Local torsion primes and the class numbers associated to an elliptic curve over $\mathbb{Q}$

TL;DR

This paper establishes a lower bound for the class number of certain number fields generated by division points of an elliptic curve over rationals, based on the Mordell-Weil rank and local torsion conditions.

Contribution

It provides a novel lower bound for class numbers of fields generated by p^n-division points of elliptic curves, under specific local torsion constraints.

Findings

Lower bound for class numbers of $Q(E[p^n])$ fields.

Relation between Mordell-Weil rank and class number growth.

Conditions on local torsion points influence class number estimates.

Abstract

Using the rank of the Mordell-Weil group $E(\mathbb{Q})$ of an elliptic curve $E$ over $\mathbb{Q}$, we give a lower bound of the class number of the number field $\mathbb{Q}(E[p^n])$ generated by $p^n$-division points of $E$ when the curve $E$ does not possess a $p$-adic point of order $p$: $E(\mathbb{Q}_p)[p] =0$.

Equations

$$\#\operatorname{Cl}_{p}(F_{n})=p^{\lambda n+\mu p^{n}+\nu},\quad\mbox{for $n\gg 0$},$$

$$\lambda \ge r-1,$$

$${\kappa }_n\ge 2n(r-1)-2\sum _{l\ne p,l\mid \Delta }{\nu }_l,$$

$$\nu_{l}:=\begin{cases}\min\set{\operatorname{ord}_{p}(\operatorname{ord}_{l}(\Delta)),n},&\mbox{if $E$ has split multiplicative reduction at $l$},\\ 0,&\mbox{otherwise}.\end{cases}$$

$$y^2+y=x^3+x^2-2x$$

$$\#{\mathrm{Cl}}_p(K_n)\ge p^{2n(r-1)}.$$

$$0\to E_1({\mathbb{Q}}_l)\to E_0({\mathbb{Q}}_l)\stackrel{\pi }{\to }{\overline{E}}_{\mathrm{ns}}({\mathbb{F}}_l)\to 0,$$

$$E({\mathbb{Q}}_l)\simeq {\mathbb{Z}}_l\oplus E({\mathbb{Q}}_l{)}_{\mathrm{tor}}$$

$$c_l:=(E({\mathbb{Q}}_l):E_0({\mathbb{Q}}_l)).$$

$$0\to E({\mathbb{Q}}_p)[p]\stackrel{\pi }{\to }\overline{E}({\mathbb{F}}_p)[p]\stackrel{\delta }{\to }\hat{E}(p{\mathbb{Z}}_p)/p\hat{E}(p{\mathbb{Z}}_p),$$

$${\Phi }^{(i)}:\mathrm{Gal}(L_n/K_n)\to E[p^n];\sigma \mapsto \sigma (Q_i)-Q_i,$$

$$\Phi :\mathrm{Gal}(L_n/K_n)\to E[p^n{]}^{\oplus r};\sigma \mapsto ({\Phi }^{(i)}(\sigma ){)}_i.$$

$$E({\mathbb{Q}}_p)\simeq {\mathbb{Z}}_p\oplus E({\mathbb{Q}}_p{)}_{\mathrm{tor}}.$$

$$E({\mathbb{Q}}_p)/[p^n]E({\mathbb{Q}}_p)\simeq \mathbb{Z}/p^n\mathbb{Z}.$$

$$\overline{P_{i}}=\overline{a_{i}}\cdot\overline{P}\quad\mbox{in $E(\mathbb{Q}_{p})/[p^{n}]E(\mathbb{Q}_{p})$}$$

$${\mathrm{ord}}_p(a_i)\le {\mathrm{ord}}_p(a_j)$$

$$(L_n{)}_{\mathfrak{P}}=(K_n{)}_{\mathfrak{p}}\left([p^n{]}^{-1}{P}_i\right).$$

$$\qquad\nu_{l}\!:=\!\begin{cases}\min\set{\operatorname{ord}_{p}(\operatorname{ord}_{l}(\Delta)),n},\ \mbox{if $E$ has split multiplicative reduction at $l$},\\ 0,\ \mbox{if $E$ has non-split multiplicative reduction at $l$}.\end{cases}$$

$$(T_n{)}_{\mathfrak{L}}([p^n{]}^{-1}{P}_j)\subset (T_n{)}_{\mathfrak{L}}([p^n{]}^{-1}{P}_i)$$

$$(L_n{)}_{\mathfrak{L}}=(T_n{)}_{\mathfrak{L}}\left([p^n{]}^{-1}{P}_i\right)$$

$$(L_n{)}_{\mathfrak{L}}=(T_n{)}_{\mathfrak{L}}([p^n{]}^{-1}{P}_i)\text{mbox}of(K_n{)}_{\mathfrak{l}}([p^n{]}^{-1}{P}_i)$$

$$E({\mathbb{Q}}_l)\simeq {\mathbb{Z}}_l\oplus E({\mathbb{Q}}_l{)}_{\mathrm{tor}}.$$

$$E({\mathbb{Q}}_l)/[p^n]E({\mathbb{Q}}_l)=0.$$

$$(L_n{)}_{\mathfrak{L}}=(K_n{)}_{\mathfrak{l}}$$

$$\quad\nu_{l}:=\begin{cases}\min\set{\operatorname{ord}_{p}(\operatorname{ord}_{l}(\Delta)),n},\ \mbox{if $E$ has split multiplicative reduction at $l$},\\ n,\ \mbox{if $p=3$, $E$ has additive reduction at $l$, and $c_{l}=3$},\\ 0,\ \mbox{otherwise},\end{cases}$$

$${\kappa }_n\ge 2n(r-1)-2\sum _{l\ne p,l\mid \Delta }{\nu }_l,$$

$$\#{\mathrm{Cl}}_p(K_n)=[{\tilde{K}}_n:K_n]\ge [L_n\cap {\tilde{K}}_n:K_n]=\frac{[L_n:K_n]}{[L_n:L_n\cap {\tilde{K}}_n]}.$$

$$[L_n:K_n]=p^{2nr}.$$

$$[L_n:L_n\cap {\tilde{K}}_n]=[L_n:L_n^I]=\#I.$$

$$\#I\le \#I_p\cdot \prod _{l\ne p,l\mid \Delta }\#I_l\le p^{2n}\cdot p^{2{\sum }_{l\ne p,l\mid \Delta }{\nu }_l}.$$

Full text

Key words: Elliptic curves, and Class number

MSC2010: 11R29, 11G05

1 Introduction

For an elliptic curve $E$ over $\mathbb{Q}$ with complex multiplication (abbreviated as CM in the following) such that $\operatorname{End}_{\mathbb{C}}(E)=\mathcal{O}_{F}$, the ring of integers of an imaginary quadratic field $F$. When $E$ has good ordinary reduction at $p>2$, the prime $p$ splits completely in $F$ as $p=\pi\overline{\pi}$ where $\pi\in\mathcal{O}_{F}$ and $\overline{\pi}$ is the complex conjugation of $\pi$. Let $F_{n}:=F(E[\pi^{n}])$ be the field generated by $\pi^{n}$-torsion points of $E$ over $F$. The extension $F_{\infty}:=\bigcup_{n}F_{n}$ of $F_{1}$ is a $\mathbb{Z}_{p}$-extension so that there exist $\lambda,\mu\in\mathbb{Z}_{\geq 0}$ and $\nu\in\mathbb{Z}$ which are all independent of $n$ such that we have

[TABLE]

where $\operatorname{Cl}_{p}(F_{n})$ is the $p$-Sylow subgroup of the ideal class group of $F_{n}$. It is known that the invariant $\lambda$ of the $\mathbb{Z}_{p}$-extension has a lower bound

[TABLE]

where $r$ is the ($\mathbb{Z}$-)rank of the group of $\mathbb{Q}$-rational points $E(\mathbb{Q})$ ([4], Sect. 5).

For an elliptic curve $E$ over $\mathbb{Q}$ which may not have CM and a prime number $p>3$, in recent papers [7] and [8], Sairaiji and Yamauchi gave a lower bound of the class number $\#\operatorname{Cl}_{p}(K_{n})$ in terms of the rank of $E(\mathbb{Q})$ associated to the field $K_{n}:=\mathbb{Q}(E[p^{n}])$ generated by $p^{n}$-torsion points $E[p^{n}]:=E(\overline{\mathbb{Q}})[p^{n}]$ under the following conditions111 In [7], the cases $p=2$ and $3$ have been studied under the additional condition: $\operatorname{Gal}(K_{n}/\mathbb{Q})\simeq GL_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$ for all $n\geq 1$. In fact, for $p>3$, $(\mathbf{Full})$ implies this condition (cf. [8], Sect. 1). :

• $(\mathbf{Red}_{l})$

$E$ has multiplicative reduction or potentailly good reduction at any prime $l\neq p$,

• $(\mathbf{Red}_{p})$

$E$ has multiplicative reduction at $p$,

• $(\mathbf{Disc})$

$p\nmid\operatorname{ord}_{p}(\Delta)$, where $\Delta$ is the minimal discriminant of $E$, and

• $(\mathbf{Full})$

$\operatorname{Gal}(K_{1}/\mathbb{Q})\simeq GL_{2}(\mathbb{Z}/p\mathbb{Z})$.

When $p>5$ the condition $(\mathbf{Red}_{p})$ automatically implies $(\mathbf{Disc})$ (cf. [8], Sect. 1). The objective of this note is to propose a condition

• $(\mathbf{Tor})$

$E(\mathbb{Q}_{p})[p]=0$

instead of using $(\mathbf{Red}_{p})$ and $(\mathbf{Disc})$ above, and give the same form of a lower bound of $\#\operatorname{Cl}_{p}(K_{n})$ as in [8]. The main theorem is the following:

Theorem 1.1** (cf. Thm. 3.3).**

Let $E$ be an elliptic curve over $\mathbb{Q}$ with minimal discriminant $\Delta$ and let $p$ be a prime number $>3$. Put $K_{n}:=\mathbb{Q}(E[p^{n}])$. Assume the conditions $(\mathbf{Tor})$ and $(\mathbf{Full})$ noted above. Then, for all $n\in\mathbb{Z}_{\geq 1}$, the exponent $\kappa_{n}$ of $\#\operatorname{Cl}_{p}(K_{n})=p^{\kappa_{n}}$ satisfies the following inequality:

[TABLE]

where $r$ is the rank of $E(\mathbb{Q})$ and

[TABLE]

Here, $\operatorname{ord}_{p}$ (resp. $\operatorname{ord}_{l}$) is the $p$-adic (resp. $l$-adic) valuation on $\mathbb{Q}$.

Remark 1.2**.**

1. $(\mathrm{i})$

The condition $(\mathbf{Full})$ means that the Galois representation $\rho:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\operatorname{Aut}(E[p])\simeq GL_{2}(\mathbb{Z}/p\mathbb{Z})$ is full (i.e., surjective). This can be checked by some criterions [9], Sect. 2.8 (see also [8], Sect. 1). 2. $(\mathrm{ii})$

In [1], for an elliptic curve $E$ over $\mathbb{Q}$, a prime number $p$ which does not satisfy $(\mathbf{Tor})$, that is, $E(\mathbb{Q}_{p})[p]\neq 0$ is called a local torsion prime for $E$. It is expected that when $E$ does not have CM, there are only finitely many local torsion primes $($[1], Conj. 1.1$)$.

A proof of Theorem 1.1 is given in Section 3 (cf. Thm. 3.3). In Section 2, we give some sufficient conditions for $(\mathbf{Tor})$. In fact, the conditions $(\mathbf{Red}_{p})$ and $(\mathbf{Disc})$ imply the condition $(\mathbf{Tor})$ (Lem. 2.3). Even though the theorem above can be applied to an elliptic curve and a prime $p$ of a wider class than [8], the proof is significantly simplified.

Closing this section, let us consider the elliptic curve $E$ over $\mathbb{Q}$ defined by

[TABLE]

(the Cremona label 389a1) which has the smallest conductor among those of $r=2$. This $E$ does not have CM and $\Delta=389$ ($E$ has multiplicative reduction at $389$). By using SAGE [2], one can confirm that the condition $(\mathbf{Full})$ holds for all primes $p$ and $(\mathbf{Tor})$ holds for any odd prime $<10^{6}$. Thus, our main theorem says that, for all odd primes $p<10^{6}$ (which may be $p=389$), we have

[TABLE]

Acknowledgement

The author would like to thank Professor F. Sairaiji and Professor T. Yamauchi who taught the author their results in [7] and [8]. Not only they generously sent the author their preprint [8], but also gave suggestions and comments which are improved the main theorem in this note. The author would like to thank Professor K. Matsuno for pointing out an error of the proof of Lemma 2.3 in an early draft of this note. The arguments in the latter part of Lemma 3.1 are due to them. The author would like to thank the referee for some comments which amend this note.

2 Local torsion primes

Throughout this note, we use the following notation:

• •

$p$ : a prime number $>2$,

• •

$E$ : an elliptic curve over $\mathbb{Q}$,

• •

$\Delta$ : the minimal discriminant of $E$ ([10], Chap. VIII, Sect. 8),

• •

$[p^{n}]:E\to E$ : the isogeny multiplication by $p^{n}$ ([10], Chap. III, Sect. 4), and

• •

$E[p^{n}]:=E(\overline{\mathbb{Q}})[p^{n}]$ : the $p^{n}$-torsion subgroup of $E(\overline{\mathbb{Q}})$.

Structure theorem on $E(\mathbb{Q}_{l})$

For a second prime number $l$ (which may be $p$), we denote also by $E$ the base change $E\otimes_{\mathbb{Q}}\mathbb{Q}_{l}$ of the elliptic curve $E$ to $\mathbb{Q}_{l}$. Define

• •

$\pi:E(\mathbb{Q}_{l})\to\overline{E}(\mathbb{F}_{l})$ : the reduction map modulo $l$ ([10], Chap. VII, Sect. 2),

• •

$\overline{E}_{\mathrm{ns}}(\mathbb{F}_{l})$: the set of non-singular points in the reduction $\overline{E}(\mathbb{F}_{l})$ (cf. [10], Chap. III, Prop. 2.5), and

• •

$E_{0}(\mathbb{Q}_{l}):=\pi^{-1}(\overline{E}_{\mathrm{ns}}(\mathbb{F}_{l}))$.

The reduction map $\pi:E(\mathbb{Q}_{l})\to\overline{E}(\mathbb{F}_{l})$ modulo $l$ induces a short exact sequence (of abelian groups)

[TABLE]

where $E_{1}(\mathbb{Q}_{l})$ is defined by the exactness (cf. [10], Chap. VII, Prop. 2.1).

Lemma 2.1**.**

1. $(\mathrm{i})$

$E_{1}(\mathbb{Q}_{l})\simeq\mathbb{Z}_{l}$. In particular, $E_{1}(\mathbb{Q}_{l})[p]=0$. 2. $(\mathrm{ii})$

• $(\mathrm{a})$

If $E$ has multiplicative reduction at $l$, then $\overline{E}_{\mathrm{ns}}(\mathbb{F}_{l})\subset\overline{E}_{\mathrm{ns}}(\mathbb{F}_{l^{2}})\simeq(\mathbb{F}_{l^{2}})^{\times}$.

• $(\mathrm{b})$

If $E$ has additive reduction at $l$, then $\overline{E}_{\mathrm{ns}}(\mathbb{F}_{l})\simeq\mathbb{F}_{l}$ as additive groups. 3. $(\mathrm{iii})$

• $(\mathrm{a})$

If $E$ has split multiplicative reduction at $l$, then $E(\mathbb{Q}_{l})/E_{0}(\mathbb{Q}_{l})\simeq\mathbb{Z}/\operatorname{ord}_{l}(\Delta)\mathbb{Z}$.

• $(\mathrm{b})$

If $E$ has non-split multiplicative reduction at $l$, then $E(\mathbb{Q}_{l})/E_{0}(\mathbb{Q}_{l})$ is a finite group of order at most $2$.

• $(\mathrm{c})$

In all other cases, namely, $E$ has good reduction or additive reduction at $l$, the quotient $E(\mathbb{Q}_{l})/E_{0}(\mathbb{Q}_{l})$ is a finite group of order at most $4$. 4. $(\mathrm{iv})$

We have an isomorphism

[TABLE]

as abelian groups, where $E(\mathbb{Q}_{l})_{\mathrm{tor}}$ is the torsion subgroup of $E(\mathbb{Q}_{l})$ which is finite.

Proof.

(i) We have $E_{1}(\mathbb{Q}_{l})\simeq\widehat{E}(l\mathbb{Z}_{l})$, where $\widehat{E}(l\mathbb{Z}_{l})$ is the group associated to the formal group $\widehat{E}$ of $E$ ([10], Chap. VII, Prop. 2.2). From [10], Chapter IV, Theorem 6.4 (b), the formal logarithm induces $\widehat{E}(l\mathbb{Z}_{l})\simeq l\mathbb{Z}_{l}\simeq\mathbb{Z}_{l}$. As $\mathbb{Z}_{l}$ is $p$-torsion free for $p>2$, we obtain $E_{1}(\mathbb{Q}_{l})[p]=0$.

(ii) The assertion follows from [10], Chapter III, Exercise 3.5.

(iii) The assertion follows from [10], Chapter VII, Theorem 6.1 for the case (a) and (c), and [11], Chapter IV, Remark 9.6 for the case (b).

(iv) As the exact sequence (2.1) splits, we have $E_{0}(\mathbb{Q}_{l})\simeq E_{1}(\mathbb{Q}_{l})\oplus\overline{E}_{\mathrm{ns}}(\mathbb{F}_{l})$. From (i), $E_{1}(\mathbb{Q}_{l})\simeq\mathbb{Z}_{l}$ and the quotients $E(\mathbb{Q}_{l})/E_{0}(\mathbb{Q}_{l}),E_{0}(\mathbb{Q}_{l})/E_{1}(\mathbb{Q}_{l})\simeq\overline{E}_{\mathrm{ns}}(\mathbb{F}_{l})$ are finite by (ii) and (iii). The assertion follows from this. ∎

Recall that the Tamagawa number $c_{l}$ at a prime $l$ for $E$ is defined by

[TABLE]

Lemma 2.2**.**

Suppose that $E$ has additive reduction at a prime $l\neq p$. We further assume the following conditions:

• $(\mathrm{a})$

$p>3$, or

• $(\mathrm{b})$

$c_{l}\neq 3$, where $c_{l}$ is the Tamagawa number at $l$ $($cf. (2.2)$)$.

Then, $E(\mathbb{Q}_{l})[p]=0$.

Proof.

As $E$ has additive reduction at $l$, we have $\overline{E}_{\mathrm{ns}}(\mathbb{F}_{l})[p]=0$ (Lem. 2.1 (ii-b)). On the other hand, $E_{1}(\mathbb{Q}_{l})[p]=0$ (Lem. 2.1 (i)) so that $E_{0}(\mathbb{Q}_{l})[p]=0$ by (2.1). As $c_{l}=\#E(\mathbb{Q}_{l})/E_{0}(\mathbb{Q}_{l})\leq 4$ (Lem. 2.1 (iii)), the quotient $E(\mathbb{Q}_{l})/E_{0}(\mathbb{Q}_{l})$ does not possess elements of order $p$ under the additional assumption (a) or (b). We obtain $E(\mathbb{Q}_{l})[p]=0$. ∎

Multiplicative reduction at $p$

Lemma 2.3**.**

Suppose the condition $(\mathbf{Red}_{p})$ in Introduction, that is, $E$ has multiplicative reduction at $p$. We further assume one of the following conditions:

• $(\mathbf{Disc})$

$p\nmid\operatorname{ord}_{p}(\Delta)$, or

• $(\mathrm{a})$

$E$* has non-split multiplicative reduction at $p$. *

Then, the condition (\mathbf{Tor})$$:E(\mathbb{Q}_{p})[p]=0 holds.

Proof.

As $E$ has multiplicative reduction at $p$, $\overline{E}_{\mathrm{ns}}(\mathbb{F}_{p})\subset\overline{E}_{\mathrm{ns}}(\mathbb{F}_{p^{2}})\simeq(\mathbb{F}_{p^{2}})^{\times}$ (Lem. 2.1 (ii)). In particular, $\overline{E}_{\mathrm{ns}}(\mathbb{F}_{p})[p]=0$. On the other hand, $E_{1}(\mathbb{Q}_{p})[p]=0$ (Lem. 2.1 (i)) and hence $E_{0}(\mathbb{Q}_{p})[p]=0$ by (2.1).

Case (a): First, we suppose that $E$ has non-split multiplicative reduction. In this case, the quotient group $E(\mathbb{Q}_{p})/E_{0}(\mathbb{Q}_{p})$ is a finite group of order at most $2$ (Lem. 2.1 (iii)) so that we obtain $E(\mathbb{Q}_{p})[p]=0$.

Case (Disc): Next, we assume $p\nmid\operatorname{ord}_{p}(\Delta)$. From Case (a) above, we may assume that $E$ has split multiplicative reduction at $p$. The assertion follows from $E(\mathbb{Q}_{p})/E_{0}(\mathbb{Q}_{p})\simeq\mathbb{Z}/\operatorname{ord}_{p}(\Delta)\mathbb{Z}$ (Lem. 2.1 (iii)). ∎

Remark 2.4**.**

When the elliptic curve $E$ over $\mathbb{Q}$ has multiplicative reduction at $2$, by considering the isomorphism $E(\mathcal{K})\simeq{\mathcal{K}}^{\times}/q^{\mathbb{Z}}$ for some unramified extension $\mathcal{K}/\mathbb{Q}_{2}$ locally, $-1\in{\mathcal{K}}^{\times}$ gives a $2$-torsion element in $E(\mathbb{Q}_{2})$. Thus the condition $(\mathbf{Tor})$ at $2$ does not hold: $E(\mathbb{Q}_{2})[2]\neq 0$.

Good reduction at $p$

Lemma 2.5**.**

Suppose that $E$ has good reduction at $p$.

1. $(\mathrm{i})$

We further assume one of the following conditions:

1. $(\mathrm{a})$

$\overline{E}(\mathbb{F}_{p})[p]=0$, or 2. $(\mathrm{b})$

$E(\mathbb{Q})_{\mathrm{tor}}\neq 0,p\geq 11$.

Then, the condition $(\mathbf{Tor})$  holds. 2. $(\mathrm{ii})$

Assume that $E$ has CM, and $p\geq 7$. Then, $(\mathbf{Tor})$ holds if and only if $\overline{E}(\mathbb{F}_{p})[p]=0$.

The lemma above essentially follows from [1], Proposition 2.1. For the sake of completeness, we give a proof.

Proof of Lem. 2.5.

(i) Case (a): We have $E_{1}(\mathbb{Q}_{p})[p]=0$ (Lem. 2.1 (i)). The condition can be checked by using the exact sequence

[TABLE]

where $\delta$ is the connecting homomorphism. The assumption $\overline{E}(\mathbb{F}_{p})[p]=0$ implies the condition $(\mathbf{Tor})$.

Case (b): Assume $E(\mathbb{Q}_{p})[p]\neq 0$. By [1], Proposition 2.1 (1), we have $E(\mathbb{Q})_{\mathrm{tor}}\simeq\mathbb{Z}/p\mathbb{Z}$. From the assumption $p\geq 11$, this contradicts with Mazur’s theorem on $E(\mathbb{Q})_{\mathrm{tor}}$ ([10], Chap. VIII, Thm. 7.5).

(ii) From (i) (the case (a)), it is enough to show that if $\overline{E}(\mathbb{F}_{p})[p]\neq 0$, then $E(\mathbb{Q}_{p})[p]\neq 0$. From Hasse’s theorem ([10], Chap. V, Thm. 1.1) and $p\geq 7$, $\#\overline{E}(\mathbb{F}_{p})=p$. We have $a_{p}(E):=p+1-\#\overline{E}(\mathbb{F}_{p})=1$. This implies $E(\mathbb{Q}_{p})[p]\neq 0$ by [1], Proposition 2.1 (3) under the assumption that $E$ has CM. ∎

When $E$ has CM, Lemma 2.5 (ii) gives a criterion for the condition $(\mathbf{Tor})$. On the other hand, Lemma 2.5 (i) says that, for $p\geq 11$, $(\mathbf{Tor})$ does not hold only if

1. $(\mathrm{a}^{\prime})$

$\overline{E}(\mathbb{F}_{p})[p]\neq 0$, and 2. $(\mathrm{b}^{\prime})$

$E(\mathbb{Q})_{\mathrm{tor}}=0$.

For our purpose, we further impose

1. $(\mathrm{c}^{\prime})$

$E$ does not have CM, and 2. $(\mathrm{d}^{\prime})$

the rank $r>1$.

The following calculations are given by using SAGE [2]. There are $1733$ elliptic curves with conductor $N<10^{4}$ satisfying ($\mathrm{b^{\prime}}$)-($\mathrm{d^{\prime}}$) above. Among them, only $50$ curves have a local torsion prime $p$ in the range $11\leq p<10^{6}$, i.e., $E(\mathbb{Q}_{p})[p]\neq 0$ listed below:

[TABLE]

Table1: local torsion primes

3 Elliptic curve over $\mathbb{Q}$

We keep the notation of the last section. We further define

• •

$K_{n}:=\mathbb{Q}(E[p^{n}])$ (cf. [10], Chap. VIII, Prop. 1.2 (d)),

• •

$r:=$ the rank of $E(\mathbb{Q})$ (which is finite by the Mordell-Weil theorem [10], Chapter VIII),

• •

$P_{1},\ldots,P_{r}\in E(\mathbb{Q})$ : generators of the free part of $E(\mathbb{Q})$, and

• •

$L_{n}:=K_{n}\!\left([p^{n}]^{-1}P_{1},\ldots,[p^{n}]^{-1}P_{r}\right)$.

Following [5], Chapter V, Section 5, for each $1\leq i\leq r$, define

[TABLE]

where $Q_{i}\in E(\overline{\mathbb{Q}})$ with $[p^{n}]Q_{i}=P_{i}$. Since $E[p^{n}]\subset E(K_{n})$, the map $\Phi^{(i)}$ does not depend on the choice of $Q_{i}$. These homomorphisms $(\Phi^{(i)})_{1\leq i\leq r}$ induce an injective homomorphism

[TABLE]

From $E[p^{n}]\simeq(\mathbb{Z}/p^{n}\mathbb{Z})^{\oplus 2}$ ([10], Chap. III, Cor. 6.4) the extension $L_{n}/K_{n}$ is an abelian extension with $[L_{n}:K_{n}]\leq p^{2nr}$.

Inertia subgroups

For any prime number $l$ and a prime ideal $\mathfrak{l}$ in (the ring of integers of) $K_{n}$ above $l$ (we write $\mathfrak{l}\mid l$ in the following), we denote by

• •

$I_{\mathfrak{l}}$ : the inertia subgroup of $\operatorname{Gal}(L_{n}/K_{n})$ at $\mathfrak{l}$ (for $L_{n}/K_{n}$ is abelian, the inertia subgroup $I_{\mathfrak{l}}$ is independent of a choice of a prime ideal in $L_{n}$ above $\mathfrak{l}$), and

• •

I_{l}:=\Braket{I_{\mathfrak{l}}\,;\mbox{prime ideal \mathfrak{l}\mid l$in$K_{n}}}: the subgroup of $\operatorname{Gal}(L_{n}/K_{n})$ generated by $I_{\mathfrak{l}}$ for all $\mathfrak{l}\mid l$.

For any prime $\mathfrak{l}\mid l$ of $K_{n}$, and a prime $\mathfrak{L}$ of $L_{n}$ above $\mathfrak{l}$ (we write $\mathfrak{L}\mid\mathfrak{l}$ ), we denote by

• •

$(K_{n})_{\mathfrak{l}}$ : the completion of $K_{n}$ at $\mathfrak{l}$, and

• •

$(L_{n})_{\mathfrak{L}}$ : the completion of $L_{n}$ at $\mathfrak{L}$.

Lemma 3.1**.**

We assume the condition $(\mathbf{Tor})$. Then, we have $\#I_{p}\leq p^{2n}$.

Proof.

By the structure theorem on $E(\mathbb{Q}_{p})$ (Lem. 2.1 (iv)),

[TABLE]

From the condition $(\mathbf{Tor})$, we have $E(\mathbb{Q}_{p})_{\mathrm{tor}}/[p^{n}]E(\mathbb{Q}_{p})_{\mathrm{tor}}=0$ and hence

[TABLE]

Let $\overline{P}\in E(\mathbb{Q}_{p})/[p^{n}]E(\mathbb{Q}_{p})$ (the residue class represented by a point $P\in E(\mathbb{Q}_{p})$) be a generator of the cyclic group $E(\mathbb{Q}_{p})/[p^{n}]E(\mathbb{Q}_{p})$ and, for each index $1\leq i\leq r$, write

[TABLE]

for some $\overline{a_{i}}\in\mathbb{Z}/p^{n}\mathbb{Z}$ $(a_{i}\in\mathbb{Z})$. Take $1\leq i\leq r$ such that

[TABLE]

for all $1\leq j\leq r$.

For any prime $\mathfrak{P}\mid p$ of $L_{n}$, we denote by $\mathfrak{p}$ the prime in $K_{n}$ below $\mathfrak{P}$. Using the chosen index $i$, we obtain

[TABLE]

Put $K_{n}^{\prime}:=K_{n}\!\left([p^{n}]^{-1}P_{i}\right)\subset L_{n}$. From the equality (3.3), the extension $L_{n}/K_{n}^{\prime}$ is unramified (at all primes in $K_{n}^{\prime}$) above $\mathfrak{p}$. As the extension $K_{n}/\mathbb{Q}$ is Galois, this extension $L_{n}/K_{n}^{\prime}$ is unramified above $p$. Since $I_{p}\cap\operatorname{Gal}(L_{n}/K_{n}^{\prime})=\set{1}$, the restriction $\Phi^{(i)}|_{I_{p}}:I_{p}\to E[p^{n}]$ of $\Phi^{(i)}$ defined in (3.1) is injective and hence $\#I_{p}\leq p^{2n}$. ∎

Lemma 3.2**.**

Let $l$ be a prime number with $l\neq p$.

1. $(\mathrm{i})$

We have $\#I_{l}\leq p^{2n}$. 2. $(\mathrm{ii})$

Suppose that $E$ has multiplicative reduction at $l$. We have $\#I_{l}\leq p^{2\nu_{l}}$, where

[TABLE] 3. $(\mathrm{iii})$

Suppose that $E$ has additive reduction at $l$. We further assume the following conditions:

• $(\mathrm{a})$

$p>3$, or

• $(\mathrm{b})$

$c_{l}\neq 3$, where $c_{l}$ is the Tamagawa number at $l$ $($cf. (2.2)$)$.

Then, we have $\#I_{l}=1$.

Proof.

(i) Take any $\mathfrak{l}\mid l$ in $K_{n}$. For a prime $\mathfrak{L}\mid\mathfrak{l}$ in $L_{n}$, let $(T_{n})_{\mathfrak{L}}:=((L_{n})_{\mathfrak{L}})^{I_{\mathfrak{l}}}$ be the inertia field of $\mathfrak{L}$ over $(K_{n})_{\mathfrak{l}}$ which is the fixed field of $I_{\mathfrak{l}}$ (cf. [6], Chap. II, Def 9.10). Since $l\neq p$ the extension $L_{n}/K_{n}$ is tamely ramified at any prime $\mathfrak{L}\mid\mathfrak{l}$ in $L_{n}$. The inertia subgroup $I_{\mathfrak{l}}=\operatorname{Gal}((L_{n})_{\mathfrak{L}}/(T_{n})_{\mathfrak{L}})$ is cyclic (cf. [6], Chap. II, Sect. 9). There exists $1\leq i\leq r$ such that

[TABLE]

for any $1\leq j\leq r$. Since $I_{\mathfrak{l}}$ does not depend on the choice of $\mathfrak{L}\mid\mathfrak{l}$ in $L_{n}$, the index $i$ above can be chosen independent of $\mathfrak{L}\mid\mathfrak{l}$. We obtain

[TABLE]

for any prime $\mathfrak{L}\mid\mathfrak{l}$.

Put $K_{n}^{\prime}:=K_{n}\!\left([p^{n}]^{-1}P_{i}\right)\subset L_{n}$. The extension $(T_{n})_{\mathfrak{L}}/(K_{n})_{\mathfrak{l}}$ of local fields is unramified from the very definition of $(T_{n})_{\mathfrak{L}}$ for any prime $\mathfrak{L}\mid\mathfrak{l}$ in $L_{n}$. Using the equality (3.4) the extension

[TABLE]

is also unramified ([6], Chap. II, Prop. 7.2). This implies that $L_{n}/K_{n}^{\prime}$ is unramified at all primes $\mathfrak{L}\mid\mathfrak{l}$ in $L_{n}$. As the extension $K_{n}/\mathbb{Q}$ is Galois, this extension $L_{n}/K_{n}^{\prime}$ is unramified above $l$. Since $I_{l}\cap\operatorname{Gal}(L_{n}/K_{n}^{\prime})=\set{1}$, the restriction $\Phi^{(i)}|_{I_{l}}:I_{l}\to E[p^{n}]$ of $\Phi^{(i)}$ defined in (3.1) is injective and hence $\#I_{l}\leq p^{2n}$.

(ii) This assertion is [8], Theorem 4.1.

(iii) By Lemma 2.1 (iv), we have

[TABLE]

From $E(\mathbb{Q}_{l})[p]=0$ (Lem. 2.2), we have

[TABLE]

Hence, $P_{i}\in[p^{n}]E(\mathbb{Q}_{l})$ for each $i$. This implies that, for any prime $\mathfrak{l}\mid l$ in $K_{n}$, $(K_{n})_{\mathfrak{l}}([p^{n}]^{-1}P_{i})=(K_{n})_{\mathfrak{l}}$ and hence

[TABLE]

for any $\mathfrak{L}\mid\mathfrak{l}$ in $L_{n}$. In particular, $L_{n}/K_{n}$ is unramified at all primes $\mathfrak{L}\mid\mathfrak{l}$ in $L_{n}$. As the extension $K_{n}/\mathbb{Q}$ is Galois, this extension $L_{n}/K_{n}$ is unramified above $l$. Hence $I_{l}=\set{1}$. ∎

Main theorem

In the rest of this section, we show the following theorem:

Theorem 3.3**.**

For a prime $p>2$, and an elliptic curve $E$ over $\mathbb{Q}$ with minimal discriminant $\Delta$, put $K_{n}:=\mathbb{Q}(E[p^{n}])$ and

[TABLE]

where $c_{l}$ is the Tamagawa number at $l$ $($cf. (2.2)$)$. We assume the following conditions:

• $(\mathbf{Full})$

$\operatorname{Gal}(K_{1}/\mathbb{Q})\simeq GL_{2}(\mathbb{Z}/p\mathbb{Z})$, and

• $(\mathbf{Tor})$

$E(\mathbb{Q}_{p})[p]=0$.

Then, for all $n\in\mathbb{Z}_{\geq 1}$, the exponent $\kappa_{n}$ of $\#\operatorname{Cl}_{p}(K_{n})=p^{\kappa_{n}}$ satisfies the following inequality:

[TABLE]

where $r$ is the rank of $E(\mathbb{Q})$.

Proof.

As in the beginning of this section, first we choose

• •

$P_{1},\ldots,P_{r}\in E(\mathbb{Q})$ : generators of the free part of $E(\mathbb{Q})$, and put

• •

$L_{n}:=K_{n}\!\left([p^{n}]^{-1}P_{1},\ldots,[p^{n}]^{-1}P_{r}\right)$.

Next, we define

• •

$\widetilde{K}_{n}$ : the Hilbert $p$-class field, that is, the maximal unramified abelian $p$-extension of $K_{n}$, and

• •

I:=\Braket{I_{l}\,;l=p\mbox{ or }l\mid\Delta}\subset\operatorname{Gal}(L_{n}/K_{n}) : the subgroup generated by the inertia subgroups $I_{p}$ and $I_{l}$ for all prime number $l\mid\Delta$.

By class field theory (cf. [6],Chap. VI, Prop. 6.9), we have

[TABLE]

From the condition $(\mathbf{Full})$ and $p>2$, $\Phi:\operatorname{Gal}(L_{n}/K_{n})\to E[p^{n}]^{\oplus r}$ defined in (3.2) is bijective ([7], Thm. 2.4222 In [8], it is considered the case where $p\geq 11$. However, the proof of Theorem 2.4 in [8] works for $p>2$. , see also [5], Chap. V, Lem. 1) and hence

[TABLE]

Since the extension $L_{n}/K_{n}$ is unramified outside $\set{p,\infty}\cup\set{l\mid\Delta}$ ([10], Chap. VIII, Prop. 1.5 (b)), we have

[TABLE]

Using the upper bound of $\#I_{l}$ given in Lemma 3.1 (for $l=p$ under the condition $(\mathbf{Tor})$) and Lemma 3.2 (for $l\neq p$), we have

[TABLE]

Finally, Theorem 3.3 follows from the following inequalities:

[TABLE]

∎