Torsion Points of Abelian Varieties with Values in Infinite Extensions over a p-adic Field

Torsion Points of Abelian Varieties with Values in Infinite Extensions over a p-adic Field

By

YoshiyasuOzeki^∗

Abstract

Let A be an abelian variety over a p-adic field K and L an algebraic infinite extension over K. We consider the finiteness of the torsion part of the group of rational pointsA(L) under some assumptions. In 1975, Hideo Imai proved that such a group is finite ifAhas good reduction andLis the cyclotomicZp-extension ofK.

In this paper, first we show a generalization of Imai’s result in the case whereAhas good ordinary reduction. Next we give some finiteness results whenA is an elliptic curve andLis the field generated by thep-th power torsion of an elliptic curve.

§1. Introduction

LetKbe a finite extension of thep-adic number fieldQ_pwith residue field k and fix an algebraic closure ¯K of K. Let A be an abelian variety over K.

If L ⊂K¯ is a finite extension overK, it is well-known that the torsion part of the L-rational points A(L) is finite (cf. [5], Thm. 7). On the other hand, in general, much less is known whether the torsion part of A(L) is finite or infinite ifL ⊂K¯ is an infinite algebraic extension over K. We are interested in understanding whether the torsion part ofA(L) is finite or infinite. As one of the known results, Imai ([4]) proved that the torsion part ofA(K(μ_p∞)) is finite if A has potential good reduction, where K(μ_p∞) is the smallest field containing K and all p-th power roots of unity. In this connection we first prove the following:

Communicated by A. Tamagawa. Received November 26, 2008. Revised April 23, 2009.

2000 Mathematics Subject Classification(s): 11G10, 11G07, 11S99, 22E50.

Key words: abelian variety, Galois representation,p-adic Lie group.

Supported by the JSPS Fellowships for Young Scientists.

∗Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan.

e-mail: y-ozeki@math.kyushu-u.ac.jp

c 2009 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

Theorem 1.1. Let Abe an abelian variety overK which has potential good ordinary reduction. LetLbe an algebraic extension ofKwith residue field k_L.

(1)Assume that the residue field ofL(μ_p∞)is a potential prime-to-pextension ofk (in the sense of Def.2.1). ThenA(L)[p^∞] is finite.

(2) If L contains K(A[p]) andK(μ_p∞), thenk_L is a potential prime-to-p ex- tension ofk if and only if A(L)[p^∞]is finite.

(3)Assume thatL(μ_p∞)is a Galois extension ofKwhose residue field is finite.

Then the torsion part ofA(L)is finite.

Here we denote by K(A[p]) the field generated by the coordinates of all p-torsion points ofA. The proof of the above Theorem is given in Theorem 2.1 and Corollary 2.1. This theorem is a generalization of Imai’s theorem under the hypothesis thatA has good ordinary reduction. We can also obtain the global case of this theorem, see Section 2.4. As an easy consequence of Theorem 1.1 (2), we see that the group A(K(μ_∞))[p^∞] is infinite for an abelian variety A as in the theorem if we replace K with its suitable finite extension (here we denote byK(μ_∞) the field obtained by adjoining all roots of unity toK). We shall point out that such a group must be finitein the global case by a result of Ribet ([7]).

Next we consider the finiteness of the torsion part ofA(L) forL=K_B,p:=

K(B[p^∞]), the field generated by the coordinates of allp-th power torsion points of a semiabelian varietyB. For example,K_Gm,pis the cyclotomic fieldK(μ_p∞), whereGmis the multiplicative group overK. Hence Imai’s theorem is a result on the torsion part of A(K_Gm,p). From such a point of view, we raise the following question:

Question. Let the notation be as above. Then is the torsion part of A(K_B,p) finite?

The torsion part ofA(K_B,p) is finite if and only ifA(K_B,p)[^∞] is finite for all primesandA(K_B,p)[^∞] = 0 for almost all primes. For any prime=p, it is easy to see that the -part ofA(K_B,p) is finite (cf. Prop. 3.1). Hence we are interested in the finiteness of thep-part ofA(K_B,p). IfA=E₁andB=E₂ are elliptic curves, we can gain various results by distinguishing the reduction types ofE₁ andE₂, see the table below.

Theorem 1.2. The finiteness ofE₁(K_E2,p)[p^∞]is as follows:

E₁ E₂ E₁(K_E2,p)[p^∞] statement ord

ord “∞”^∗1 Prop.3.4

ss finite Cor.3.2

mult finite Cor.3.2

ss

FCM

ord CM finite^∗2 Prop.3.6

non-CM finite Prop.3.6

ss FCM “finite” Prop.3.7

non-FCM finite Prop.3.6

non- FCM

ord CM finite Prop.3.6

non-CM finite Prop.3.6

ss FCM finite Prop.3.6

non-FCM finite or∞^∗3 Prop.3.8

split mult any ∞ Prop.3.9

non-split mult any “finite” Prop. 3.10

Here “ord”, “ss”, “mult”, “CM” and “FCM” in the above table stand for ordinary, supersingular, multiplicative, complex multiplication and formal complex multiplication, respectively. The definition of “formal complex multi- plication” is given in Section 3.2. The symbols∗1,∗2 and∗3in the table imply the followings:

∗₁· · · E₁,E₂ : good ordinary reduction

⇒E₁(K_E2,p)[p^∞] is infinite in many cases.

∗2· · · E₁ : good supersingular reduction with formal complex multiplica- tion

E₂ : good ordinary reduction with complex multiplication

⇒E₁(K_E2,p)[p^∞] is finite in all cases.

∗3· · · E₁,E₂: good supersingular reduction without formal complex mul- tiplication

⇒E₁(K_E2,p)[p^∞] may be finite or infinite (case by case).

For more precise information of Theorem 1.2, see the corresponding state- ments of the table.

§2. Finiteness Theorems for Abelian Varieties

In this section, we prove Theorem 1.1 given in the Introduction and con- sider the global case for this theorem.

Let p be a prime number. Let A be an abelian variety over a field F of dimension d and A^∨ the dual abelian variety of A. Fix an algebraic closure

F¯ ofF and a separable closure F^sepof F in ¯F. Put G_F := Gal(F^sep/F), the absolute Galois group of F. For any algebraic extension L over F and any integer n > 0, We denote by A(L)[pⁿ] the kernel of the multiplication-by-pⁿ map of the L-rational points A(L) of A and put A(L)[p^∞] := ∪

n>0A(L)[pⁿ].

We denote by F_A[pn] = F(A[pⁿ]) the field generated by the coordinates of A(F^sep)[pⁿ] = A( ¯F)[pⁿ]. Put F_A,p = F(A[p^∞]) := ∪

n>0F(A[pⁿ]). Then the fieldF_A,pcontainsF(μ_p∞), the field adjoining allp-th power roots of unity to F. The natural continuous representation associated to the Tate moduleT_p(A) ofAis denoted by

ρ_A,p:G_F →GL(T_p(A)) GL_h(Z_p) for someh≥0 and denote its residual representation by

¯

ρ_A,p:G_F ^ρ→^A,pGL_h(Zp)^mod→^pGL_h(Fp) ( GL(A(F^sep)[p])).

By abuse of notation, we shall considerρ_A,p as the representation ofV_p(A) = T_p(A)⊗_ZpQp. We note that the definition field of the representationρ_A,p(resp.

¯

ρ_A,p) is the fieldF_A,p (resp.F_A[p]).

§2.1. Some properties of torsion points

In this subsection, we collect some (in-)finiteness properties of the torsion part of abelian varieties which are easy to prove.

First we note the following proposition, which plays an important role throughout this paper:

Proposition 2.1. A(L)[p^∞] is finite if and only if the fixed subgroup T_p(A)^GL of T_p(A)by the absolute Galois groupG_L of Lis0.

Proof. This immediately follows from the definition of Tate module and the fact that eachA(L)[pⁿ] is finite.

The field F(A[p]) is a finite Galois extension of F whose Galois group Gal(F(A[p])/F) is a subgroup of GL_h(Fp) for the integer h ≥ 0 such that A( ¯F)[p] (Z/pZ)^⊕h. Putg_h,p:= (p^h−1)(p^h−p)· · ·(p^h−p^h−1), the order of GL_h(F_p).

Proposition 2.2. (1)If the absolute Galois groupG_F ofFis an inverse limit of finite groups of order prime top, then

A(F_A[p])[p^∞] =A(F^sep)[p^∞].

(2)If the absolute Galois groupG_F of F is an inverse limit of finite groups of order prime tog_h,p, then

A(F)[p^∞] =A(F^sep)[p^∞].

Proof. (1) If the group A(F^sep)[p^∞] is 0, there is nothing to prove and hence we may assume thatA(F^sep)[p^∞]= 0. Hence, for someh >0,ρ_A,p|_GFA[p]

has values in the kernel of the reduction mapGL_h(Z_p)→GL_h(F_p), which is a pro-pgroup. SinceG_FA[p] is an inverse limit of finite groups of order prime to p, the representationρ_A,p|G_FA[p] is trivial and thus we have

A( ¯F)[p^∞] =A( ¯F)[p^∞]^kerρA,p⊂A( ¯F)[p^∞]^GFA[p] =A(F_A,p)[p^∞].

This completes the proof of (1).

(2) We use the same argument as (1) except only that we do not need to considerF_A[p] sinceρ_A,p|_GF is already trivial.

If there is a Galois equivariant homomorphism among two Tate modules of abelian varieties, we can see some infiniteness properties about torsion points of abelian varieties.

Proposition 2.3. Let AandB be abelian varieties overF. If Hom_Zp[GF](T_p(B), T_p(A))= 0,

thenA(F_B,p)[p^∞]is infinite.

Proof. Take f to be a non-trivial element of Hom_Zp[GF](T_p(B), T_p(A)).

ThenT_p(B)/ker(f) is a non-zero subspace ofT_p(A) with trivialG_FB,p-action.

This implies the desired statement.

Definition 2.1. LetLbe an algebraic extension ofF.

(1) We say that L is a prime-to-p extension of F if L is a union of finite extensions overF of degree prime-to-p.

(2) We say thatLis apotential prime-to-pextension ofF ifLis a prime-to-p extension over some finite extension field ofF.

Proposition 2.4. Assume that A has the following property: for any finite Galois extension F of F, the torsion part of A(F) is finite. Then A(L)[p^∞]is finite for any potential prime-to-pGalois extension L ofF.

Proof. The assertion follows from the facts that A(L)[p^∞] =A(L∩F_A,p)[p^∞] andL∩F_A,pis a finite Galois extension ofF.

Note that we can apply Proposition 2.4 ifF is one of the following fields:

(i) a finitely generated field over a prime field, (ii) a finite extension ofQp.

Proposition 2.5. LetF be a finite field of characteristicp >0andF_p the maximal pro-p-extension ofF inF¯. Assume thatA( ¯F)[p]is not trivial.

(1) Suppose that A( ¯F)[p] is rational over F. ThenA(L)[p^∞] is finite if and only ifLis a potential prime-to-pextension overF. Furthermore in the other case,A(L)[p^∞] =A( ¯F)[p^∞].

(2)F_A,p=F_p(A[p]).

Proof. (1) First we note thatF_p=F_A,pbecause A( ¯F)[p]= 0 andF is a finite field of characteristicp. IfL is a potential prime-to-pextension over F, Prop. 2.4 implies thatA(L)[p^∞] is finite. In the other case,LcontainsF_p and henceA(L)[p^∞] is infinite.

(2) If we putF :=F(A[p]), we see thatF_A,p coincides withF_p, the maximal pro-p-extension ofF. SinceF_p =F_p(A[p]), we have done.

§2.2. Finiteness theorems for abelian varieties with good ordinary reduction

LetKbe a finite extension ofQp with integer ringOK and residue fieldk.

LetI_K be the inertia subgroup ofG_K. If an abelian varietyAoverKhas good reduction, we denote by ˜Athe reduction ofAoverk. For anyp-divisible group Gover O_K, we denote its Tate module, Tate comodule, connected component and maximal ´etale quotient byT_p(G), Φ_p(G),G⁰andG^´et, respectively and put V_p(G) :=T_p(G)⊗ZpQp.

Before proving Theorem 1.1 given in the Introduction, we shall show the proposition related with the matrix of the representation attached to abelian varieties by an analogous proof due to Conrad ([1], Thm. 1.1).

Proposition 2.6. Let A be an abelian variety over K of dimension d which has good reduction.

(1)The representationρ_A,p has the form

S_AU_A 0 T_A

with respect to a suitable basis ofT_p(A). Here, for some integer0≤f ≤d, (i) S_A:G_K→GL_2d−f(Zp) is a continuous homomorphism,

(ii) T_A:G_K→GL_f(Zp)is an unramified continuous homomorphism and (iii) U_A:G_K →M_2d−f,f(Zp)is a map.

(2)IfAhas good ordinary reduction overK, thenf =dandS_A|_IK is conjugate with the direct sum of thep-adic cyclotomic charactersε^⊕d.

(3)IfAhas good ordinary reduction over K, the mapS_A is conjugate with the mapε·(^tT_A∨)⁻¹.

Here, explicitly, the map ε^⊕d:G_K →GL_d(Zp) is given by the equation ε^⊕d(σ) = diag(ε(σ))∈GL_d(Zp),

the diagonal matrix with coefficients ε(σ) for all σ ∈ G_K, and for any map T :G_K→GL_d(Zp), we denote by^tT the mapG_K→GL_d(Zp) defined by

tT(σ) :=^t(T(σ))∈GL_d(Zp)

for allσ∈G_K, where^t(T(σ)) is the transposed matrix ofT(σ).

Proof of Proposition 2.6. (1) Let A be the N´eron model of A over OK

andA(p) thep-divisible group associated toA. The connected-´etale sequence ofA(p) induces the exact sequence

0→V_p(A(p)⁰)→V_p(A(p))→V_p(A(p)^´et)→0

ofQ_p[G_K]-modules. The desired decomposition ofρ_A,pcan be obtained by this sequence.

(2) Assume the reduction type of A over OK is ordinary. Let Kˆ^ur be the completion of the maximal unramified extension K^ur over K and OKˆ^ur the integer ring ofKˆ^ur. ThenA_OKurˆ :=A × _OKSpf(OKˆ^ur) is isomorphic to G^⊕d_m overOKˆ^ur, where Ais the formal completion ofA along its zero section and G_m is the formal multiplicative group overO_Kˆur (cf. [6], Lem. 4.26 and Lem.

4.27). This implies the assertion (2).

(3) Consider the following two exact sequences asG_K-modules;

0→V_p(A(p)^´et)^∨→V_p(A(p))^∨→V_p(A(p)⁰)^∨→0,

0→V_p(A^∨(p)⁰)(−1)→V_p(A^∨(p))(−1)→V_p(A^∨(p)^´et)(−1)→0,

whereM^∨= Hom_Qp(M,Q_p) is the dual of aQ_p[G_K]-moduleM andM(i) is the i-th Tate twist ofM. Note that V_p(A(p))^∨ V_p(A^∨(p))(−1) asG_K-modules.

By taking the functor H⁰(I_K,−) of the above exact sequences and using the assertion (2), we can see that

V_p(A(p)^´et)^∨ (V_p(A(p))^∨)^IK (V_p(A^∨(p))(−1))^IK V_p(A^∨(p)⁰)(−1) as G_K-modules, since I_K acts on V_p(A(p)^´et)^∨ and V_p(A^∨(p)⁰)(−1) trivially, and I_K acts on V_p(A(p)⁰)^∨ and V_p(A^∨(p)^´et)(−1) by (ε^⊕d)⁻¹. We know that the groupG_K acts onV_p(A(p)^´et)^∨(1) byε·(^tT_A)⁻¹and also acts onV_p(A^∨(p)⁰) byS_A∨. Therefore, we see thatε·(^tT_A)⁻¹ is conjugate withS_A∨ and thus we finish the proof of the assertion (3) after replacingAwithA^∨.

Remark.

(1) The integer f is equal to the dimension of V_p( ˜A) and the map S_A is the natural continuous homomorphism

G_K→GL(T_p(A(p)⁰)) GL_2d−f(Z_p) and the mapT_A is the natural continuous homomorphism

G_K→GL(T_p( ˜A)) GL_f(Zp).

(2) Suppose thatAhas good ordinary reduction overK. We denote the semi- simplification ofρ_A,p byρ^ss_A,p. By Proposition 2.6, the representationρ^ss_A,p|GL

factors throughG_kL for any algebraic extension L of K such thatL contains allp-th power roots of unity.

Now we can show our main finiteness theorem which is given in the Intro- duction.

Theorem 2.1. Let Abe an abelian variety overK which has potential good ordinary reduction. LetLbe an algebraic extension of K.

(1)Assume that the residue field ofL(μ_p∞)is a potential prime-to-pextension ofk. Then A(L)[p^∞] is finite.

(2)Assume thatL(μ_p∞)is a Galois extension ofKwhose residue field is finite.

Then the torsion part ofA(L)is finite.

Proof. (1) By extendingKandL, we may assume thatAhas good ordi- nary reduction overK and LcontainsK(μ_p∞). Putd:= dim(A) and denote

byk_L the residue field of L. We have the following form of the representation ρ_A,p for a suitable basis of theZp-moduleT_p(A):

S_AU_A

0 T_A

,

with certainS_A = ε·(^tT_A∨)⁻¹ : G_K →GL_d(Z_p), T_A : G_K →GL_d(Z_p) and U_A:G_K→M_d(Z_p) as in Proposition 2.6. In this proof, we shall fix the above basis and identify V_p(A) with Q^⊕2d_p = M_2d,1(Qp). Since L contains all p-th power roots of unity, we know that the mapsS_A|GL andT_A|GL factor through the absolute Galois groupG_kL := Gal(k^sep/k_L) of k_L. Thus the maps S_A|GL

and T_A|GL are determined by a topological generator σ_L of G_kL. To prove V_p(A)^GL = 0, it suffices to show that the matricesS_A(σ_L) andT_A(σ_L) do not have eigenvalue 1. First we assume that this assertion aboutT_A(σ_L) is false.

Then the Tate moduleV_p( ˜A)^GL is not zero and hence ˜A(k_L)[p^∞] is infinite. By Proposition 2.4, this contradicts the assumption that the fieldk_L is potential prime-to-pextension ofk. Next we show thatS_A(σ_L) does not have eigenvalue 1. SinceLcontainsK(μ_p∞) andAhas good ordinary reduction, the mapS_A|GL

coincides with (^tT_A∨|_GL)⁻¹. Hence if we assumeS_A(σ_L) has eigenvalue 1, then T_A∨(σ_L) also has eigenvalue 1. But this induces a contradiction by the same argument as the above.

(2) The criterion of N´eron-Ogg-Shafarevich implies that A(L)⊂A(L(μ_p∞)) =A(L(μ_p∞)∩K^ur),

where the symbol means the prime-to-ppart, and thusA(L)is finite because L(μ_p∞) has a finite residue field. Note that the above equality A(L(μ_p∞)) = A(L(μ_p∞)∩K^ur)comes from the assumption thatL(μ_p∞) is a Galois extension of K. Therefore, the assertion (1) implies that the torsion points of A(L) is finite.

Corollary 2.1. LetAbe an abelian variety overKwhich has good ordi- nary reduction. LetLbe a Galois extension ofK with residue fieldk_L. Assume thatL containsK(μ_p∞)andK(A[p]). Then the followings are equivalent:

(1)A(L)[p^∞] is finite, (2)A^∨(L)[p^∞] is finite, (3) ˜A(k_L)[p^∞]is finite, (4) ˜A^∨(k_L)[p^∞]is finite,

(5)k_L is a potential prime-to-pextension overk.

Note that the Weil-paring implies thatLcontainsK(μ_p∞) andK(A[p]) if and only ifLcontainsK(μ_p∞) andK(A^∨[p]).

Proof. The equivalence of assertions (3), (4) and (5) follows from Proposition 2.5, thus it is enough to show that (1) is equivalent to (3), (4) and (5). By Theorem 2.1 (1), the condition (5) implies the condition (1).

Let us assume that the condition (4) is not satisfied. Then we know that T_A∨(σ_L) has eigenvalue 1, where T_A∨ is a natural unramified homomor- phism G_K → GL(T_p( ˜A^∨)) and σ_L is a topological generator of G_kL. Hence ε·(^tT_A∨)⁻¹(σ_L) has also eigenvalue 1. By using this fact and Proposition 2.6, we can see thatT_p(A)^GL is not trivial. Therefore, the group A(L)[p^∞] is infinite.

§2.3. K^ur-rational points

We continue to use the same notationp,K,k,Aanddas in the previous subsection. In this subsection, we consider some relations of ρ_A,p : G_K → GL(V_p(A)) GL_2d(Qp) with A(K^ur)[p^∞]. Our goal in this subsection is to prove that

Theorem 2.2. Let E be an elliptic curve over K which has good ordi- nary reduction. Then the followings are equivalent;

(1)ρ_E,p is abelian, (2)ρ_E,p|IK is abelian, (3)E(K^ur)[p^∞]is infinite.

Note that it is known that the condition (1) is equivalent to the condition (4)E has complex multiplication overK.

See [9], A.2.4 for more information.

Now we start with an argument by proving the fact below;

Proposition 2.7. Let A be an abelian variety overK which has good ordinary reduction. Ifρ_A,p|_IK is abelian, then A(K^ur)[p^∞]is infinite.

Proof. By Proposition 2.6, we have the following form of the representa- tionρ_A,p|_IK for a suitable basis of theQ_p-moduleV_p(A):

ε^⊕dU

0 I_d

,

where U is a map I_K → M_d(Q_p) and I_d is the unit matrix of d×d. Since ρ_A,p|IK is abelian, we see thatU = (ε^⊕d−I_d)U₀for someU₀∈M_d(Qp). Then,

onI_K, ε^⊕dU

0 I_d

U₀1_d 1_d

=

U₀1_d 1_d

,

where 1_d =^t(1,1,· · · ,1)∈M_d,1(Qp). Consequently, we know thatV_p(A)^IK = 0 and this implies the desired result.

In the rest of this subsection we always assume that Ahas good ordinary reduction overK. LetK^ab be the maximal abelian extension ofK in ¯K. The exact sequence

0→ker(r)→V_p(A)→^r V_p( ˜A)→0 gives the exact sequence

0→ker(r)→V_p(A)^GKab →^r V_p( ˜A) (∗)

of Qp[Gal(K^ab/K)]-modules. Here we remark that G_Kab acts trivially on ker(r) and V_p( ˜A). We define the representation ρ^ur_A,p by the natural action of Gal(K^ab/K^ur) on V_p(A)^GKab =V_p(A(K^ab));

ρ^ur_A,p: Gal(K^ab/K^ur)→GL(V_p(A)^GKab) =GL(V_p(A(K^ab))).

Now we define the integere(A) to bee(A) := dim_QpV_p(A(K^ab))−dim_Qpker(r) = dim_QpV_p(A(K^ab))−d. Then ρ^ur_A,p is ap-adic representation of dimensiond+ e(A). Clearly 0≤e(A)≤d. Furthermore, the above sequence (∗) implies that ρ^ur_A,phas the following shape for a suitable basis of theQp-moduleV_p(A(K^ab)):

ρ^ur_A,p

ε^⊕d U 0 I_e(A)

,

whereU is a map Gal(K^ab/K^ur)→M_d,e(A)(Qp) and I_e(A)is the unit matrix ofe(A)×e(A).

(I) The casee(A)= 0.

In this case, we see thatU = (ε^⊕d−I_d)U₀for someU₀∈M_d,e(A)(Qp) since ρ^ur_A,p is abelian. Then, on Gal(K^ab/K^ur),

ε^⊕d U

0 I_e(A)

U₀1_e(A) 1_e(A)

=

U₀1_e(A) 1_e(A)

,

where 1_e(A)=^t(1,1,· · ·,1)∈M_e(A),1(Q_p). This equation shows the fact that V_p(A(K^ab))^Gal(Kab/Kur)= 0, that is,A(K^ur)[p^∞] is infinite.

(II) The casee(A) = 0.

In this case, we have ρ^ur_A,p =ε^⊕d and hence V_p(A(K^ab))^Gal(Kab/Kur)= 0, that is,A(K^ur)[p^∞] is finite.

Consequently we obtained the following lemma:

Lemma 2.1. Let the notation be as above. Then e(A) = 0if and only ifA(K^ur)[p^∞] is finite.

Now we can finish the proof of Theorem 2.2.

Proof of Theorem 2.2. SinceE is an elliptic curve, by combining Propo- sition 2.7 with Lemma 2.1, we can show the desired statement by the following way: e(E) = 0 ⇔ e(E) = 1 ⇔ dim_QpV_p(E(K^ab)) = 2 ⇔ V_p(E(K^ab)) = V_p(E)⇔G_Kab⊂G_KE,p⇔ρ_E,p is abelian.

§2.4. Global cases

Consider “global cases” of Theorem 2.1. LetAbe an abelian variety over a number fieldK. It is well-known that the groupA(L) is a finitely generated commutative group for a finite extension fieldLofKby the theorem of Mordell- Weil-N´eron-Lang. In particular its torsion subgroup is finite. In the case where Lis any algebraic extension of K, there are many results on the finiteness of torsion points ofA(L).

Letvbe a finite place ofK. For any finite extensionKofKand any finite place v of K above v, we denote the completion of K at v by K_v. More generally, for any algebraic extensionL and any placewabovev, we denote

L_w:= ∪

KK_v,

whereK runs through all the finite extensions ofK inLand v is the unique place ofK underw. Note that the residue fieldk_Lw ofL_w is ∪

Kk_K v.

As corollaries of Theorem 2.1, we can see the “global cases” below imme- diately.

Corollary 2.2. Let K, L, Abe as above. Assume that there exist places v ofK above pandw_∞ of L(μ_p∞)above v satisfying the following properties:

(i)The residue fieldk_w∞ ofL(μ_p∞)atw_∞ is a potential prime-to-pextension of the residue fieldk_v of K atv.

(ii)Ahas potential good ordinary reduction atv.

ThenA(L)[p^∞]is finite.

Corollary 2.3. Let K, L, Abe as above. Assume thatL(μ_p∞)is a Ga- lois extension of K, and there exist places v of K above pandw_∞ of L(μ_p∞) abovev satisfying the following properties:

(i)The residue fieldk_w∞ ofL(μ_p∞)atw_∞ is finite.

(ii)Ahas potential good ordinary reduction atv.

Then the torsion part of A(L)is finite.

If we always assume that L contains all p-th power roots of unity, these corollaries are generalizations of a result of Greenberg. See [3], Prop. 1.2 (ii).

§3. Finiteness of Torsion Points for Elliptic Curves

We use the same notations as defined at the beginning of the previous section (soon we will supposeK to be a finite extension of Qp). In particular A is an abelian variety over a field K. In this section, we give results on the Question which is proposed in the Introduction. At the beginning, we shall remark the following proposition related with “a torsion problem of two abelian varieties”.

Proposition 3.1. LetAandB be abelian varieties over a fieldK. As- sume thatK has the following property: the torsion part ofA(K)is finite for any finite Galois extensionK ofK. Then, for two different prime numbers₁ and₂, the groupA(K_B,2)[^∞₁ ]is finite.

Proof. This follows immediately from Proposition 2.4.

In view of the above proposition, we will be interested in the finiteness of A(K_B,2)[^∞₁ ] with₁=₂.

From now on, throughout this Section, we always denote by K a finite extension of Qp. Some of results on such the (in-)finiteness properties can be checked immediately, by using the results given in the previous section, as follows.

Proposition 3.2. Let E be elliptic curves over K which has potential multiplicative reduction or potential good supersingular reduction. We assume thatA has potential good ordinary reduction overK. Then the torsion part of A(K_E,p)is finite.

Proof. The two Lie groups ρ_E,p(G_K) and ρ_E,p(I_K) have the same Lie algebras (cf. [8]) and hence the residue field ofK_E,pis a finite field. Therefore, the residue field of K_E,p is also finite. Consequently we finish the proof by Theorem 2.1 (2).

Proposition 3.3. LetA andB be abelian varieties overK which have good ordinary reductions. Assume that A( ¯K)[p] is rational over K. Then A(K_B,p)is infinite.

Proof. Since B(K_B,p)[p^∞] is infinite, the residue field k_B,p of K_B,p is not a potential prime-to-pextension ofk by Theorem 2.1. BecauseA( ¯K)[p] is rational overK, Corollary 2.1 shows thatA(K_B,p)[p^∞] is infinite.

In the rest of this section, we discuss the following question:

Question. Let A and B be abelian varieties over K. When is A(K_B,p)[p^∞] finite?

Now we are interested in the case whereA(andB) are elliptic curves. Let us consider the above Question by distinguishing the reduction type ofA.

§3.1. Good ordinary reduction case

LetA=E₁ and B =E₂ be two elliptic curves overK. We have already proved in Corollary 3.2 that the torsion part ofE₁(K_E2,p)[p^∞] is finite ifE₁has good ordinary reduction over K and E₂ has good supersingular reduction or multiplicative reduction overK. We consider the infiniteness ofE₁(K_E2,p)[p^∞] under the condition that E₁ and E₂ have good ordinary reduction over K.

One of the result for the infiniteness ofE₁(K_E2,p)[p^∞] has given in Proposition 3.3. However, in this 1-dimensional case, we will show more precise criterion in Theorem 3.4 and Corollary 3.1.

We shall give some notation that we need. In the rest of this subsection we always assume thatE₁ and E₂ have good ordinary reduction overK. Let E˜₁and ˜E₂ be the reduction ofE₁andE₂overk, respectively. For each elliptic curvesE_i, Put

χ_i=ρ_E˜

i,p:G_K →GL(T_p( ˜E_i)) GL₁(Zp) =Z^×_p, and

¯ χ_i= ¯ρ_E˜

i,p:G_K→GL( ˜E_i(k^sep)[p]) GL₁(F_p) =F^×_p.

Clearly χ_i and ¯χ_i are unramified characters by their definitions. It can be checked that eachχ_i is of infinite order. We know that eachρ_Ei,phas the form

εχ⁻¹_i u_i

0 χ_i

with respect to a suitable basis of T_p(E_i). We fix such a basis and identify T_p(E_i) withZ^⊕2_p .

Proposition 3.4. Let the notations be as above and p≥3. Consider the following four conditions:

(a) The groupE₁(K_E2,p)[p^∞]is infinite.

(b)G_KE

2,p ⊂ker(χ₁).

(c) ker(χ₂)⊂ker(χ₁).

(d) Im( ¯χ₁)⊂Im( ¯χ₂).

Then there is the following relation: (a)⇔(b)⇐(c)⇔(d).

Proof. If we assume that the condition (a) is satisfied, there is a 1- dimensional G_KE

2,p-invariant subspace W of V_p(E₁) = M_2,1(Qp). Take any non-zero element (^x_y) inW. Sincexoryis a non-zero element which is invariant under the multiplication byχ₁(σ) for allσ∈G_KE

2,p, we see thatV_p( ˜E₁)^GKE2,p

has non-trivial subspace and henceV_p( ˜E₁)^GKE2,p =V_p( ˜E₁). This implies the condition (b). Conversely we assume that the condition (b) is satisfied. Then the condition (a) follows from the fact that

ρ_E1,p|G_KE2,p =

εχ⁻¹₁ u₁ 0 χ₁

=

1u₁ 0 1

.

Because G_KE

2,p ⊂ker(χ₂), the condition (c) implies (b). Finally let us show that conditions (c) and (d) are equivalent. Sinceχ₁andχ₂are unramified, we may consider each characterχ_ias a character

G_k →^χi Z^×_p Z/(p−1)Z×Zp,

whereG_kis an absolute Galois group ofk. By pr₁:Z/(p−1)Z×Zp→Z/(p−1)Z and pr₂ :Z/(p−1)Z×Z_p→Z_p, we denote natural projections. Furthermore we denote byk_χi, k_χ¯i andk_pthe corresponding field overkofχ_i,χ¯_i= pr₁◦χ_i and pr₂◦χ_ionG_k, respectively. Then we havek_χi =k_χ¯ik_p. Since the extension degree ofk_χ¯i overk is prime-to-p, we see that ker(χ₂)⊂ker(χ₁) if and only if k_χ¯1 ⊂k_χ¯2, which implies the condition (d).

Let k_E˜

i[p] be the smallest field extension of kover which the elements of E˜_i(k^sep)[p] is rational.

Corollary 3.1. Let E₁ and E₂ be elliptic curves over K which have good ordinary reduction. Assume thatp≥3.

(1)If k_E˜

1[p]⊂k_E˜

2[p], thenE₁(K_E2,p)[p^∞] is infinite.

(2)If the mapχ¯₁ is trivial, thenE₁(K_E2,p)[p^∞] is infinite.

(3)If the mapχ¯₂ is surjective, thenE₁(K_E2,p)[p^∞]is infinite.

Proof. All statements will immediately follow from Proposition 3.4.

§3.2. Good supersingular reduction case

LetA=E₁ and B=E₂ be elliptic curves over K. In this subsection we consider the case whereE₁ has (potential) good supersingular reduction over K. We recall the structure of the Lie algebra associated to an elliptic curve (cf.

[9], Appendix of Chapter 4).

Proposition 3.5. Let E be an elliptic curve overK. Putg:=

Lie(ρ_E,p(G_K))andi:= Lie(ρ_E,p(I_K)) (these are Lie subalgebras of End_Qp(V_p(E))).

(1) If E has good ordinary reduction with complex multiplication, then gis a split Cartan subalgebra ofEnd_Qp(V_p(E))and iis a 1-dimensional subspace of g.

(2)If E has good ordinary reduction without complex multiplication, thengis the Borel subalgebra ofEnd_Qp(V_p(E))corresponding to the kernel of the natural reduction map V_p(E) → V_p( ˜E) and i is a 2-dimensional subspace of g with dim(i/[i,i]) = 1.

(3) If E has good supersingular reduction with formal complex multiplication, thengis a non-split Cartan subalgebra ofEnd_Qp(V_p(E))andi=g.

(4)IfEhas good supersingular reduction without formal complex multiplication, theng= End_Qp(V_p(E))andi=g.

(5)If thej-invariant ofE has negativep-adic valuation, then gcoincides with n_X for some 1-dimensional subspace X of V_p(E) and i = g. Here n_X is the subspace of End_Qp(V_p(E)) generated by all u ∈ End_Qp(V_p(E)) satisfying that u(V_p(E))⊂X.

Here, for any elliptic curveE overK which has good supersingular reduction, we say that E hasformal complex multiplication over K if an endomorphism ring of thep-divisible groupE(p) over OK has rank 2 as a Zp-module, where E(p) is thep-divisible group associated with the N´eron modelEofEoverO_K. We also say thatE hasformal complex multiplication if E×KK has formal complex multiplication defined over some algebraic extensionK of K. Then the quadratic field End_OK(E(p))⊗ZpQp is called the field of formal complex multiplication. We can takeK for at most degree 2 extension ofK.

Our first result in this subsection is:

Proposition 3.6. Let E₁ and E₂ be elliptic curves over K. Suppose thatE₁has potential good supersingular reduction overK. ThenE₁(K_E2,p)[p^∞] is finite if one of the following conditions is satisfied: