iwasawa theory for elliptic curves at supersingular primes

DOI: 10.1007/s00222-002-0265-4Invent. math. 152, 1–36 (2003)

Iwasawa theory for elliptic curves at supersingularprimes

Shin-ichi Kobayashi

Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo,153-8914 Japan (e-mail: [email protected])

Oblatum 17-VI-2002 & 2-IX-2002Published online: 18 December 2002 – Springer-Verlag 2002

Abstract. We give a new formulation in Iwasawa theory for elliptic curvesat good supersingular primes. This formulation is similar to Mazur’s at goodordinary primes. Namely, we define a new Selmer group, and show that it isof Λ-cotorsion. Then we formulate the Iwasawa main conjecture as that thecharacteristic ideal is generated by Pollack’s p-adic L-function. We showthat this main conjecture is equivalent to Kato’s and Perrin-Riou’s mainconjectures. We also prove an inequality in the main conjecture by usingKato’s Euler system. In terms of the λ- and the µ-invariants of our Selmergroup, we specify the numbers λ and µ in the asymptotic formula for theorder of the Tate-Shafarevich group by Kurihara and Perrin-Riou.

1. Introduction

Let p be an odd prime. Let F = Q and F∞/F the cyclotomic Zp-extensionwith n-th layer Fn . We denote Gal(F∞/F) by Γ . We identify Zp[[Γ ]] withthe ring of power series Zp[[X]].

In 1970’s, Mazur formulated the Iwasawa theory for elliptic curves atgood ordinary primes. In his formulation, the main conjecture states thatthe Selmer group over F∞ is Zp[[Γ ]]-cotorsion and the characteristic idealis generated by the Mazur-Swinnerton-Dyer p-adic L-function. On thecontrary, at supersingular primes, the Selmer group is no longer Zp[[Γ ]]-cotorsion, and the p-adic L-function is not an element of Zp[[Γ ]] ⊗ Qp.Hence, at supersingular primes, in spite of the significant works by Katoand Perrin-Riou, we had no formulation analogue to Mazur’s.

The aim of this paper is to consider Iwasawa theory for elliptic curvesat good supersingular primes in a way similar with Mazur’s at good ordi-nary primes. More precisely, let E be an elliptic curve over Q with good

2 S. Kobayashi

supersingular reduction at p. Suppose ap = 0. This condition is automati-cally satisfied for a supersingular p greater than 3, however, the conditionis crucial for us. Our methods will probably have to be modified quite sig-nificantly to treat the excluded cases. Then we define new Selmer groupsSel±(E/Fn) by putting moderately stronger local conditions at p on thedefinition of the usual (p-primary) Selmer group Sel(E/Fn).

Definition 1.1. We define the n-th even (odd) Selmer group by

Sel±(E/Fn) := Ker

(Sel(E/Fn) −→ H1

(Fn,p, E[p∞])

E±(Fn,p) ⊗Qp/Zp

)

where Fn,p is the completion of Fn at the place over p and

E+(Fn,p) := P ∈ E(Fn,p) | Trn/m+1 P ∈ E(Fm,p) for even m (0 ≤ m < n)E−(Fn,p) := P ∈ E(Fn,p) | Trn/m+1 P ∈ E(Fm,p) for odd m (0 ≤ m < n)where Trn/m+1 : E(Fn,p) → E(Fm+1,p) is the trace map.

We remark that the group E±(Fn,p) essentially appeared in Knospe [8]and Perrin-Riou [13]. Let Sel±(E/F∞) := lim−→ n Sel±(E/Fn). Then Zp[[Γ ]]acts naturally on the Pontryagin dual of Sel±(E/F∞).

Theorem 1.2. The Pontryagin dual of the even (odd) Selmer groupSel±(E/F∞)∨ is a finitely generated torsion Zp[[Γ ]]-module.

We formulate the main conjecture as follows.

Conjecture (Main Conjecture). Suppose ap = 0 (e.g. p ≥ 5). Then thecharacteristic ideal of the Pontryagin dual of Sel±(E/F∞) is generated byPollack’s p-adic L-function L±

p (E, X):

Char(

Sel±(E/F∞)∨ ) = (L±

p (E, X)).

Pollack’s p-adic L-function [18] is a p-adic function which interpolatesspecial values of the Hasse-Weil L-function similarly to the usual p-adicL-function. However, this p-adic L-function lives in Zp[[Γ ]] (cf. Sect. 3).We will see that our main conjecture is equivalent to Kato’s and Perrin-Riou’s main conjectures (cf. [6], [7], [14]).

By using Kato’s Euler system [7], we show that

Theorem 1.3. Suppose E/Q does not have complex multiplication. Thenfor almost all prime p, we have

Char(

Sel±(E/F∞)∨ ) ⊇ (L±

p (E, X)).

Iwasawa theory for elliptic curves at supersingular primes 3

For the more precise statement of the theorem, see Sect. 4. Recently,Pollack and Rubin informed the author that the above main conjecture istrue for CM elliptic curves ([19]).

We also obtain an asymptotic formula for the p-adic order of theTate-Shafarevich group X(E/Fn) using the λ- and the µ-invariants ofSel±(E/F∞)∨.

Theorem 1.4. Assume that X(E/Fn) is finite for all n. Let en =ordp X(E/Fn). Let λ± and µ± be the λ- and the µ-invariantsof Sel±(E/F∞)∨. Then for sufficiently large n, we have

en =[ n−2

2 ]∑m=0

pn−1−2m −[n

2

]+

[n

2

]λ+ +

[n + 1

2

]λ− − n r∞

+[ n

2 ]∑m=1

φ(

p2m)µ+ +

[ n+12 ]∑

m=1

φ(

p2m−1)µ− + ν

where r∞ = rank E(F∞), φ(pk) = pk−pk−1 and ν is an integer independentof n. Here [x] denotes the largest integer not greater than x.

Similar formulas have been obtained by Kurihara and Perrin-Riou [17]with unspecified rational numbers λ and µ. In the above formula, these λand µ are specified as λ± and µ± as in Pollack’s analytic counterpart.(The sign in the notation of our λ- and µ-invariants are opposite to that ofPollack’s.)

The key to our theory is a construction of the even (odd) Colemanmaps. These are Λ-valued and send Kato’s zeta element to Pollack’s p-adicL-function. The group E±(Fn,p) is defined so as to be equal to the exactannihilator with respect to the Tate pairing of the kernel of the n-th even(odd) Coleman map. Once these maps are constructed, the main theoremsare proved in the same way as in the good ordinary case (cf. Sect. 7). Weconstruct these Coleman maps by modifying Kurihara’s Pn-pairing in [9]with Pollack’s idea on the p-adic L-function. We outline the construction.

Let kn = Qp(ζpn+1). We denote Λn = Zp[Gn] and Λ = Zp[[G∞]] whereGn = Gal(kn/Q) and G∞ = lim←− Gn . Let T be the p-adic Tate module of E

and H1Iw(T ) = lim←− n H1(kn, T ). We construct the even (odd) Coleman map

Col± : H1Iw(T ) → Λ as follows.

Using Honda’s theory of formal groups, we construct a canonical system(cn)n, cn ∈ E(kn) ⊗ Zp ⊆ H1(kn, T ) satisfying

Trn+1/n (cn+1) − ap cn + cn−1 = 0. (1.1)

(Perrin-Riou [13] also constructed such a system using the theory of normfields by Fontaine and Wintenberger.) If ap = 0, we have Trn+1/n (cn+1) =

4 S. Kobayashi

−cn−1. Hence, according to the parity of n, the elements (cn)n give kindsof norm compatible systems. Then we consider a morphism

Pn : H1(kn, T ) −→ Λn, z −→ (−1)[ n+22 ] ∑

σ∈Gn

(cσ

n , z)

nσ

where ( , )n is the pairing induced by the cup product. We show thatthe image of Pn is contained in the ideal generated by a certain productof cyclotomic polynomials. Eliminating the product and using the normrelation of (cn)n, we can take the limit of Pn for n of the same parity. Thenwe get the even (odd) Coleman map

Col± : H1Iw(T ) −→ Λ.

Acknowledgements. This paper is based on the author’s thesis. It is a pleasure to thank histhesis advisor, Takeshi Saito, who read the manuscript carefully and pointed out mathe-matical mistakes. He also thank Ken-ichi Bannai and Seidai Yasuda for discussions and forreading the manuscript carefully. He would like to thank Robert Pollack for showing him thepreprint [18]. He is grateful to Kazuya Kato for showing him the preprint [7]. He expresseshis sincere gratitude to Masato Kurihara. The author learned a lot of things through themanuscript of the paper [9], including the Pn-pairing and the idea to use Honda’s theory.

2. The even (odd) Selmer group

Let p be an odd prime number. We fix a generator (ζpn ) of Zp(1), namely,for each n ≥ 0, ζpn is a primitive pn-th root of unity such that ζ

ppn+1 = ζpn .

We denote Kn = Q(ζpn+1), K−1 = Q and K∞ = ∪n Kn .Let E be an elliptic curve over Q with good reduction at p. We assume

that ap = 0. When p ≥ 5, the condition ap = 0 is equivalent to the conditionthat E has supersingular reduction at p.

The (p-primary) Selmer group is defined by

Sel(E/Kn) := Ker

(H1(Kn, E[p∞]) res−→

∏v

H1(Kn,v, E[p∞])

E(Kn,v) ⊗Qp/Zp

)

and Sel(E/K∞) := lim−→ n Sel(E/Kn), where v ranges over all places ofKn and Kn,v is the completion of K at v. We regard E(Kn,v) ⊗ Qp/Zp asa subgroup of H1(Kn,v, E[p∞]) by the Kummer map.

We define subgroups E±(Kn,v) of E(Kn,v) by

E+(Kn,v) = P ∈ E(Kn,v) | Trn/m+1 P ∈ E(Km,v) for even m (0 ≤ m < n),E−(Kn,v) = P ∈ E(Kn,v) | Trn/m+1 P ∈ E(Km,v) for odd m (−1 ≤ m < n),where Trn/m+1 : E(Kn,v) → E(Km+1,v) is the trace map.


Definition 2.1. The even (odd) Selmer group is defined by

Sel±(E/Kn) := Ker

(H1(Kn, E[p∞]) −→

∏v

H1(Kn,v, E[p∞])E±(Kn,v) ⊗Qp/Zp

)

and Sel±(E/K∞) := lim−→ n Sel±(E/Kn).

Following Kurihara [9], we also define the zero Selmer group by

Sel0(E/Kn) := Ker

(H1

(Kn, E[p∞]) −→

∏v

H1(Kn,v, E[p∞])

)

and Sel0(E/K∞) := lim−→ n Sel0(E/Kn).

We denote the Pontryagin dual of Selmer groups by “X”. For ex-ample, X(E/Kn) is the dual of Sel(E/Kn) and X±(E/K∞) is the dualof Sel±(E/K∞).

Let Gn = Gal(Kn/Q) and G∞ = lim←− Gn. Then Λn = Zp[Gn] actsnaturally on X±(E/Kn) and Λ = Zp[[G∞]] on X±(E/K∞).

Theorem 2.2. The Pontryagin dual of the even (odd) Selmer groupX±(E/K∞) is a finitely generated torsion Λ-module.

Proof. This theorem is proved in Sect. 7 (cf. Theorem 7.3).

3. Pollack’s p-adic L-function

We first recall the classical p-adic L-function of an elliptic curve E/Q ata good prime p. Let α be a root of the p-Euler factor X2 − ap X + p suchthat ordp(α) < 1. Let L = Qp(α). For an integer h ≥ 1, we let

Hh,L =∑

n≥0

an Xn ∈ L[[X]] | limn→∞

|an|p

nh= 0

where | |p denotes the multiplicative valuation of L normalized by |p|p = 1p .

Then H∞,L = ∪h Hh,L is a ring.

Let κ : G∞∼→ Z×

p be the cyclotomic character. Then we have thedecomposition

G∞ = ∆ × Γ, where ∆ ∼= Z/(p − 1)Z and Γ ∼= Zp.

We fix a topological generator γ ∈ Γ . Then we identify Zp[[Γ ]] withZp[[X]] and Λ with Zp[∆][[X]] by identifying γ with 1 + X.

6 S. Kobayashi

Theorem 3.1 (Amice-Velu [1], Vishik [23]). Let η : ∆ → Z×p be a char-

acter with sign δ = η(−1). We regard η as a character of G∞ by takingη(γ) = 1. Let Ω±

E be the real and imaginary Neron periods. Then thereexists a unique element

Lp(E, η, α, X) ∈ H1,L

having the following interpolation properties. (If η = 1, we write Lp(E, η,α, X) simply by Lp(E, α, X).)

Let χ : Gn → µp∞ be a character and let χ(γ) = ζ and ψ = χη.We denote the Hasse-Weil L-function with the Dirichlet character ψ byL(E, ψ, s).

Then, if the conductor of ψ is pn+1 = 1, we have

Lp(E, η, α, ζ − 1) = pn+1

τ(ψ) αn+1

L(E, ψ, 1

)Ωδ

E

(3.2)

where ψ = ψ−1 and τ(ψ) is the Gauss sum∑

a∈Gnψ(a) ζa

pn+1 . Here, we

identify Gn with (Z/pn+1Z)×.If ψ = 1, we have

Lp(E, α, 0) = (1 − α−1)2 L(E, 1)

ΩE. (3.3)

We call the power series Lp(E, η, α, X) the p-adic L-function. (Usually,the function Lp(E, η, α, κ(γ)s−1 −1) of variable s ∈ Zp is called the p-adicL-function.) The p-adic L-function Lp(E, η, α, X) is in Zp[[X]]⊗ L if andonly if ap is a unit.

Now we assume that ap = 0. Then the both roots α, α of X2 − ap X + p= X2 + p satisfy ordp(α) < 1, and we have two p-adic L-functionsLp(E, η, α, X) and Lp(E, η, α, X). Then by the interpolation property(3.2), we have

Lp(E, η, α, X) ± Lp(E, η, α, X) = 0

at X = ζ−1 where ζ is a primitive pn -th root of unity such that (−1)n = ±1.Pollack compared the sum and the difference of two p-adic L functions

with the following two functions log+p (1 + X) and log−

p (1 + X), which alsohave trivial zeros of the form ζ − 1.

log+p (1+X) = 1

p

∞∏m=1

Φ2m(1 + X)

p, log−

p (1+X) = 1

p

∞∏m=1

Φ2m−1(1 + X)

p

where Φn(X) = ∑p−1i=0 X pn−1i is the pn-th cyclotomic polynomial. We

remark that logp(1 + X) = p2 X log+p (1 + X) log−

p (1 + X).


Theorem 3.2 (Pollack). There exist power series L±p (E, η, X) in Zp[[X]]

satisfying

Lp(E, α, η, X) = L−p (E, η, X) log+

p (1 + X) + L+p (E, η, X) log−

p (1 + X) α,

Lp(E, α, η, X) = L−p (E, η, X) log+

p (1 + X) + L+p (E, η, X) log−

p (1 + X) α.

Proof. See Pollack [18], Theorem 6.1 and 6.6.

We call L±p (E, η, X) Pollack’s p-adic L-function. If η is trivial, we sim-

ply write L±p (E, X) for L±

p (E, η, X). We remark that our sign of Pollack’sp-adic L-function is opposite to that in [18].

Pollack’s p-adic L-function is characterized by the following interpola-tion properties.

Let χ : Gn → µp∞ be a non-trivial character of conductor pn+1 and letψ = ηχ and χ(γ) = ζ . For an even n, we have

L+p (E, η, ζ − 1) = (−1)

n+22

pn+1

τ(ψ)

∏k

Φk(ζ)−1 L

(E, ψ, 1

)Ωδ

E

(3.4)

where the product is take over odd k (1 ≤ k ≤ n).For an odd n, we have

L−p (E, η, ζ − 1) = (−1)

n+12

pn+1

τ(ψ)

∏k

Φk(ζ)−1 L

(E, ψ, 1

)Ωδ

E

(3.5)

where the product is take over even k (1 ≤ k ≤ n).We also have

L+p (E, 0) = 2

L(E, 1)

ΩE, L−

p (E, 0) = (p − 1)L(E, 1)

ΩE(3.6)

and for η = 1

L+p (E, η, 0) = − p

τ(η)

L(E, η, 1)

ΩδE

, L−p (E, η, 0) = 0. (3.7)

In Sect. 8, we reconstruct Pollack’s p-adic L-function, without men-tioning the usual p-adic L-functions, as the power series satisfying theabove interpolation properties. They are given as the images of Kato’s zetaelements by the even (odd) Coleman maps (Theorem 6.3).

8 S. Kobayashi

4. Main conjecture

Now we formulate the main conjecture when ap = 0.Let η : ∆ → Z×

p be a character. For a Zp[∆]-module M, let Mη denotethe η-component of M. If we denote εη = 1

∆

∑τ∈∆ η−1(τ) τ, Mη is given

by εηM. If η is trivial, we replace the superscript η by ∆. For a finitelygenerated torsion Zp[[Γ ]]-module N, we denote its characteristic ideal byChar(N).

By Theorem 2.2, X±(E/K∞)η is a torsion Zp[[Γ ]]-module.

Conjecture (Even main conjecture). For every η, we have

Char(X+(E/K∞)η

) = (L+

p (E, η, X)).

Conjecture (Odd main conjecture). If η is trivial, we have

Char(X−(E/K∞)∆

) = (L−

p (E, X)).

If η is non-trivial, we have

Char(X−(E/K∞)η

) =( 1

XL−

p (E, η, X)).

(By (3.7), the p-adic L-function L−p (E, η, X) is divisible by X.)

Theorem 4.1. There exists an integer n ≥ 0 such that

Char(X+(E/K∞)η

) ⊇ (pn L+

p (E, η, X)),

Char(X−(E/K∞)∆

) ⊇ (pn L−

p (E, X)),

Char(X−(E/K∞)η

) ⊇( pn

XL−

p (E, η, X))

for η = 1.

If the p-adic representation Gal(Q/Q) → GLZp(T ) is surjective, then wecan take n = 0. Here T = Tp E is the p-adic Tate module of E.

Proof. This theorem is proved in Sect. 7.

5. Kato’s main conjecture

We recall Kato’s main conjecture and his results for the Euler system. Inthis section, we do not need to make any assumption on the reduction of Eat p. However, for simplicity, we assume that E has good reduction at p.

Let T be the p-adic Tate module of E and V = T ⊗ Qp. Let j be thenatural morphism Spec Kn → Spec OKn [1/p]. We denote

Hq(T ) = lim←− n Hq(Spec OKn [1/p], j∗T )

where Hq(Spec OKn [1/p], j∗T ) is the etale cohomology group. We alsowrite Hq(V ) = Hq(T ) ⊗Q.


Theorem 5.1 (Kato [7]). i) H2(T ) is a finitely generated torsion Λ-module.

ii) H1(T ) is a torsion free Λ-module, and H1(V ) is a free Λ ⊗Q-moduleof rank 1.

iii) If T/pT is irreducible as a two dimensional representation of Gal(Q/Q)over Fp, then H1(T ) is a free Λ-module of rank 1.

Proof. See Kato [7], Theorem 12.4. Theorem 5.2 (Kato [7]). i) There exist two systems of elements,

z± = (z±

n

)n

∈ H1(V ), z±n ∈ H1(OKn [1/p], T ) ⊗Q

such that for a character ψ of Gn of conductor pn+1, we have

∑σ∈Gn

ψ(σ) exp∗ωE

(σ(z±

n

)) = δ(1 − ap ψ(p) p−1 + ψ(p)2 p−1) L(E, ψ, 1)

Ω±E

where exp∗ωE

is the dual exponential map (cf. Sect. 8.7) and δ = 1 if ψ(−1)

is equal to the sign of z±n , otherwise, δ = 0.

ii) Let Z(V ) be the Λ ⊗Q-submodule of H1(V ) generated by z+ and z−.Then H1(V )/Z(V ) is a torsion Λ ⊗Q-module.

iii) Let η : ∆ → Z×p be a character. Then

Char(

H2(V )η) ⊇ Char

(H1(V )η/Z(V )η

).

iv) Let Z(T ) be the Λ-submodule of H1(V ) generated by z+ and z−.Suppose that T/pT is irreducible as a two dimensional representation ofGal(Q/Q) over Fp. Then

Z(T ) ⊆ H1(T ) in H1(V ).

v) Suppose that the homomorphism

Gal(Q/Q) −→ GLZp(T )

is surjective. Then,

Char(

H2(T )η) ⊇ Char

(H1(T )η/Z(T )η

).

Proof. See Kato [7], Theorem 12.5. For iv), we need Theorem 12.6 ofKato [7]. See also the argument in Sect. 13.14 of [7]. We also use the factthat the Neron period differs from the canonical period by a p-adic unit (cf.Greenberg-Vatsal [3]).

10 S. Kobayashi

Remark 5.3. i) When E has supersingular reduction at p, the condition inTheorem 5.1 iii) and 5.2 iv) is satisfied.ii) The signs in Theorem 5.2 i) are related with the action of the complexconjugate on the homology group of E. They are not connected with thesigns of even (odd) Selmer groups.iii) If E has no complex multiplication, then by the result of Serre [22], thecondition in Theorem 5.2 v) is satisfied for almost all p.

In [6], Kato formulated the Iwasawa main conjecture for motives. Asa special case, his main conjecture can be read as follows (cf. Conjec-ture 12.10 in [7]).

Conjecture (Kato’s main conjecture). Let η : ∆ → Z×p be a character.

Then, Z(T )η ⊆ H1(T )η and

Char(

H2(T )η) = Char

(H1(T )η/Z(T )η

).

We may see that this is a generalization of the classical Iwasawa mainconjecture relating the ideal class groups and the groups of the global unitsmodulo the cyclotomic units.

Kato’s main conjecture is known to be equivalent to Perrin-Riou’s mainconjecture [14], [16]. We see that the even and the odd main conjectures arealso equivalent to Kato’s main conjecture (Theorem 7.4).

6. The even and the odd Coleman maps

We review the even (odd) Coleman maps constructed in Sect. 8.Let kn denote Qp(ζpn+1).

Definition 6.1. We define H1±(kn, T ) as the exact annihilator with respectto the Tate pairing

H1(kn, V/T ) × H1(kn, T ) → Qp/Zp

of the subgroup E±(kn) ⊗Qp/Zp ⊆ H1(kn, V/T ).

We let H1Iw(T ) = lim←− n H1(kn, T ) and H1

Iw,±(T ) = lim←− n H1±(kn, T ).

Let Φn(X) = ∑p−1i=0 X pn−1i be the pn-th cyclotomic polynomial. We

define ω±0 (X) = X and

ω+n (X) = X

∏2≤m≤n, m:even

Φm(1 + X), (6.8)

ω−n (X) = X

∏1≤m≤n, m:odd

Φm(1 + X). (6.9)


We also let Λ±n = Zp[∆][X]/(ω±

n (X)), I±n = XΛ±

n and I = XΛ.In Sect. 8, we construct the n-th even (odd) Coleman map

Col±n : H1(kn, T )/H1±(kn, T ) −→ Λ±

n

and its limit the even (odd) Coleman map

Col± : H1Iw(T )/H1

Iw,±(T ) −→ Λ.

They are morphisms of Λ-modules and have the following properties.

Theorem 6.2. The even (odd) Coleman maps induce isomorphisms

Col+n : H1(kn, T )/H1+(kn, T )

−→ Λ+n , (6.10)

Col−n : H1(kn, T )∆/H1−(kn, T )∆ −→ (Λ−

n )∆, (6.11)

Col−n : H1(kn, T )η/H1−(kn, T )η −→ (I−

n )η for η = 1. (6.12)

Taking the limit of the above isomorphisms, we have

Col+ : H1Iw(T )/H1

Iw,+(T )−→ Λ, (6.13)

Col− : H1Iw(T )∆/H1

Iw,−(T )∆ −→ Λ∆, (6.14)

Col− : H1Iw(T )η/H1

Iw,−(T )η −→ Iη for η = 1. (6.15)

Proof. This theorem is proved in Sect. 8. Theorem 6.3. Let z± ∈ H1(T ) be Kato’s zeta element in Theorem 5.2. Bythe localization map, we regard z± as an element of H1

Iw(T ). Then, theimage of Kato’s zeta element by the even (odd) Coleman map is Pollack’sp-adic L-function:

Col±(

zη(−1))εη = η(−1)L±

p (E, η, X) εη.

Proof. This theorem is proved in Sect. 8.7. Theorem 6.3 gives another construction of Pollack’s p-adic L-function.

7. Proofs of main theorems

Admitting Theorem 6.2 and 6.3, we prove Theorem 2.2 and 4.1. We alsoprove the equivalence of our main conjecture and Kato’s main conjecture.The proofs are basically the same as the good ordinary case (cf. Kato [7]).

12 S. Kobayashi

Let S be a finite set of places of Q containing p, all bad places of E,and the archimedean place. Let Sn (resp. S∞) be the set of places of Kn(resp. K∞) over S. Let Hq(Gn,S, T ) denote the Galois cohomology groupof Gn,S = Gal(Kn,S/Kn) where Kn,S is the maximal unramified extensionof Kn outside Sn. We let

Hq/S(T ) := lim←− n Hq(Gn,S, T ), Hq

Iw,v(T ) := lim←− n Hq(Kn,v, T )

where the limit is taken with respect to the corestrictions.We have E(Kn,v) ⊗ Qp/Zp = 0 for the places v not lying above p.

By definition, the exact annihilator of E±(kn) ⊗Qp/Zp with respect to thelocal Tate pairing is H1±(kn, T ). The exact annihilator of E(kn) ⊗Qp/Zp isE(kn) ⊗ Zp. Hence from the Tate-Poitou exact sequence, we have

0 →X0(E/Kn) → H2(Gn,S, T ) → ⊕v∈Sn H2(Kn,v, T ), (7.16)

H1(Gn,S, T ) → H1(kn, T )/H1±(kn, T ) → X±(E/Kn) → X0(E/Kn) → 0,

(7.17)

H1(Gn,S, T ) → H1(kn, T )/E(kn) ⊗ Zp → X(E/Kn) → X0(E/Kn) → 0.(7.18)

(cf. Perrin-Riou [16], Appendix A. 3.2.)Taking the limit of the above exact sequences, we have

0 → X0(E/K∞) → H2/S(T ) → ⊕v∈S∞H2

Iw,v(T ), (7.19)

H1/S(T ) →H1

Iw(T )/H1Iw,±(T ) → X±(E/K∞) → X0(E/K∞) → 0.

(7.20)

Proposition 7.1 (Kurihara). i) The canonical mapping H1(T ) → H1/S(T )

is an isomorphism.ii) H2(T ) is isomorphic to X0(E/K∞) as Λ-module.

Proof. The proposition is due to Kurihara [9]. It follows from the local-ization sequence of etale cohomology groups

0 →H1(OKn [1/p], j∗T ) → H1(OKn [1/S], T ) → ⊕v H0(k(v), H1(In,v, T ))

→ H2(OKn [1/p], j∗T ) → H2(OKn [1/S], T ) → ⊕v H2(Kn,v, T )

where v ranges over the places over S′ = S \ p, k(v) is the residue fieldof v and In,v is the inertia subgroup of the absolute Galois group of Kn,v.Then we use the fact lim←− n H0(k(v), H1(In,v, T )) = 0 and (7.19). We also

remark that H1(OKn[1/S], T ) = H1(Gn,S, T ). We identify H1(T ) = H1

/S(T ).


Corollary 7.2. X0(E/K∞) is a torsion Λ-module.

Proof. This follows from Theorem 5.1 and Proposition 7.1. Theorem 7.3. i) We have an exact sequence

0 → H1(T ) → H1Iw(T )/H1

Iw,±(T ) → X±(E/K∞) → X0(E/K∞) → 0.

(7.21)

ii) The even (odd) Selmer group Sel±(E/K∞) is Λ-cotorsion.

Proof. The exact sequence is nothing but (7.20) except for the injectivityof H1(T ) → H1

Iw(T )/H1Iw,±(T ). We consider the composition of this map

and the even (odd) Coleman map:

H1(T ) → H1Iw(T )/H1

Iw,±(T ) → Λ.

(cf. Theorem 6.2.) Since H1(T ) is a free Λ-module of rank 1 (cf. Theo-rem 5.1 iii)), this morphism is injective if and only if it is a non-zero map.Hence i) follows from Theorem 6.3 and Rohrlich’s theorem [20], whichassures L±

p (E, η, X) = 0.For ii), by Theorem 6.3, the η-component of the cokernel of the mor-

phism H1(T ) → H1Iw(T )/H1

Iw,±(T ) is killed by L±p (E, η, X). Hence the

cokernel is Λ-torsion. Since X0(E/K∞) is Λ-torsion by Corollary 7.2, ii)follows from the exact sequence (7.21). Theorem 7.4. The three conjectures, namely, Kato’s main conjecture(Sect. 5), the even main conjecture and the odd main conjecture (Sect. 4)are equivalent.

Proof. By Theorem 6.2, 6.3 and (7.21), we have three exact sequences

0 → H1(T )η/Z(T )η → Λη/(L+

p (E, η, X)) → X+(E/K∞)η → X0(E/K∞)η → 0,

0 → H1(T )∆/Z(T )∆ → Λ∆/(L−

p (E, X)) → X−(E/K∞)∆ → X0(E/K∞)∆ → 0,

0 → H1(T )η/Z(T )η → Iη/(L−

p (E, η, X)) → X−(E/K∞)η → X0(E/K∞)η → 0.

The last sequence is for a non-trivial η. The theorem follows from thesesequences and Proposition 7.1 ii).

Theorem 4.1 also follows from Theorem 5.2 iii), iv), v) and the exactsequences in the proof of Theorem 7.4.

14 S. Kobayashi

8. The construction of the even (odd) Coleman maps

8.1. A review of Honda’s theory

We recall Honda’s theory of formal groups. We restrict ourselves to com-mutative formal groups over Zp of dimension 1.

Let P be the Zp-submodule of Qp[[X]] consisting of elements f(X) =∑∞k=1 ak Xk satisfying kak ∈ Zp for all k. Let ϕ(X) = (1 + X)p − 1. We

define an endomorphism ϕ of P by

ϕ(∑

ak Xk)

:=∑

akϕ(X)k.

We denote the n-iterated composition of the endomorphism ϕ of P by ϕ(n).For u(t) = ∑∞

=0 bt ∈ Zp[[t]], we define an endomorphism u(ϕ) of P by

u(ϕ)( f )(X) :=∞∑

=0

b f(ϕ()(X)

).

Lemma 8.1. The action of ϕ induces the Frobenius operator F : ∑ak Xk →∑

ak X pk on P modulo pZp[[X]].Proof. This follows from the congruence

(X + pY )m

m≡ Xm

mmod pZp[X, Y ].

(cf. Honda [4], Lemma 4.) Definition 8.2. For u(t) ∈ Zp[[t]], we say f ∈ P is of the Honda type uwhen u(ϕ)( f ) ≡ u(F)( f ) ≡ 0 mod pZp[[X]] and f ′(0) = 1.

For a formal group F over Zp of dimension 1, we denote the logarithmby logF and the exponential by expF . We remark that logF (X) ∈ P . Wesay a polynomial u(t) = a0 + a1t + · · · + ahth ∈ Zp[t] is an Eisensteinpolynomial if ah is a unit, p | ai for all i = h and a0 = p. Then Hondashowed that there is a one-to-one correspondence between the isomorphismclasses of commutative formal groups overZp of dimension 1 and Eisensteinpolynomials in Zp[t]. More precisely,

Theorem 8.3 (Honda).

i) Let F be a formal group of dimension 1 over Zp. Then there existsa unique Eisenstein polynomial u such that logF is of the Honda typeu.

ii) Let F and G be formal groups over Zp of dimension 1 whose logarithmsare of the same Honda type u. Then expG logF (X) is an element of

Zp[[X]] and gives an isomorphism F∼→ G.


iii) Let u be an Eisenstein polynomial. Then there exist an element of P ofthe Honda type u. If f ∈ P is of the Honda type u, there exists a formalgroup over Zp with the logarithm f .

Proof. The theorem is a combination of theorems and propositions inHonda [5]. See Theorem 2 and 4, Proposition 2.6 and 3.5 in Honda [5].

8.2. A formal group Fss

We consider the power series

logFss(X) :=

∞∑k=0

(−1)k ϕ(2k)(X)

pk=

∞∑k=0

(−1)k (1 + X) p2k − 1

pk.

This is an element of P and of the Honda type t2+ p. Hence by Theorem 8.3,there exists a formal group Fss whose logarithm is given by logFss

(X).

Theorem 8.4. Let E be an elliptic curve overQp with good reduction at p.Let E be the formal group of the minimal model of E over Zp. Then logE(X)

is of the Honda type t2 − apt + p.

Proof. See Honda [5], Theorem 9, p. 240. Corollary 8.5. Suppose ap = 0. Then the Artin-Hasse type power seriesexpE logFss

(X) ∈ Zp[[X]] gives an isomorphism Fss → E.

Proof. This follows from Theorem 8.3 and 8.4. Hence the p-adic Tate module of any elliptic curve with ap = 0 is

isomorphic to the p-adic Tate module of Fss.

8.3. Torsion subgroups of Fss

Proposition 8.6. Over the integer ring of the unramified quadratic exten-sion of Qp, the formal group Fss is isomorphic to the Lubin-Tate formalgroup of height two with the parameter −p.

Proof. This is well-known since Fss is isomorphic to the formal group ofan elliptic curve. However, we give another proof. It suffices to show thatthe multiplication [−p] of Fss satisfies that i) [−p]X ≡ −pX modulodegree 2 and ii) [−p]X ≡ X p2

modulo p. The condition i) follows fromgeneral properties of commutative formal groups. For ii), the multiplica-tion [−p] of Fss is given by expFss

(−p logFss(X)), which is in Zp[[X]]

by Honda. Since (ϕ(2) + p) logFss(X) ≡ 0 modulo pZp[[X]], we have

expFss(−p logFss

(X)) ≡ expFss(ϕ(2) logFss

(X)) ≡ expFss(logFss

(ϕ(2)(X)))

≡ ϕ(2)(X) mod pZp[[X]]. (cf. expFss(pZp[[X]]) = pZp[[X]].)

16 S. Kobayashi

Proposition 8.7. The formal group Fss has no p-power torsion point onthe cyclotomic field kn.

Proof. Let L be the composite field of kn and the unramified quadraticextension F of Qp. Let mL be the maximal ideal of the integer ring OL .We denote the field obtained by adjoining all p-torsion points of Fss to Fby F(Fss[p]), which is an abelian extension of F of degree p2 − 1 by theLubin-Tate theory. Suppose that there is a p-torsion point in Fss(mL). SinceF(Fss[p])/F is abelian, all subextensions are normal over F. Therefore Lcontains all p-torsion points, namely, L ⊇ F(Fss[p]). However, the degreeof L/F is φ(pn) = pn − pn−1, which is not a multiple of p2 − 1. Hence wehave a contradiction.

8.4. Norm subgroups of Fss

Let mn be the maximal ideal of Zp[ζpn+1] and m−1 = pZp. For m ≥ n, wedenote the trace (norm) map Fss(mm) → Fss(mn) by Trss

m/n.Since the logarithm of a formal group over Zp is bijective on pZp, there

exists a unique element ε ∈ pZp such that logFss(ε) = p

p+1 .

Definition 8.8. We define

cn = ε[+]Fss (ζpn+1 − 1)

where [+]Fss is the addition of Fss. We also define c−1 = ε and c−2 = [2] ε.

Lemma 8.9. For a non-negative integer n, we have

Trssn+1/n (cn+1) = − cn−1, Trss

k0/Qp(c0) = − c−2.

Proof. By Proposition 8.7, logFssis injective on Fss(mn). Hence it suffices

to show that the image of cn by logFsssatisfies the above relation. We have

Trn+1/n logFss(cn+1) = Trn+1/n

p

p + 1+

[ n+22 ]∑

k=0

(−1)k ζpn+2−2k − 1

pk

= p2

p + 1− p + p

[ n+2

2 ]∑k=1

(−1)k ζpn+2−2k − 1

pk

= − logFss(cn−1).

The other case may be proved similarly.


Definition 8.10. We define the n-th norm subgroup Css(mn) byP ∈ Fss(mn)

∣∣ Trssn/m+1 P ∈ Fss(mm) for m (−1 ≤ m < n, m ≡ n mod 2)

for n ≥ 0 and Css(m−1) = Fss(m−1).

Proposition 8.11. For n ≥ 0, we have logFss(mn) ⊆ mn + kn−1. In particu-

lar, there exist canonical isomorphisms of Zp-modules

Fss(mn)/Fss(mn−1) ∼= logFss(mn)/logFss

(mn−1) ∼= mn/mn−1.

By these isomorphism, cσn is sent to cσ

n itself. In particular, (cσn )σ∈Gn generates

Fss(mn)/Fss(mn−1) as Zp-module.

Proof. As a set, Fss(mn) is the maximal ideal mn , and we write x ∈Fss(mn) of the form x = ∑

i ai ζipn+1 , ai ∈ Zp. Then x pk ≡ ∑

i ai ζi pk

pn+1

mod pZp[ζpn+1] and yk := ∑i ai ζ

i pk

pn+1 ∈ mn−1 for k ≥ 1. Therefore by thecongruence in the proof of Lemma 8.1, we have

(x + 1)p2k − 1

pk≡

(x pk + 1

)pk − 1

pk≡ (yk + 1)pk − 1

pkmod mn.

Hence we have (x+1)p2k−1pk ∈ mn + kn−1. For sufficiently large k0, we have∑∞

k=k0

(x+1)p2k−1pk ∈ mn, and logFss

(x) ∈ mn + kn−1.Since logFss

is injective on Fss(mn) (cf. Proposition 8.7) and is compati-ble with the Galois action, we have logFss

mn∩kn−1 = logFssmn−1. Therefore

we have an injection

logFss(mn)/logFss

(mn−1) → (mn + kn−1)/kn−1∼= mn/mn−1.

By direct calculations, we have logFss(cσ

n ) ≡ cσn mod kn−1. Since (cσ

n )σ∈Gn

generates mn/mn−1 with respect to the usual addition, the above injectionis in fact a bijection. Proposition 8.12. i) The n-th norm subgroup Css(mn) is generated by(cσ

n )σ∈Gn as Zp-module.ii) For a non-negative integer n, we have Css(mn) ∩ Css(mn−1) = Fss(m−1)and Fss(mn) = Css(mn) + Css(mn−1). In other words, we have an exactsequence

0 → Fss(m−1) → Css(mn) ⊕ Css(mn−1) → Fss(mn) → 0 (8.22)

where the first map is the diagonal embedding by inclusions, and the secondmap is (a, b) → a[−]Fss b.

18 S. Kobayashi

Proof. We first show that Css(mn) ∩ Css(mn−1) = Fss(m−1). Let P ∈Css(mn) ∩ Css(mn−1). We show that if P ∈ Fss(mm) then P ∈ Fss(mm−1).Suppose that n ≡ m−1 mod 2. Then by the definition of Css(mn), we havepn−m P = Trss

n/m P ∈ Fss(mm−1). Therefore for σ ∈ Gal(km/km−1), we havepn−m(Pσ − P) = 0. Hence, by Proposition 8.7, P ∈ Fss(mm−1). The casen ≡ m mod 2 is shown similarly by considering n − 1 instead of n in theabove argument.

For the moment, let C ′ss(mn) be theZp-submodule generated by (cσ

n )σ∈Gn .We prove Css(mn) = C ′

ss(mn) and Fss(mn) = Css(mn) + Css(mn−1) byinduction. Suppose they are true for n −1. Then by the inductive hypothesisand Lemma 8.9, we have Fss(mn−1) = Css(mn−1) + Css(mn−2) and

Css(mn−2) = C ′ss(mn−2) ⊆ C ′

ss(mn) ⊆ Css(mn).

Therefore, by Proposition 8.11, we have

Fss(mn) = C ′ss(mn) + Fss(mn−1) = C ′

ss(mn) + Css(mn−1).

In particular, C ′ss(mn) + Css(mn−1) ⊇ Css(mn). Since the intersection

Css(mn) ∩ Css(mn−1) = Fss(m−1), we have C ′ss(mn) = Css(mn).

Corollary 8.13. Let qn = ∑nk=0 (−1)n−k pk and let η : ∆ → Z×

p a char-acter. Then rankZp Css(mn)

η = qn + δ where δ = 1 if η = 1 and n is odd,otherwise, δ = 0.

Proof. Since the Zp-rank of Fss(mn)η is pn , the corollary follows from

Proposition 8.12 ii) by induction. Remark 8.14. Let N = Trss

n+1/n. By Lemma 8.9 and Proposition 8.12, wecan also show that

dimFp Fss(mn)η/NFss(mn+1)

η =

qn − 1 if η = 1 and n is even,qn otherwise.

8.5. The construction of Coleman maps

Let E/Qp be an elliptic curve with good reduction and ap = 0. Let T be theTate module of E and V = T ⊗Qp. Then by Corollary 8.5, the Artin-Hassetype power series expE logFss

(X) induces an isomorphism i : Fss → E.We regard Fss(mn) ⊆ H1(kn, T ) by the composition of i and the Kummermap. Then the cup product induces a pairing

( , )n : Fss(mn) × H1(kn, T ) → H2(kn,Zp(1)) ∼= Zp. (8.23)

For x ∈ Fss(mn), we define a morphism Pn, x : H1(kn, T ) → Zp[Gn] by

z −→∑σ∈Gn

(xσ , z)n σ.

This morphism is compatible with the natural Galois action.


Lemma 8.15. The morphisms Pn, x are compatible:

H1(kn, T )Pn, x−−−→ Zp[Gn]

H1(kn−1, T )Pn−1, Nx−−−−→ Zp[Gn−1]

where the vertical maps are the corestriction and the projection.

Proof. The lemma follows from the definition of Pn, x and basic propertiesof the cup product.

We define two sequences (c+n )n≥0, (c−

n )n≥0, c+n , c−

n ∈ Fss(mn), by

c+n : −c0, −c0, c2, c2, −c4, . . . , c−

n : c−1, −c1, −c1, c3, c3, . . . .

Namely,

c+n =

(−1)

n+22 cn if n is even,

(−1)n+1

2 cn−1 if n is odd,c−

n =

(−1)n+1

2 cn if n is odd,

(−1)n2 cn−1 if n is even.

(cf. Definition 8.8.)

Definition 8.16. We define the n-th norm subgroup and the n-th even normsubgroup and the n-th odd norm subgroup of E(mn) by

CE(mn) = P ∈ E(mn) | Trn/m+1 P ∈ E(mm) for m (−1 ≤ m < n, m ≡ n mod 2),E+(mn) = P ∈ E(mn) | Trn/m+1 P ∈ E(mm) for even m (0 ≤ m < n),E−(mn) = P ∈ E(mn) | Trn/m+1 P ∈ E(mm) for odd m (−1 ≤ m < n),

where Trn/m+1 : E(mn) → E(mm+1) is the trace map of E.

The relations between these groups are

E+(mn) =CE(mn) if n is even,CE(mn−1) if n is odd,

E−(mn) =CE(mn) if n is odd,CE(mn−1) if n is even.

By Proposition 8.12 i), the group E±(mn) is generated by the image ofc±

n by i as Zp[Gn]-module.

Lemma 8.17. The Kummer map induces an injection E±(mn)⊗Qp/Zp →H1(kn, V/T ).

Proof. It suffices to show that the natural map E±(mn) ⊗ Qp/Zp →E(mn)⊗Qp/Zp is injective. Suppose we have P = pk Q where P ∈ E±(mn)

and Q ∈ E(mn). Let R = Trn/m+1 Q. Then if (−1)m = ±1, we havepk(Rσ − R) = 0 for all σ ∈ Gal(km+1/km). Hence by Proposition 8.7, wehave R ∈ E(mm) and Q ∈ E±(mn).

20 S. Kobayashi

Let H1±(kn, T ) be the exact annihilator with respect to the Tate pairing

H1(kn, V/T ) × H1(kn, T ) → Qp/Zp

of the subgroup E±(mn) ⊗Qp/Zp ⊆ H1(kn, V/T ).

We define P±n = Pn, c±

n.

Proposition 8.18. The kernel of P±n is H1±(kn, T ).

Proof. By the definition of P±n , we have an exact sequence

0 → Ker P±n → H1(kn, T ) → Hom(E±(mn), Zp)

where the last morphism is induced by the cup product. Taking the Pontrya-gin dual of this sequence and by Lemma 8.17, we have

0 → E±(mn) ⊗Qp/Zp → H1(kn, V/T ) → (Ker P±

n

)∨ → 0.

The proposition follows from this sequence. Next we study the image of P±

n .Let Φn(X) be the pn-th cyclotomic polynomial. We let ωn(X) =

(1 + X)pn − 1, ω±0 (X) = 1 and

ω+n (X) =

∏2≤m≤n, m:even

Φm(1 + X), ω−n (X) =

∏1≤m≤n, m:odd

Φm(1 + X).

(8.24)

(cf. (6.8) and (6.9).)We identify Λn = Zp[Gn] with Zp[∆][X]/(ωn(X)) by sending γn to

1 + X, where γn is the image of γ in Λn .

Proposition 8.19. The image of P±n is contained in ω∓

n (X) Λn.

Proof. We prove the proposition for P+n . The other case is proved similarly.

We consider a morphism

Zp[∆][X]/(ωn(X)) = Zp[Gn] Π ψ−−−→ ∏ψ Qp,

where ψ : Gn → Qp×

ranges over all characters of Gn of conductorpm+1 with odd m (ψ is naturally extended to Zp[Gn] → Qp). Since ψ(γn)is a primitive pm -th root of unity, the kernel of the above morphism isgenerated by ω−

n (X). Hence to prove the proposition, it suffices to showthat ψ P+

n = 0.


By Lemma 8.9 and 8.15, we have p−[ n+12 ]ψ P+

n = p−[ m+12 ]ψ P+

m Corn/m where Corn/m is the corestriction map. Therefore we may assumethat m = n. Then

ψ P+n (z) = (−1)

n+12

∑σ∈Gn

(cσ

n−1, z)

nψ(σ)

= (−1)n+1

2

∑τ∈Gn−1

(cτ

n−1, z)

n

∑σ∈Gn, σ≡τ

ψ(σ) = 0

where σ is the image of σ by the natural projection Gn → Gn−1. Since ωn(X) = ω∓

n (X) ω±n (X), the multiplication by ω∓

n (X) induces anisomorphism

Λ±n := Zp[∆][X]/(ω±

n (X)) −→ ω∓

n (X) Λn. (8.25)

Corollary 8.20. There exists a unique morphism Col±n which makes thefollowing diagram commutative.

H1(kn, T ) Col±n

Λ±n

× ω∓n

H1(kn, T )

H1±(kn, T )

P±

nΛn.

Proof. This follows from Proposition 8.18, 8.19 and (8.25). We call Col±n the n-th even (odd) Coleman map.

Proposition 8.21. The even (odd) Coleman maps are compatible:

H1(kn+1, T ) Col±n+1

Λ±n+1

H1(kn, T ) Col±n

Λ±n

where the vertical maps are the corestriction and the projection.

Proof. We prove the proposition for Col+n . The compatibility for Col−n isproved similarly. For an odd n, we consider a diagram

H1(kn+1, T )P+

n+1−−−→ ω−n+1(X) Λn+1

−−−→ Λ+n+1 proj

H1(kn, T )

P+n−−−→ ω−

n (X) Λn−−−→ Λ+

n

(8.26)

22 S. Kobayashi

where the vertical maps are the corestriction and the projection. ByLemma 8.9, 8.15 and ω−

n+1(X) = ω−n (X), the diagram (8.26) is commuta-

tive. For an even n, we consider a diagram

H1(kn+1, T )P+

n+1−−−→ ω−n+1(X) Λn+1

−−−→ Λ+n+1 × 1

p

H1(kn, T )

P+n−−−→ ω−

n (X) Λn−−−→ Λ+

n

(8.27)

where the left vertical map is the corestriction and the right vertical map isthe projection. Since Φn+1(1+ X) = ∑p−1

i=0 (1+ X)pn i ≡ p modulo ωn(X),we have ω−

n+1(X) = p ω−n (X) in Λn . The commutativity of the right square

in (8.27) follows from this fact. The commutativity of the left square followsfrom the fact c+

n+1 = c+n and Lemma 8.15.

Definition 8.22. We define the even (odd) Coleman map

Col± : H1Iw(T ) −→ Λ

as the limit of the even (odd) Coleman map Col±n : H1(kn, T ) → Λ±n .

We remark that

lim←− n Λ±n = lim←− n Zp[∆][X]/(ω±

n (X)) = Zp[∆][[X]] = Λ.

8.6. The image of the even (odd) Coleman map

Proposition 8.23. The Coleman maps Col+n and Col+ are surjective.

Proof. We first show that the corestriction maps H1(km, T ) → H1(kn, T )are surjective for all m ≥ n. By the Tate duality, it suffices to show thatthe kernel of the restriction map H1(kn, V/T ) → H1(km, V/T ) is zero.By the inflation-restriction sequence, the kernel is equal to H1(Gal(km/kn),H0(km, V/T )), and H0(km, V/T ) = 0 by Proposition 8.7.

Next we show that the 0-th even Coleman map Col+0 = P+0 : H1(k0, T )

→ Zp[∆] is surjective. Then, by Nakayama’s lemma, Col+n and Col+ aresurjective.

By Proposition 8.12 i) and Corollary 8.13, the elements (cσ0 )σ∈∆ become

a basis of Fss(m0) as Zp-module. Then P+0 is described as

H1(k0, T ) → Hom(Fss(m0),Zp) ∼= Zp[∆]where the first map is induced by the pairing (8.23) and the last identificationis f → −∑

σ∈∆ f(cσ0 ) σ . The morphism H1(k0, T ) → Hom(Fss(m0),Zp)

is a surjection since its Pontryagin dual is the injection Fss(m0)⊗Qp/Zp →H1(k0, V/T ).


For characters η : ∆ → Z×p , we decompose the odd Coleman maps to

the η-components

εηCol−n : H1(kn, T )η −→ εηΛ−n .

Let In be the augmentation ideal of Zp[Gn] = Zp[∆][Γn] :

In = Ker ( Zp[∆][Γn] −→ Zp[∆] )

where Γn is the quotient of Γ isomorphic to Z/pnZ and the map is inducedby Γn → 1. We denote I := lim←− n In = XΛ and I±

n := XΛ±n .

Proposition 8.24. If η is trivial, then εηCol−n and εηCol− are surjective. Ifη is non-trivial, the image of εηCol−n is the augmentation ideal (I−

n )η andthe image of εηCol− is Iη.

Proof. The proof is similar to that of Proposition 8.23. The difference is,for the component of a non-trivial η, we have to prove the surjectivity of thefirst Coleman map

εηCol−1 : H1(k1, T )η −→ (Λ−1 )η

to the augmentation ideal. We have Λ−1 = Λ1 and Col−1 = P−

1 .For σ ∈ G1, let dσ = εη cσ

1 . Then dστ = η(τ) dσ for τ ∈ ∆ and byLemma 8.9,

∑σ∈Γ1

dσ = εη c−1 = 0. Therefore by Proposition 8.12 i) andCorollary 8.13, (dσ )σ∈Γ1, σ =1 becomes a basis of Css(m1)

η. Hence we havean isomorphism

Hom(Css(m1)

η,Zp) ∼= Iη

1 , f → −εη

∑σ∈Γ1

f(dσ ) σ.

Then the morphism εηCol−1 = −εη P1, c1 is described as

H1(k1, T )η → Hom(Fss(m1)

η,Zp) → Hom

(Css(m1)

η,Zp) ∼= Iη

1

where the first map is induced by the pairing (8.23) and the second map isthe projection by the decomposition Fss(m1)

η = Css(m1)η ⊕ Css(m0)

η (cf.Proposition 8.12). The surjectivity of H1(k1, T )η → Hom(Fss(m1)

η,Zp)is proved as in the proof of Proposition 8.23.

Now Theorem 6.2 follows from Proposition 8.18, 8.23 and 8.24.

24 S. Kobayashi

8.7. Coleman maps in terms of the dual exponential maps

We give a description of Coleman maps in terms of the dual exponentialmaps. Then it turns out that our Coleman maps are similar to the explicitColeman map in the appendix in Rubin [21]. It also turns out that Pn, x inSect. 8.5 is obtained by Kurihara’s Pn-pairing (Kurihara [9], Sect. 3).

We recall the dual exponential maps following Rubin [21]. For every nlet tan(E/kn) denote the tangent space of E/kn at the origin, and considerthe Lie group exponential map

expE : tan(E/kn) → E(kn) ⊗Qp.

Fix a minimal Weierstrass model of E and let ωE denote the Neron differ-ential. Then the cotangent space cotan(E/kn) is kn ωE , and we let ω∗

E bethe corresponding dual basis of tan(E/kn). Then we have a commutativediagram

logFss(mn) Fss(mn)oo

logFss

expElogFss

logE(mn)

·ω∗

E

E(mn)oo

logE E1(kn)

tan(E/kn) expE E(kn) ⊗Qp

(8.28)

There is a dual exponential map

exp∗E : H1(kn, V ) −→ cotan(E/kn) = kn ωE,

which has a property

(x, z)n = Trkn/Qp logE(x) exp∗ωE

(z) (8.29)

for every x ∈ E(mn) and z ∈ H1(kn, V ). Here ( , )n is the Tate pairing andexp∗

ωE= ω∗

E exp∗E . (See formula (2) in Sect. 5 of Rubin [21].)

Proposition 8.25. The morphism Pn, x in Sect. 8.5 is described in terms ofthe dual exponential map as follows.

Pn, x(z) =∑σ∈Gn

(xσ , z

)n

σ

=∑σ∈Gn

(Trkn/Qp logFss

(xσ

)exp∗

ωE(z)

)σ

=( ∑

σ∈Gn

logFss

(xσ

)σ) ( ∑

σ∈Gn

exp∗ωE

(zσ

)σ−1

).

Proof. The formula follows from the diagram (8.28) and the formula (8.29).See also the appendix of Rubin [21].


Proposition 8.26. Let ψ be a character of Gn of conductor pm+1. Then wehave

p−[ n+12 ] ∑

σ∈Gn

logFss

(c+

n

)σψ(σ) =

(−1)

m+22 p−[ m+1

2 ] τ(ψ) if m is even,

0 if m is odd,

p−[ n2 ] ∑

σ∈Gn

logFss

(c−

n

)σψ(σ) =

(−1)

m+12 p−[ m

2 ] τ(ψ) if m is odd,

0 if m is even.

Here τ(ψ) is the Gauss sum∑

σ∈Gmψ(σ) ζσ

pm+1 .

Proof. Since p−[ n+12 ] logFss

(c+n ) and p−[ n

2 ] logFss(c−

n ) are norm compati-ble, we may assume that m = n. Then the formula is obtained by directcalculations as in the proof of Lemma 8.9.

Now we prove Theorem 6.3.

Proof of Theorem 6.3. It suffices to show that Col± ( zη(−1) ) εη has interpo-lation properties (3.4) and (3.5).

Let χ : Gn → µp∞ be a Dirichlet character of conductor pn+1. Letψ = ηχ. Suppose n is even. Then by Corollary 8.20, we have ψ P+

n =ω−

n (ζ − 1) πψ Col+n where χ(γn) = ζ and πψ : Λ+n → Qp, X → ζ − 1.

Hence by Proposition 8.25, 8.26 and Theorem 5.2, εη Col+n ( zη(−1) ) has theproperties (3.4) and (3.5). We remark that τ(ψ) τ(ψ) = ψ(−1) pn+1. Theother case is shown similarly by considering odd n.

9. A control theorem for the even (odd) Selmer group

Lemma 9.1. The morphisms Sel±(E/Kn) → Sel±(E/K∞) are injective.

Proof. The lemma follows from the inflation-restriction sequence forH1(Kn, V/T ) → H1(K∞, V/T ) and the fact that E(K∞)p∞ = 0 (cf. Propo-sition 8.7). Proposition 9.2. Let η : ∆ → Z×

p be a character. The natural maps

lim←− m H1+(km, T ) → H1

+(kn, T )/ω+n (γ − 1)H1(kn, T ) (9.30)

lim←− m H1−(km, T )∆ → H1

−(kn, T )∆/ω−n (γ − 1)H1(kn, T )∆ (9.31)

lim←− m H1−(km, T )η → H1

−(kn, T )η/ω−n (γ − 1)H1(kn, T )η (9.32)

are surjective. The last morphism is for a non-trivial η.

26 S. Kobayashi

Proof. We prove the surjectivity for (9.30). The other cases may be provedsimilarly.

Since H1(kn, T )/H1+(kn, T ) ∼= Λ/(ω+n (X)) by Theorem 6.2, we have

ω+n (γ − 1) H1(kn, T ) ⊂ H1

+(kn, T ).

For simplicity, we write H1+,n = H1+(kn, T ), and consider a commutativediagram

lim←− m H1+,m

ω+n (γ − 1)

lim←− m H1(km, T )

ω+n (γ − 1)

lim←− m H1(km, T )/H1+,m

ω+n (γ − 1)

0

0 H1+,n

ω+n (γ − 1)H1(kn, T )

H1(kn, T )

ω+n (γ − 1)H1(kn, T )

H1(kn, T )

H1+,n.

By Theorem 6.2, the right vertical map is an isomorphism. Sincelim←− m H1(km, T ) → H1(kn, T ) is surjective by Proposition 8.7, the mid-dle vertical map is surjective. Hence the proposition follows from the snakelemma.

For simplicity, we write X±∞ = X±(E/K∞) and Xn = X(E/Kn). Wealso abbreviate ω±

n (γ − 1) to ω±n , and similarly for ω±

n .

Theorem 9.3. The natural maps

X+∞/ω+

n X+∞ −→ X+

n /ω+n X+

n ,

X−,∆∞ /ω−

n X−,∆∞ −→ X−,∆

n /ω−n X−,∆

n ,

X−,η∞ /ω−

n X−,η∞ −→ X−,η

n /ω−n X−,η

n for η = 1

are surjective and have finite kernel of bounded order as n varies.

Proof. We prove the theorem for the even Selmer groups. The other casesmay be proved similary. We prove the Pontryagin dual of the theorem bysimilar arguments as in Greenberg [2, Sect. 3]. For a Λ-module M, we let

Mω+n =0 =

x ∈ M∣∣ ω+

n (γ − 1) x = 0.

We also let H+(Kn,v) = H1(Kn,v, V/T )ω+n =0/(E+(Kn,v) ⊗Qp/Zp). Then,

taking the dual of Proposition 9.2, the morphism

H+(kn) → lim−→ m H+(km) (9.33)


is injective. Let us consider a diagram

0 Sel+(E/Kn)ω+

n =0

sn

H1(Kn, V/T )ω+n =0

hn

Πv H+(Kn,v)

gn

0 Sel+(E/K∞)ω+n =0 H1(K∞, V/T )ω+

n =0 Πv H+(K∞,v)

The morphism sn is injective by Lemma 9.1. If the superscript ω+n = 0 is

replaced by ωn = 0, then the morphism hn is bijective by the inflation-restriction sequence. Hence a posteriori, hn is bijective. By Lemma 3.3 inGreenberg [2], for v | p, the kernel of rv : H+(Kn,v) → lim−→ m H+(Km,v)

is finite of bounded order. For v | p, the kernel of rv is zero by (9.33). Hencethe kernel of gn is finite of bounded order as n varies. The theorem followsfrom the snake lemma.

The following is half of the p-adic Birch and Swinnerton-Dyer conjec-ture, which is already proved in Kato [7].

Theorem 9.4.

ordX=0 L±p (E, X) ≥ rank E(Q).

In particular, we have

ordX=0 Lp(E, X) ≥ rank E(Q).

Proof. By Theorem 9.3, we have a pseudo-isomorphism

X±,∆∞ /(γ − 1)X±,∆

∞ ∼= X±,∆0 /(γ − 1)X±,∆

0 = X±,∆0 = X∆

0 .

We compare the Zp-rank of the both sides. By the half of the even (odd)main conjecture, the rank of the left hand side is less than or equal toordX=0 L±

p (E, X), and the rank of the right hand side is greater than orequal to the rank of E(Q).

10. An asymptotic formula for the order ofX(E/Kn)

We give an asymptotic formula for the p-adic order of the Tate-ShafarevichgroupX(E/Kn) in terms of the λ- and µ-invariants of the even and theodd Selmer groups. Similar formulas have been obtained by Kurihara andPerrin-Riou [17] for unspecified rational numbers λ and µ.

28 S. Kobayashi

10.1. The even (odd) λ- and µ-invariants

Let η : ∆ → Z×p be a character. We denote the λ- and the µ-invariants of

X±(E/K∞)η by λη± and µ

η±. As in the good ordinary case, we can bound

the corank of the Selmer groups by these invariants.

Let Sel1(E/Kn) := Ker

(Sel(E/Kn) −→ H1(kn, E[p∞])

E(Qp) ⊗Qp/Zp

).

By (8.22), we have an exact sequence

0 → Sel1(E/Kn)i→ Sel+(E/Kn) ⊕ Sel−(E/Kn)

j→ Sel(E/Kn)

where i is the diagonal embedding and j is (x, y) → x − y.

Proposition 10.1. The cokernel of j is finite.

Proof. Since Sel(E/Kn)∨ is a finitely generated Zp-module, it suffices to

show that the image of the morphism j contains the p-divisible part ofSel(E/Kn). Since Trk/k−1 is given by the pk-th cyclotomic polynomialΦk(γ), we have ω−

n (γ − 1) Q ∈ Sel+(E/Kn) for every Q ∈ Sel(E/Kn).Similarly, ω+

n (γ − 1) Q ∈ Sel−(E/Kn) for every Q ∈ Sel(E/Kn). Sinceω+

n (X) and ω−n (X) are mutually prime inQp[X], there exist A(X), B(X) ∈

Zp[X] such that A(X) ω−n (X) − B(X) ω+

n (X) = pm for some integer m.Let P be a p-divisible element of Sel(E/Kn) and let Q ∈ Sel(E/Kn) besuch that P = pm Q. We put P+ = A(γ − 1) ω−

n (γ − 1) Q and P− =B(γ − 1) ω+

n (γ − 1) Q. Then P± ∈ Sel±(E/Kn) and P = (P+ − P−). Corollary 10.2. For every n, we have

corank Zp Sel(E/Q)η + corank Zp Sel(E/Kn)η ≤ λ

η+ + λ

η−.

In particular, we have rank E(Q)η + rank E(K∞)η ≤ λη+ + λ

η−.

Proof. This follows from Lemma 9.1, Proposition 10.1 and the facts thatSel1(E/Kn) ⊇ Sel(E/Q) and Sel(E/Kn) ⊇ E(Kn) ⊗Qp/Zp.

10.2. An asymptotic formula

The (p-primary) Tate-Shafarevich group is defined by

X(E/Kn) := Ker

(H1(Kn, E[p∞])E(Kn) ⊗Qp/Zp

res−→∏v

H1(Kn,v, E[p∞])E(Kn,v) ⊗Qp/Zp

).

By definition, we have an exact sequence

0 → E(Kn) ⊗Qp/Zp → Sel(E/Kn) →X(E/Kn) → 0. (10.34)

In the following, we always assume that Xn := X(E/Kn) is finite forall n. We first show elementary lemmas.


Lemma 10.3. For an exact sequence of Λ-modules 0 → A → B → C→ 0, we have

0 → AΓn → BΓn → CΓn → AΓn → BΓn → CΓn → 0.

Proof. For a generater γn ∈ Γn, we consider a commutative diagram

0 A

× (γn−1)

B

× (γn−1)

C

× (γn−1)

0

0 A B C 0.

The lemma follows from the snake lemma. Let (Mn)n=1,2,... be a projective system of finitely generated Zp-modules.

If πn : Mn → Mn−1 has finite kernel and cokernel, then we define

∇Mn := lengthZp

(Ker πn) − lengthZp

(Coker πn) + dimQp Mn−1 ⊗Qp.

Lemma 10.4. i) Suppose we have an exact sequence 0 → (M′n) →

(Mn) → (M′′n ) → 0. If two of ∇Mn, ∇M′

n, ∇M′′n are defined, then the

other is also defined and ∇Mn = ∇M′n + ∇M′′

n .

ii) Suppose that Mn are constant. If Mn is finite or the transition mapMn → Mn−1 is given by the multiplication by p, then ∇Mn = 0.

Proof. This is straightforward. We write ωn(X) = (1 + X)pn − 1 simply by ωn .

Lemma 10.5. i) For f ∈ R := Zp[[X]], we let R f,n = R/( f, ωn). Weconsider a natural projective system (R f,n)n. If f(ζpn − 1) = 0, then thenumber ∇R f,n is defined and

∇R f,n = ord ζpn −1 f(ζpn − 1).

ii) Let M be a finitely generated torsion R-module with the characteristicpolynomial f , and let Mn = M/ωn M. We consider a natural projectivesystem (Mn)n. Then for sufficiently large n, ∇Mn is defined and

∇Mn = ord ζpn −1 f(ζpn − 1) = λ(M) + φ(pn)µ(M),

where λ(M) and µ(M) are the λ- and the µ-invariants of M.

Proof. Let h be the common divisor of f and ωn−1, and we write f = gh.Then R f,n−1 ⊗Qp = ⊕m≤n−1 R ⊗Qp /( f,Φm) and

dimQp R f,n−1 ⊗Qp =∑Φm |h

φ(pm) = deg h = ord ζpn −1 h(ζpn − 1).

30 S. Kobayashi

On the other hand, by direct computation, the kernel of the projectionπn : R f,n → R f,n−1 is isomorphic to R/(g,Φn) and

lengthZp

Ker πn = lengthZp

R/(g,Φn) = ord ζpn −1 g(ζpn − 1).

Hence we have

∇R f,n = lengthZp

Ker πn + dimQp R f,n−1 ⊗Qp = ord ζpn −1 f(ζpn − 1).

Let us prove ii). By the structure theorem, we have a pseudo-isomorphism

0 → A → M→ ⊕i R/ fi → B → 0

where A and B are finite groups. Then by Lemma 10.3, we have

(⊕i R/ fi)Γn → BΓn → (Im )Γn → ⊕i R/( fi, ωn) → BΓn → 0,

(Im )Γn → AΓn → Mn → (Im )Γn → 0.

The groups AΓn , BΓn and BΓn stabilize for sufficiently large n. By thenoetherian property, (⊕i R/ fi)

Γn and (Im )Γn also stabilize. Hence forsufficiently large n, by Lemma 10.4, we have

∇(⊕i R/( fi, ωn)) = ∇((Im )Γn) = ∇Mn.

Then ii) follows from i) in this lemma. We consider a projective system (X(E/Kn)

η)n , which is the dual ofthe inductive system (Sel(E/Kn)

η)n by the restrictions. For sufficientlylarge n, the Mordell-Weil group over Kn stabilize (e.g. Corollary 10.2) andby (10.34) we have

∇X(E/Kn)η = length

Zp

(X

ηn/X

η

n−1

) + rankZp E(K∞)η.

We let

Y(E/Kn) = Coker(H1(Gn,S, T ) → H1(kn, T )/E(kn) ⊗ Zp

).

Then by (7.18), we have an exact sequence

0 → Y(E/Kn) → X(E/Kn) → X0(E/Kn) → 0. (10.35)

Let us determine the numbers ∇Y(E/Kn)η and ∇X0(E/Kn)

η.We let

Y′(E/Kn) = Coker(H1(T )Γn → H1(kn, T )/E(kn) ⊗ Zp

),

Zn = Im(

H1(Gn,S, T ) → H1(kn, T )/E(kn) ⊗ Zp).


Then we have a commutative diagram

H1(T )Γn

gn

H1(kn, T )/E(kn) ⊗ Zp Y′(E/Kn)

0

0 Zn H1(kn, T )/E(kn) ⊗ Zp

Y(E/Kn) 0.

(10.36)

We consider projective systems (Y(E/Kn)η)n , (Y′(E/Kn)

η)n and(Coker gn)n by the corestrictions.

Proposition 10.6. i) ∇ (Coker gn)η = 0 for n 0. In particular,

∇ Y(E/Kn)η = ∇ Y′(E/Kn)

η.

ii) Let f η

0 be the characteristic polynomial of X0(E/K∞)η. Then for suffi-ciently large n, we have

∇X0(E/Kn)η = ord ζpn −1 f η

0 (ζpn − 1).

Proof. Let Z′n = Im ( H1(Gn,S, T ) → ⊕v H1(Kn,v, T ) ) where v ranges

over all places over S. By the Tate-Poitou sequence, we have a commutativediagram

H1(T )

⊕vH1Iw,v(T )

f∞

H1(G∞,S, V/T )∨

X0(E/K∞)

0

0 Z′n

⊕v H1(Kn,v, T ) fn

H1(Gn,S, V/T )∨ X0(E/Kn) 0.

Here v ranges over the places over S. From the above, we have diagrams

H1(T )Γn

g′n

⊕vH1Iw,v(T )Γn

hn

(Im f∞)Γn

in

0

0 Z′n

⊕v H1(Kn,v, T ) Im fn 0,

(Im f∞)Γn

in

(H1(G∞,S, V/T )∨)Γn

jn

X0(E/K∞)Γn

sn

0

0 Im fn H1(Gn,S, V/T )∨ X0(E/Kn)

0.

32 S. Kobayashi

By Lemma (10.3), we also have an exact sequence

(H1(G∞,S, V/T )∨)Γn tn

X0(E/K∞)Γn (Im f∞)Γn (H1(G∞,S, V/T )∨)Γn .

We consider the dual of hn and the inflation-restriction sequence. Thensince Γn has cohomological dimension 1, the morphism hn is injective.Similarly, since E(K∞ )p∞ = 0 by Proposition 8.7, we see that jn is bijective.Hence we have

0 Coker tn Coker g′n

Coker hn Ker sn

0.

We consider a commutative diagram

H1(T )Γn

1+γn+···+γp−1

n

g′n

H1(T )Γn+1

g′n+1

Z′n

Res Z′n+1.

Then the restriction induces a morphism Coker g′n → Coker g′

n+1. Sim-ilarly, we have a commutative diagram

0 Coker tn

Coker g′n

Coker hn

Ker sn

0

0 Coker tn+1 Coker g′n+1

Coker hn+1 Ker sn+1 0,

(10.37)

where the vertical maps are induced by restrictions.We show that for sufficiently large n, all vertical maps in (10.37) are

isomorphisms. Since diagram (10.37) consists of finitely generated Zp-modules, it suffices to show that the vertical maps on Coker tn and Coker hnare isomorphisms for n 0. By the noetherian property, the groups(H1(G∞,S, V/T )∨)Γn and X0(E/K∞)Γn stabilize. Therefore for large nthe restriction map Coker tn → Coker tn+1 is an isomorphism. The dualof Coker hn is ⊕v(E(K∞,v)p∞)Γn , which is finite of bounded order as nvaries (cf. Greenberg [2], Lemma 3.3) and the projection (Coker hn+1)

∨ →(Coker hn)

∨ is surjective. Hence for large n the restriction Coker hn →Coker hn+1 is an isomorphism.

By definition, there is a canonical projection Z′n → Zn. This induces

a surjection Coker g′n → Coker gn . Since Coker g′

n is a finitely generatedZp -module and stabilize for large n, Coker gn also stabilize and the restrictionCoker gn → Coker gn+1 is an isomorphism for n 0. We consider thecomposition Cor Res : Coker gn−1

∼= Coker gn → Coker gn−1, which isthe multiplication by p. Then by Lemma 10.4 ii), we have ∇(Coker gn)

η = 0


and ∇ Y(E/Kn)η = ∇ Y′(E/Kn)

η by (10.36). We have already seen thatKer sn is finite and stabilize for large n. In particular, ∇Ker sn = 0. Henceby Lemma 10.5 ii), we have

∇X0(E/Kn)η = ∇(

X0(E/K∞)ηΓn

) = ord ζpn −1 f η

0 (ζpn − 1). Next we compute ∇Y′(E/Kn)

η. Let Y′′(E/Kn) be the cokernel of

H1(T )Γn −→ H1(kn, T )

H1+(kn, T )

⊕ H1(kn, T )

H1−(kn, T )

(10.38)

where the map is the diagonal by localization.By taking the Pontryagin dual of (8.22) after tensoring with ⊗Qp/Zp,

we have an exact sequence

0 → H1(kn, T )

E(kn) ⊗ Zp→ H1(kn, T )

H1+(kn, T )

⊕ H1(kn, T )

H1−(kn, T )

→ H1(Qp, T )

E(Qp) ⊗ Zp→ 0.

Thus we have

0 → Y′(E/Kn) → Y′′(E/Kn) → H1(Qp, T )/E(Qp) ⊗ Zp → 0.(10.39)

Since H1(Qp, T )/E(Qp) ⊗ Zp∼= Zp, we have

∇Y′(E/Kn)η =

∇Y′′(E/Kn)∆ − 1 if η = 1,

∇Y′′(E/Kn)η if η = 1.

(10.40)

Using the even and the odd Coleman maps, we identify

H1(kn, T )/H1+(kn, T ) ∼= Λ+

n ,

H1(kn, T )∆/H1−(kn, T )∆ ∼= Λ−,∆

n , H1(kn, T )η/H1−(kn, T )η ∼= I−,η

n .

The last identification is for a non-trivial η.We consider commutative diagrams with exact rows

0 Zp GΛ+,∆

n ⊕ Λ−,∆n

F

proj

Λ∆n

proj

Λ∆n /

(ω+

n , ω−n

)

proj

0

0 Zp GΛ

+,∆n−1 ⊕ Λ

−,∆n−1

FΛ∆

n−1 Λ∆

n−1/(ω+

n , ω−n

) 0,

(10.41)

and for η = 1

0 Λ+,ηn ⊕ I−,η

nF

proj

Ληn

proj

Ληn/

(ω+

n , ω−n

)

proj

0

0 Λ+,η

n−1 ⊕ I−,η

n−1FΛ

η

n−1 Λ

η

n−1/(ω+

n , ω−n

) 0.

(10.42)

34 S. Kobayashi

Here

F : ( f, g) → ω−n f + ω+

n g, G : a → (ω+

n a, −ω−n a

).

Lemma 10.7. i) The projections Λn/(ω+n , ω−

n ) → Λn−1/(ω+n , ω−

n ) andΛn/(ω

+n , ω−

n ) → Λn−1/(ω+n , ω−

n ) are bijective.

ii) Let un : H1(T )∆ → Λ+,∆n ⊕ Λ−,∆

n be the morphism induced by (10.38).Then Im un ∩ Im G = Im un−1 ∩ Im G = 0.

Proof. i) follows from the fact that ωn−1 is a multiple of ω+n or ω−

n . Let v =(vm)m be an element H1(T )∆ and let un−1(v) = (v+, v−) ∈ Λ

+,∆n−1 ⊕ Λ

−,∆n−1 .

Then by the definition and the compatibility of the odd Coleman map, wehave

v− ≡ ε∆

∑σ∈G0

(cσ−1, v0

)0 ≡ ε∆ (p − 1) × (ε, v0)0 mod X.

Similarly, we have v+ ≡ 2 ε∆ (ε, v0)0 mod X. Hence (p − 1) v+ ≡ 2 v−.Suppose that un−1(v) ∈ Im G. We write (v+, v−) = (ω+

n a,−ω−n a) for

a ∈ Zp. Then (p − 1)v+ ≡ 2 v− mod X if and only if a = 0. Let e be a basis of the free Λ-module H1(T ) of rank 1. We put f η

+ :=εηCol+(e), f η

− := εηCol−(e) and

Mηn := Λ

ηn(

ω−n f η

+ + ω+n f η

−) , Mη

n−1 := Λη

n−1(ω−

n f η+ + ω+

n f η−

) .

(If η is trivial, we replace the superscript η by ∆.) Then by (10.41), (10.42)and Lemma 10.7, we have

0 Zp Y′′(E/Kn)∆

M∆n

Λ∆n /

(ω+

n , ω−n

) 0

0 Zp Y′′(E/Kn−1)∆ M∆

n−1 Λ∆

n−1/(ω+

n , ω−n

) 0,

(10.43)

and for η = 1

0 Y′′(E/Kn)η

Mηn

Ληn/

(ω+

n , ω−n

) 0

0 Y′′(E/Kn−1)η Mη

n−1 Λ

η

n−1/(ω+

n , ω−n

) 0.

(10.44)

Hence together with (10.40), we have ∇Y′(E/Kn)η = ∇Mη

n for every η.


Proposition 10.8. For sufficiently large n, we have

∇X(E/Kn)η =

deg ω−

n + λη+ + φ(pn)µ

η+ if n is even,

deg ω+n + λ

η− + φ(pn)µ

η− + ξη if n is odd

where ξη = 1 if η = 1, otherwise, ξη = 0.

Proof. By Lemma 10.5 i), we have

∇ Mηn = deg ω∓

n + ord ζpn −1 f η±(ζpn − 1).

Here the signs are taken according to the parity of n. On the other hand,we have Char(X−(E/K∞)η) = ( f η

0 (X) f η−(X)/X) if η = 1, otherwise,

Char(X±(E/K∞)η) = ( f η

0 (X) f η±(X)). Hence the proposition follows from

(10.35), Proposition 10.6 and ∇Y′(E/Kn)η ∼= ∇Mη

n . By summing up the formula

∇X(E/Km)η = lengthZp

(X

ηm/X

η

m−1

) + rank E(K∞)η

for m (0 ≤ m ≤ n), we have the following.

Theorem 10.9. Assume thatX(E/Kn) are finite for all n. Let η : ∆ → Z×p

be a character and let

eηn = ordp(X(E/Kn)

η).

Then, for sufficiently large n, we have

en =[ n−2

2 ]∑m=0

pn−1−2m −[n

2

]+

[n

2

]λ

η+ +

[n + 1

2

] (λ

η− + ξη

) − n rη∞

+[ n

2 ]∑m=1

φ(

p2m)µ

η+ +

[ n+12 ]∑

m=1

φ(

p2m−1) µη− + νη

where νη is an integer independent of n, rη∞ = rank E(K∞)η, and ξη = 1 if

η = 1, otherwise, ξη = 0.

By comparing with Greenberg’s conjecture in the good ordinary caseand by numerical data, Pollack conjectured that his analytic µ-invariantsare always equal to zero. By the even (odd) main conjecture, our µ

η+ and

µη− would be also equal to zero.

Corollary 10.10. Assume that µη+ = µ

η− = 0. Then for sufficiently large n,

we have

eηn =

[ληn + µpn + ν

η+]

if n is even,[ληn + µpn + ν

η−]

if n is odd

36 S. Kobayashi

where λη = λη++λ

η−+ξη−12 − rη∞, µ = p

p2−1and ν

η+, ν

η− are some rational

constants.

The above formula was conjectured by Kurihara in [10].

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iwasawa theory for elliptic curves at supersingular primes

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