heegner point to the first derivative of a p-adic L-function of E.
Here, we explain our main theorem for odd primes (see Theorem 4.6 for the general
case). Let G be a d-dimensional, commutative formal group over Z
p
of finite height h.Wefix
an isomorphism G
∼
=
Spf(Z
p
[[X
1
,X
2
,...,X
d
]]) between formal schemes over Z
p
. Then,
we can identify G(Z
p
[[T ]]) with the subset (p, T )
⊕d
of Z
p
[[T ]]
⊕d
and G(m
n
) with m
⊕d
n
not as Z
p
-modules but as sets (cf. Section 2). We denote by M the Dieudonné module of G,
which is a free Z
p
-module of rank h. Then, M has operators Frobenius F and Verschiebung
V .Letdet(t − V) = t
h
+ b
h−1
t
h−1
+ ··· +b
0
∈ Z
p
[t], the logarithm log
G
of G and
tr
m/n
: Q
p
(ζ
p
m
) → Q
p
(ζ
p
n
) the usual trace map. Our main theorem is the following:
T
HEOREM 1.1 (cf. Theorem 4.6). Fo r e ach sy stem (c
n
)
n
∈
∞
n=1
G(m
n
) satisfying
(1.2) tr
n+h/n
(c
n+h
) + b
h−1
tr
n+h−1/n
(c
n+h−1
) +···+b
0
(c
n
) = 0
in Q
p
(ζ
p
n
) for all n ≥ 1, the following conditions are equivalent:
(a) There exists a power series f(T) ∈ G(Z
p
[[T ]]) such that
f(ζ
p
n
− 1) = c
n
for all n ≥ 1;
(b) c
p
n+1
≡ c
n
mod pZ
p
[ζ
p
n+1
]
⊕d
for all n ≥ 1.
R
EMARK 1.2. (1) We prove our main theorem for formal groups over unramified
rings.
(2) In [7], Perrin-Riou constructed systems satisfying (1.2) similarly as in the case G =
ˆ
E. See Section 3 for details.
(3) In the case G =
ˆ
E,wehavedet(t − V) = t
2
− a
p
t + p and
n
ˆ
E(m
n
)
tor
= 0.
Therefore, we see that (1.2) is equivalent to (1.1) and that our main theorem coincides
with Kobayashi’s theorem [6, Theorem 3.15]. In this paper, we modify his proof. The
key is to use Knospe [4, Proposition 2.1].
Acknowledgment. This paper is based on Master’s thesis of the author. He would like to express
his sincere gratitude to his adviser Professor Shinichi Kobayashi for suggesting the problem to him and
for giving much advice. The author would also like to thank the referee for giving helpful comments.
2. Preliminaries. In this section, we fix notation and recall Dieudonné modules of
formal groups over unramified rings.
Let p be a prime number and k a perfect field of characteristic p. We denote by W the
ring W(k) of Witt vectors with coefficients in k and by K its fractional field with absolute
A GENERALIZATION OF THE THEORY OF COLEMAN POWER SERIES 311
Frobenius σ . We fix a prime element π of Z
p
and ϕ(T ) ∈ Z
p
[[T ]] such that
ϕ(T ) ≡ πT mod T
2
Z
p
[[T ]] ,ϕ(T)≡ T
p
mod πZ
p
[[T ]] .
As is well known, ϕ induces a Lubin-Tate formal group F over Z
p
whose multiplication-by-π
map [π]
F
is given by ϕ(T ).Forn ≥ 0, we denote by F[π
n
] the kernel of the endomorphism
[π
n
]
F
of F and put K
n
= K(F [π
n
]). We denote by m
n
the maximal ideal of the valuation
ring O
K
n
of K
n
. We fix a system (π
n
) ∈
∞
n=1
F[π
n
]\F[π
n−1
] such that [π]
F
(π
n+1
) = π
n
.
Let G be a d-dimensional, commutative formal group over W of finite height h.Wealso
fix an isomorphism G
∼
=
Spf(W [[X]]) between formal schemes over W ,whereW [[X]] =
W [[X
1
,...,X
d
]], and denote by X ⊕ Y ∈ W [[X, Y ]]
⊕d
the d-dimensional formal group
law. Then, for a commutative W -algebra A which is complete under the topology induced
by an ideal I of A containing p, we can identify G(I ) with the set of W -homomorphisms
f : W [[X]] → A such that f(X
i
) ∈ I for 1 ≤ i ≤ d. Thus, we can identify G(I ) with I
⊕d
as a set by f → (f (X
i
))
i
. In other words, we identify G(I) with the Z
p
-module I
⊕d
whose
addition is induced by the formal group law X ⊕ Y .
In order to define Dieudonné modules, we recall the formal differential module of
W [[X]] over W . We denote by D the space of derivations of W [[X]] over W and by D
∗
the dual W [[X]]-module of D. Then, we have D
∗
=
1≤i≤d
W [[X]]dX
i
.Herewedenote
by dg the map D → W[[X]] defined by D → Dg for g ∈ W [[X]]. We say that ω ∈ D
∗
is an
exact form if ω = df =
1≤i≤d
(∂f/∂X
i
)dX
i
for some f ∈ K[[X]]. For an exact form ω,
we denote by F
ω
∈ K[[X]] a unique power series such that ω = dF
ω
and F
ω
(0) = 0. We put
Z ={ω ∈ D
∗
; ω is exact,F
ω
(X ⊕ Y)− F
ω
(X) − F
ω
(Y ) ∈ pW[[X, Y ]]} ,
B ={df ∈ D
∗
; f ∈ pW[[X]]} ,
L ={ω ∈ D
∗
; F
ω
(X ⊕ Y) = F
ω
(X) + F
ω
(Y )} .
We define the Dieudonné module M of G by M := Z/B. Since it depends only on the special
fiber
G of G, we also call it the Dieudonné module of G. It is known that M is a free W -module
of rank h.LetW [F,V ] be the Dieudonné ring. Namely, F and V satisfy the relations
FV = VF = p, Fx = x
σ
F, Vx= x
σ
−1
V
for x ∈ W . We remark that W [F,V ] acts on M as follows: For ω ∈ Z, we put F
ω
(X) =
α
a
α
X
α
.Hereα ranges over all (i
1
,...,i
d
) ∈ Z
⊕d
≥0
\{(0,...,0)},andX
α
denotes the
product X
i
1
1
···X
i
d
d
for α = (i
1
,...,i
d
).WemakeF and V act on M by
F(ω+ B) = d
F
σ
ω
X
p
+ B =
1≤i≤d
∂
∂X
i
F
σ
ω
X
p
dX
i
+ B,
V(ω+ B) = d
α
pa
σ
−1
pα
X
α
+ B =
1≤i≤d
∂
∂X
i
α
pa
σ
−1
pα
X
α
dX
i
+ B.
Here pα denotes (pi
1
,...,pi
d
) for α = (i
1
,...,i
d
). Thus, M is a left W [F,V ]-module. We
call F Frobenius operator and V Verschiebung operator, respectively, on M.
312 K. OTA
According to Honda [3, Proposition 1.3 and Lemma 1.4], we see that L ⊂ Z.Byabuse
of notation, we also denote by L the image of L in M. Furthermore, one can show that M is
aleftW [F,V ]-module generated by the W -submodule L of M (cf. [3, Lemma 4.3]).
We define a W [[T ]]-submodule P of K[[T ]] and its quotient
P by
P =
f(T)=
∞
n=0
a
n
T
n
∈ K[[T ]] ; na
n
∈ W for n ≥ 0,f(0) ∈ pW
,
P = P/pW [[T ]] .
We remark that ϕ acts on P by ϕ(f )(T ) = f
σ
(ϕ(T )). Then, by [3, Lemma 2.1], we see that
ϕ induces Frobenius operator F of
P, which is defined by
F
n
a
n
T
n
+ pW[[T ]]
=
n
a
σ
n
T
pn
+ pW[[T ]] .
By an argument similar to that in the proof of [7, Lemme 1.2], one can show that there exists
a unique σ
−1
-linear operator ψ of P such that ψ ◦ ϕ = p and
ϕ ◦ ψ(f )(T ) =
η∈F [π ]
f(T ⊕
F
η) ,
where ⊕
F
denotes the addition in F.
P
ROPOSITION 2.1. For f(T)∈ P and n ≥ 1, we have
(a) ϕ(f )(π
n+1
) = f
σ
(π
n
);
(b) ψ(f
σ
)(π
n
) = tr
n+1/n
(f (π
n+1
)).
Here,tr
m/n
is the usual trace map from K
m
to K
n
for m ≥ n.
P
ROOF. For example, see [7, Lemme 1.4]. 2
Moreover, ψ induces Verschiebung operator V of
P, which is defined by
V
n
a
n
T
n
+ pW[[T ]]
=
n
pa
σ
−1
pn
T
n
+ pW[[T ]] .
Thus,
P is a left W [F,V ]-module. According to Fontaine [2], we have a canonical Z
p
-linear
isomorphism
G(k[[T ]])
∼
=
Hom
W [F,V]
(M, P).
In order to construct this isomorphism, we fix some notation.
We denote by ω
1
,...,ω
d
the canonical invariant differentials of G, which is also called
the canonical base of L in [3]. Here, we recall that we have identified G with the formal
group law X ⊕ Y over W through the fixed isomorphism G
∼
=
Spf(W [[X]]). We denote by
log
G
= (F
ω
i
(X)) ∈ K[[X]]
⊕d
the logarithm of G, which is also called the transformer in [3].
We put exp
G
(X) = log
−1
G
∈ K[[X]]
⊕d
. For a local ring A, we denote by m
A
the maximal
ideal.
A GENERALIZATION OF THE THEORY OF COLEMAN POWER SERIES 313
First, we construct G(k[[T ]]) → Hom
W [F,V ]
(M, P).For(f
i
(T )) ∈ G(k[[T ]]) =
m
⊕d
k[[T ]]
= Tk[[T ]]
⊕d
, we take any lift (f
i
(T )) in G(W [[T ]]) = m
⊕d
W [[T ]]
= (p, T )
⊕d
such
that f
i
(0) = 0. We define an element of Hom
W [F,V]
(M, P) by
ω → F
ω
(f
1
(T ), . . . , f
d
(T )) mod pW[[T ]] .
Note that this morphism is independent of the choice of (f
i
(T )) by [3, Lemma 2.1]. Thus, we
have a Z
p
-morphism
G(k[[T ]]) → Hom
W [F,V]
(M, P).
For v ∈ Hom
W [F,V]
(M, P), we take any lift F
i
(T ) of v(ω
i
) such that F
i
(0) = 0. We define
an element
f(T)of G(k[[T ]]) by
f(T)= exp
G
(F
1
(T),...,F
d
(T )) mod G(pW [[T ]]).
Note that the power series exp
G
(F
1
(T ), . . . , F
d
(T )) is an element of G(W [[T ]]) by
[3, Lemma 2.4]. By [3, Proposition 3.3 and Lemma 4.1], we also see that this is indepen-
dent of the choice of (F
i
(T )). Thus, we have the inverse morphism.
In the following, we often identify G(k[[T ]]) with Hom
W [F,V ]
(M, P) by this isomor-
phism.
3. Construction of systems after Perrin-Riou. In this section, we shall construct
systems satisfying a relation similar to (1.2) in the same way as in [7]. First, we recall a
lift Hom
W [F,V ]
(M, P) → Hom
W
(M, P) constructed in [7]. It plays important roles in this
paper. We keep the same notation as in the previous section.
P
ROPOSITION 3.1. For ea ch x ∈ Hom
W [F,V]
(M, P), there exists a unique lift ˆx ∈
Hom
W
(M, P) of x such that
(3.3) ψ(ϕ(ˆx(m)) −ˆx(Fm)) = 0
for all m ∈ M, or equivalently, ψ(ˆx(m)) =ˆx(V m) for all m ∈ M.
P
ROOF. This is [7, Proposition 3.2]. In [7], only the case where ϕ(T ) = (1 + T)
p
− 1
is treated. However, we can also prove this proposition for our ϕ by the same argument as in
[7]. 2
We construct systems satisfying a relation similar to (1.2) by using the lift.
Let x be an element of G(k[[T ]])
∼
=
Hom
W [F,V]
(M, P). In the case p>2, the power
series exp
G
◦( ˆx(ω
i
)) is an element of G(W [[T ]]) = m
⊕d
W [[T ]]
by [3, Lemma 2.4]. Thus, we
have a lift
ι : G(k[[T ]]) → G(W [[T ]]), x → exp
G
◦( ˆx(ω
i
)) .
Hence, we have
G(W [[T ]]) = ι
(
G(k[[T ]])
)
⊕ Hom
W
(L, pW[[T ]])
(cf. [7, Théorème 4.1]). For f(T) = ι(x) = exp
G
◦( ˆx(ω
i
)) ∈ ι
(
G(k[[T ]])
)
G(W [[T ]]),
we put
c
n
= f
σ
−n
(π
n
) =
exp
G
◦( ˆx(ω
i
))
σ
−n
(π
n
) ∈ G
σ
−n
(m
n
).