Curves with few bad primes
over cyclotomic $\mathbb{Z}_{\ell}$ -extensions

Samir Siksek Mathematics Institute
University of Warwick
CV4 7AL
United Kingdom s.siksek@warwick.ac.uk and Robin Visser Mathematics Institute
University of Warwick
CV4 7AL
United Kingdom Robin.Visser@warwick.ac.uk

Abstract.

Let $K$ be a number field, and $S$ a finite set of non-archimedean places of $K$ , and write $\mathcal{O}_{S}^{\times}$ for the group of $S$ -units of $K$ . A famous theorem of Siegel asserts that the $S$ -unit equation $\varepsilon+\delta=1$ , with $\varepsilon$ , $\delta\in\mathcal{O}_{S}^{\times}$ , has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over $K$ with good reduction outside $S$ . Now instead of a number field, let $K=\mathbb{Q}_{\infty,\ell}$ which denotes the $\mathbb{Z}_{\ell}$ -cyclotomic extension of $\mathbb{Q}$ . We show that the $S$ -unit equation $\varepsilon+\delta=1$ , with $\varepsilon$ , $\delta\in\mathcal{O}_{S}^{\times}$ , has infinitely many solutions for $\ell\in\{2,3,5,7\}$ , where $S$ consists only of the totally ramified prime above $\ell$ . Moreover, for every prime $\ell$ , we construct infinitely many elliptic or hyperelliptic curves defined over $K$ with good reduction away from $2$ and $\ell$ . For certain primes $\ell$ we show that the Jacobians of these curves in fact belong to infinitely many distinct isogeny classes.

Key words and phrases:

Shafarevich conjecture, Abelian varieties, cyclic fields, cyclotomic fields, integral points

2010 Mathematics Subject Classification:

Primary 11G10, Secondary 11G05

Siksek is supported by the EPSRC grant Moduli of Elliptic curves and Classical Diophantine Problems (EP/S031537/1). Visser is supported by an EPSRC studentship (EP/V520226/1)

1. Introduction

Let $\ell$ be a rational prime and $r$ a positive integer. Write $\mathbb{Q}_{r,\ell}$ for the unique degree $\ell^{r}$ totally real subfield of $\cup_{n=1}^{\infty}\mathbb{Q}(\mu_{n})$ , where $\mu_{n}$ denotes the set of $\ell^{n}$ -th roots of $1$ . We let $\mathbb{Q}_{\infty,\ell}=\cup_{r}\mathbb{Q}_{r,\ell}$ ; this is the $\mathbb{Z}_{\ell}$ -cyclotomic extension of $\mathbb{Q}$ , and $\mathbb{Q}_{r,\ell}$ is called the $r$ -th layer of $\mathbb{Q}_{\infty,\ell}$ . Now let $K$ be a number field, and write $K_{\infty,\ell}=K\cdot\mathbb{Q}_{\infty,\ell}$ and $K_{r,\ell}=K\cdot\mathbb{Q}_{r,\ell}$ . To ease notation we shall sometimes write $K_{\infty}$ for $K_{\infty,\ell}$ . We write $\mathcal{O}_{\infty}$ (or $\mathcal{O}_{\infty,\ell}$ ) for the integers in $K_{\infty}$ (i.e. the integral closure of $\mathbb{Z}$ in $K_{\infty}$ ), and write $\mathcal{O}_{r}$ (or $\mathcal{O}_{r,\ell})$ for the integers of $K_{r,\ell}$ . Clearly $\mathcal{O}_{\infty,\ell}=\cup_{r}\mathcal{O}_{r,\ell}$ . The motivation for the present paper is a series of conjectures and theorems that suggest that the arithmetic of curves (respectively abelian varieties) over $K_{\infty}$ is similar to the arithmetic of curves (respectively abelian varieties) over $K$ . One of these is the following conjecture of Mazur [10], which in essence says that the Mordell–Weil theorem continues to hold over $K_{\infty}$ .

Conjecture (Mazur).

Let $A/K_{\infty}$ be an abelian variety. Then $A(K_{\infty})$ is finitely generated.

Another is a conjecture of Parshin and Zarhin [24, page 91] which is the analogue of Faltings’ theorem (Mordell conjecture) over $K_{\infty}$ .

Conjecture (Parshin and Zarhin).

Let $X/K_{\infty}$ be a curve of genus $\geq 2$ . Then $X(K_{\infty})$ is finite.

A third is the following theorem of Zarhin [25, Corollary 4.2], which asserts that the Tate homomorphism conjecture (also a theorem of Faltings [5] over number fields) continues to hold over $K_{\infty}$ .

Theorem (Zarhin).

Let $A$ , $B$ be abelian varieties defined over $K_{\infty,\ell}$ , and denote their respective $\ell$ -adic Tate modules by $T_{\ell}(A)$ , $T_{\ell}(B)$ . Then the natural embedding

\operatorname{Hom}_{K_{\infty}}(A,B)\otimes\mathbb{Z}_{\ell}\hookrightarrow\operatorname{Hom}_{\operatorname{Gal}(\overline{K_{\infty}}/K_{\infty})}(T_{\ell}(A),T_{\ell}(B))

is a bijection.

Mazur’s conjecture is now known to hold for certain elliptic curves. For example, if $E$ is an elliptic curve defined over $\mathbb{Q}$ then $E(\mathbb{Q}_{\infty})$ is finitely generated thanks to theorems of Kato, Ribet and Rohrlich [7, Theorem 1.5]. From this one can deduce [7, Theorem 1.24] that $X(\mathbb{Q}_{\infty})$ is finite for curves $X/\mathbb{Q}$ of genus $\geq 2$ equipped with a non-constant morphism to an elliptic curve $X\rightarrow E$ defined over $\mathbb{Q}$ . We also note that the conjecture of Parshin and Zarhin follows easily from Mazur’s conjecture and Faltings’ theorem. Indeed, using the Abel-Jacobi map we can deduce from Mazur’s conjecture that $X(K_{\infty})=X(K_{r})$ for suitably large $r$ , and we know that $X(K_{r})$ is finite by Faltings’ theorem.

It is natural to wonder whether other standard conjectures and theorems concerning the arithmetic of curves and abelian varieties over number fields continue to hold over $K_{\infty}$ . The purpose of this paper is to give counterexamples to potential generalizations of certain theorems of Siegel and Shafarevich to $K_{\infty}$ . A theorem of Siegel (e.g. [1, Theorem 0.2.8]) asserts that $(\mathbb{P}^{1}-\{0,1,\infty\})(\mathcal{O}_{K,S})$ is finite for any number field $K$ and any finite set of primes $S$ . We show that the corresponding statement over $\mathbb{Q}_{\infty,\ell}$ is false, at least for $\ell=2$ , $3$ , $5$ , $7$ . We denote by $\upsilon_{\ell}$ the totally ramified prime of $\mathbb{Q}_{\infty,\ell}$ above $\ell$ (the precise meaning of primes in infinite extensions of $\mathbb{Q}$ is clarified in Section 2).

Theorem 1.

Let $\ell=2$ , $3$ , $5$ or $7$ . Let

(1)

S\;=\;\begin{cases}\{\upsilon_{\ell}\}&\text{if $\ell=2$, $5$, $7$}\\ \emptyset&\text{if $\ell=3$}.\end{cases}

Let $\mathcal{O}_{S}$ denote the $S$ -integers of $\mathbb{Q}_{\infty,\ell}$ . Then $(\mathbb{P}^{1}-\{0,1,\infty\})(\mathcal{O}_{S})$ is infinite.

Remarks.

•

If $S=\emptyset$ then $\mathcal{O}_{S}=\mathcal{O}_{\infty}$ is the set of integers of $\mathbb{Q}_{\infty,\ell}$ . In [6] it is shown that $(\mathbb{P}^{1}-\{0,1,\infty\})(\mathcal{O}_{\infty})=\emptyset$ for $\ell\neq 3$ . The obstruction given in [6] for $\ell\neq 3$ is local in nature. In essence, Theorem 1 complements this result, showing that we can obtain infinitely many integral or $S$ -integral points in the absence of the local obstruction. The proof of Theorem 1 is constructive.
•

Theorem 1 strongly suggests that the conjecture of Parshin and Zarhin does not admit a straightforward generalization to the broader context of integral points on hyperbolic curves. We also remark that there is a critical difference over $K_{\infty}$ between complete curves $X$ of genus $\geq 2$ and $\mathbb{P}^{1}-\{0,1,\infty\}$ . For the former, the group of $K_{\infty}$ -points of the Jacobian is expected to be finitely generated by Mazur’s conjecture. For the latter, the analogue of the Jacobian is the generalized Jacobian which is $\mathbb{G}_{m}\times\mathbb{G}_{m}$ , and its group of $K_{\infty}$ -points is $(\mathbb{G}_{m}\times\mathbb{G}_{m})(K_{\infty})=\mathcal{O}_{\infty}^{\times}\times\mathcal{O}_{\infty}^{\times}$ , which is infinitely generated.

Variants of the proof of Theorem 1 give the following.

Theorem 2.

Let $\ell=2,3$ or $5$ . Let $S=\{\upsilon_{\ell}\}$ and write $\mathcal{O}_{S}$ for the $S$ -integers of $\mathbb{Q}_{\infty,\ell}$ . Let

k\;\in\;\begin{cases}\{1,2,3,4,5,6,7,8,10,12,24\}&\text{if $\ell=2,3$},\\ \{1,2,4\}&\text{if $\ell=5$}.\end{cases}

Then $(\mathbb{P}^{1}-\{0,k,\infty\})(\mathcal{O}_{S})$ is infinite.

Shafarevich’s conjecture asserts that for a number field $K$ , a dimension $n$ , a degree $d$ , and a finite set of places $S$ , there are only finitely many isomorphism classes of polarized abelian varieties defined over $K$ of dimension $n$ with degree $d$ polarization and with good reduction away from $S$ . This conjecture was proved by Shafarevich for elliptic curves (i.e. $n=1$ ) and by Faltings [5] in complete generality. If we replace $K$ by $\mathbb{Q}_{\infty,\ell}$ then the Shafarevich conjecture no longer holds. For example, consider

E_{\varepsilon}\;:\;\varepsilon Y^{2}=X^{3}-X

where $\varepsilon\in\mathcal{O}_{\infty}^{\times}$ . This elliptic curve has good reduction away from the primes above $2$ . Moreover, $E_{\varepsilon}$ , $E_{\delta}$ are isomorphic over $\mathbb{Q}_{\infty}$ if and only if $\varepsilon/\delta$ is a square in $\mathcal{O}_{\infty}^{\times}$ . As $\mathcal{O}_{\infty}^{\times}/(\mathcal{O}_{\infty}^{\times})^{2}$ is infinite, we deduce that there are infinitely many isomorphism classes of elliptic curves over $\mathbb{Q}_{\infty}$ with good reduction away from the primes above $2$ . It is however natural to wonder if a sufficiently weakened version of the Shafarevich conjecture continues to hold over $\mathbb{Q}_{\infty}$ . Indeed, the curves $E_{\varepsilon}$ in the above construction form a single $\overline{\mathbb{Q}}$ -isomorphism class. This it is natural to ask if, for suitable $\ell$ and finite set of primes $S$ , does the set of elliptic curves over $\mathbb{Q}_{\infty}$ with good reduction outside $S$ form infinitely many $\overline{\mathbb{Q}}$ -isomorphism classes?

Theorem 3.

Let $\ell=2$ , $3$ , $5$ , or $7$ . Let $S$ be given by (1) and let $S^{\prime}=S\cup\{\upsilon_{2}\}$ where $\upsilon_{2}$ is the unique prime of $\mathbb{Q}_{\infty,\ell}$ above $2$ . Then, there are infinitely many $\overline{\mathbb{Q}}$ -isomorphism classes of elliptic curves defined over $\mathbb{Q}_{\infty,\ell}$ with good reduction away from $S^{\prime}$ and with full 2-torsion in $\mathbb{Q}_{\infty,\ell}$ . Moreover, these elliptic curves form infinitely many distinct $\mathbb{Q}_{\infty,\ell}$ -isogeny classes.

Remarks

•

By [6, Lemma 2.1], a rational prime $p\neq\ell$ is inert in $\mathbb{Q}_{\infty,\ell}$ if and only if $p^{\ell-1}\not\equiv 1\pmod{\ell^{2}}$ . It follows from this that $2$ is inert in $\mathbb{Q}_{\infty,\ell}$ for $\ell=3$ , $5$ , $7$ and $11$ .
•

Faltings’ proof [5] of the Mordell conjecture can be considered to have three major steps. In the first step, Faltings proves the Tate homomorphism conjecture. In the second step, Faltings derives the Shafarevich conjecture from the Tate homomorphism conjecture, and in the final step Faltings uses the ‘Parshin trick’ to deduce the Mordell conjecture from the Shafarevich conjecture. Although Zarhin has extended the Tate homomorphism conjecture to $K_{\infty}$ , Theorem 3 suggests that there is no plausible strategy for proving the conjecture of Parshin and Zarhin by mimicking Faltings’ proof of the Mordell conjecture.

It is natural to wonder if the isogeny classes appearing in the proof of Theorem 3 are finite or infinite. Rather reassuringly they turn out to be finite.

Theorem 4.

Let $E$ be an elliptic curve over $\mathbb{Q}_{\infty,\ell}$ without potential complex multiplication. Then the $\mathbb{Q}_{\infty,\ell}$ -isogeny class of $E$ is finite.

The original version of Shafarevich’s conjecture [20], (also proved by Faltings [5, Korollar 1]) states that for a given number field $K$ , a genus $g$ and a finite set of places $S$ , there are only finitely many isomorphism classes of genus $g$ curves $C/K$ with good reduction away from $S$ . Again this statement becomes false if we replace $K$ by $\mathbb{Q}_{\infty,\ell}$ , for any prime $\ell$ .

Theorem 5.

Let $g\geq 2$ and let $\ell=3$ , $5$ , $7$ , $11$ or $13$ . There are infinitely many $\overline{\mathbb{Q}}$ -isomorphism classes of genus $g$ hyperelliptic curves over $\mathbb{Q}_{\infty,\ell}$ with good reduction away from $\{\upsilon_{2},\upsilon_{\ell}\}$ .

Theorem 6.

Let $\ell\geq 11$ be an odd prime and let $g=\lfloor\frac{\ell-3}{4}\rfloor$ . There are infinitely many $\overline{\mathbb{Q}}$ -isomorphism classes of genus $g$ hyperelliptic curves over $\mathbb{Q}_{\infty,\ell}$ with good reduction away from $\{\upsilon_{2},\upsilon_{\ell}\}$ . Moreover, if

\ell\in\{11,23,59,107,167,263,347,359\},

then the Jacobians of these curves form infinitely many distinct $\mathbb{Q}_{\infty,\ell}$ -isogeny classes.

The paper is structured as follows. In Section 2 we recall basic results on units and $S$ -units of the cyclotomic field $\mathbb{Q}(\zeta_{\ell^{n}})$ . In Sections 3–6 we employ identities between cyclotomic polynomials to give constructive proofs of Theorems 1 and 2. Section 7 gives a proof of Theorem 4, making use of a deep theorem of Kato to control the $\mathbb{Q}_{\infty,\ell}$ -points on certain modular curves. Section 8 uses the integral and $S$ -integral points on $\mathbb{P}^{1}-\{0,1,\infty\}$ furnished by Theorem 1 to construct infinite families of elliptic curves over $\mathbb{Q}_{\infty,\ell}$ for $\ell=2$ , $3$ , $5$ , $7$ , with good reduction away from $\{\upsilon_{2},\upsilon_{\ell}\}$ , which are used to give a proof of Theorem 3. Sections 9 and 10 give proofs of Theorems 5 and 6, making use of the relation, due to Kummer, between the class number of $\mathbb{Q}(\zeta_{\ell^{n}})^{+}$ , and the index of cyclotomic units in the full group of units.

We are grateful to Minhyong Kim for drawing our attention to the conjecture of Parshin and Zarhin, and to Alain Kraus and David Loeffler for useful discussions.

2. Units and $S$ -units of $\mathbb{Q}(\zeta)$

Let $K$ be a subfield of $\overline{\mathbb{Q}}$ . We denote the integers of $K$ (i.e. the integral closure of $\mathbb{Z}$ in $K$ ) by $\mathcal{O}(K)$ . Let $p$ be a rational prime. By a prime of $K$ above $p$ we mean a map $\upsilon:K\rightarrow\mathbb{Q}\cup\{\infty\}$ satisfying the following

•

$\upsilon(p)=1$ , $\upsilon(0)=\infty$ ;
•

$\upsilon|_{K^{\times}}:K^{\times}\rightarrow\mathbb{Q}$ is a homomorphism;
•

$\upsilon(1+b)=0$ whenever $\upsilon(b)>0$ .

Suppose $K=\cup K_{n}$ where $K_{0}\subset K_{1}\subset K_{2}\subset\cdots$ is a tower of number fields (i.e. finite extensions of $\mathbb{Q}$ ), with $K_{0}=\mathbb{Q}$ . One sees that the primes of $K$ above $p$ are in $1$ – $1$ correspondence with sequences $\{\mathfrak{p}_{n}\}$ where

•

$\mathfrak{p}_{n}$ is a prime ideal of $\mathcal{O}(K_{n})$ ;
•

$\mathfrak{p}_{n+1}\mid\mathfrak{p}_{n}\mathcal{O}(K_{n+1})$ ;
•

$\mathfrak{p}_{0}=p\mathbb{Z}$ .

Indeed, from $\upsilon$ one obtains the corresponding sequence $\{\mathfrak{p}_{n}\}$ via the formula $\mathfrak{p}_{n}=\{\alpha\in\mathcal{O}(K_{n})\;:\;\upsilon(\alpha)>0\}$ . Given a sequence $\{\mathfrak{p}_{n}\}$ , we can recover the corresponding $\upsilon$ by letting

\upsilon(\alpha)=\operatorname{ord}_{\mathfrak{p}_{n}}(\alpha)/\operatorname{ord}_{\mathfrak{p}_{n}}(p)

whenever $\alpha\in K_{n}^{\times}$ . Given a finite set of primes $S$ of $K$ , we define the $S$ -integers of $K$ to be the set $\mathcal{O}(K,S)$ of all $\alpha\in K$ such that $\upsilon(\alpha)\geq 0$ for every prime $\upsilon\notin S$ . We let $\mathcal{O}(K,S)^{\times}$ be the unit group of $\mathcal{O}(K,S)$ ; this is precisely the set of $\alpha\in K^{\times}$ such that $\upsilon(\alpha)=0$ for every prime $\upsilon\notin S$ . If $S=\emptyset$ then $\mathcal{O}(K,S)=\mathcal{O}(K)$ are the integers of $K$ and $\mathcal{O}(K,S)^{\times}=\mathcal{O}(K)^{\times}$ are the units of $K$ .

Fix a rational prime $\ell$ . For a positive integer $n$ , let $\zeta_{\ell^{n}}$ denote a primitive $\ell^{n}$ -th root of $1$ which is chosen so that

\zeta_{\ell^{n+1}}^{\ell}=\zeta_{\ell^{n}}.

Let $\Omega_{n,\ell}=\mathbb{Q}(\zeta_{\ell^{n}})$ ; this has degree $\varphi(\ell^{n})$ where $\varphi$ is Euler totient function. Let

\Omega_{\infty,\ell}=\bigcup_{n=1}^{\infty}\Omega_{n,\ell}.

The prime $\ell$ is totally ramified in each $\Omega_{n,\ell}$ , and we denote by $\lambda_{n}$ the unique prime ideal of $\mathcal{O}({\Omega_{n,\ell}})$ above $\ell$ . Thus

(2)

\ell\cdot\mathcal{O}({\Omega_{n,\ell}})\,=\,\lambda_{n}^{\varphi(\ell^{n})}.

We write $\upsilon_{\ell}$ for the unique prime of $\Omega_{\infty,\ell}$ above $\ell$ . For now fix $n\geq 1$ if $\ell\neq 2$ and $n\geq 2$ if $\ell=2$ . We recall that $\lambda_{n}=(1-\zeta_{\ell^{n}})\cdot\mathcal{O}({\Omega_{n,\ell}})$ . If $\ell\nmid s$ then $(1-\zeta_{\ell^{n}}^{s})\cdot\mathcal{O}({\Omega_{n,\ell}})=\lambda_{n}$ ; we can see this by applying the automorphism $\zeta_{\ell^{n}}\mapsto\zeta_{\ell^{n}}^{s}$ to (2).

Lemma 7.

Let $s$ be an integer and let $t=\operatorname{ord}_{\ell}(s)$ . Suppose $t<n$ . Then

(1-\zeta_{\ell^{n}}^{s})\cdot\mathcal{O}({\Omega_{n,\ell}})\;=\;\lambda_{n}^{\ell^{t}}.

Moreover,

\upsilon_{\ell}(1-\zeta_{\ell^{n}}^{s})=\frac{1}{\ell^{n-1-t}(\ell-1)}.

Proof.

Write $\zeta=\zeta_{\ell^{n}}$ . Note that $\zeta^{s}$ is a primitive $\ell^{n-t}$ -th root of $1$ . Thus

(1-\zeta^{s})\cdot\mathcal{O}(\Omega_{n-t,\ell})\;=\;\lambda_{n-t}.

As $\ell$ is totally ramified in $\Omega_{n,\ell}$ , we have

(1-\zeta^{s})\cdot\mathcal{O}(\Omega_{n,\ell})\;=\;\lambda_{n}^{[\Omega_{n,\ell}\,:\,\Omega_{n-t,\ell}]}\;=\;\lambda_{n}^{\ell^{t}}.

For the final part of the lemma,

\upsilon_{\ell}(1-\zeta^{s})=\frac{\operatorname{ord}_{\lambda_{n}}(1-\zeta^{s})}{\operatorname{ord}_{\lambda_{n}}(\ell)}=\frac{\ell^{t}}{\varphi(\ell^{n})}=\frac{1}{\ell^{n-1-t}(\ell-1)}.

∎

Cyclotomic units and $S$ -units

Write $V_{n}$ for the subgroup of $\mathcal{O}(\Omega_{n},\{\upsilon_{\ell}\})^{\times}$ generated by

\left\{\pm\zeta_{\ell^{n}},\quad 1-\zeta_{\ell^{n}}^{k}\;:\;1\leq k<\ell^{n}\right\}

and let

C_{n}=V_{n}\cap\mathcal{O}(\Omega_{n})^{\times}.

The group $C_{n}$ is called [22, Chapter 8] the group of cyclotomic units in $\Omega_{n}$ . We will often find it more convenient to work with group $V_{n}$ .

Lemma 8.

The abelian group $V_{n}/\langle\pm\zeta_{\ell}^{n}\rangle$ is free with basis

(3)

\left\{1-\zeta_{\ell^{n}}^{k}\;:\;1\leq k<\ell^{n}/2,\quad\ell\nmid k\right\}.

Proof.

The torsion subgroup of $V_{n}$ is the torsion subgroup of $\Omega_{n}^{\times}$ which is $\langle\pm\zeta_{\ell^{n}}\rangle$ . Thus $V_{n}/\langle\pm\zeta_{\ell^{n}}\rangle$ is torsion free. By definition of $V_{n}$ , the group $V_{n}/\langle\pm\zeta_{\ell^{n}}\rangle$ is generated by $1-\zeta_{\ell^{n}}^{k}$ with $\ell^{n}\nmid k$ . Write $k=\ell^{r}d$ with $\ell\nmid d$ ; thus $r<n$ . Suppose $r\geq 1$ . Then,

\begin{split}1-\zeta_{\ell^{n}}^{k}&=1-\zeta_{\ell^{n}}^{\ell^{r}d}\\ &=\prod_{i=0}^{\ell^{r}-1}(1-\zeta_{\ell^{n}}^{d}\zeta_{\ell^{r}}^{i})\qquad\text{using}\quad 1-X^{\ell^{r}}=\prod_{i=0}^{\ell^{r}-1}(1-\zeta_{\ell^{r}}^{i}X)\\ &=\prod_{i=0}^{\ell-1}(1-\zeta_{\ell^{n}}^{d+i\ell^{n-r}}).\end{split}

It follows that $V_{n}/\langle\pm\zeta_{\ell^{n}}\rangle$ is generated by $1-\zeta_{\ell^{n}}^{k}$ with $\ell\nmid k$ . If $\ell^{n}/2<k<\ell^{n}$ and $\ell\nmid k$ then

(4)

1-\zeta_{\ell^{n}}^{k}=-\zeta_{\ell^{n}}^{k}(1-\zeta_{\ell^{n}}^{\ell^{n}-k}).

Thus (3) certainly generates $V_{n}/\langle\pm\zeta_{\ell}^{n}\rangle$ . Note that (3) has cardinality $\varphi(\ell^{n})/2$ where $\varphi$ is the Euler totient function. It therefore suffices to show that $V_{n}$ has rank $\varphi(\ell^{n})/2$ . A well-known theorem [22, Theorem 8.3] states that $C_{n}$ has finite index in $\mathcal{O}(\Omega_{n})^{\times}$ and thus, by Dirichlet’s unit theorem, $C_{n}$ has rank $-1+\varphi(\ell^{n})/2$ . We note that $C_{n}$ is the kernel of the surjective homomorphism $V_{n}\rightarrow\mathbb{Z}$ , sending $\mu$ to $\operatorname{ord}_{\lambda_{n}}(\mu)$ . Thus $V_{n}$ has rank $\varphi(\ell^{n})/2$ completing the proof. ∎

Lemma 9.

Let $n\geq 2$ if $\ell\neq 2$ and $n\geq 3$ if $\ell=2$ . Then $V_{n-1}\subset V_{n}$ . Moreover,

\prod_{\begin{subarray}{c}1\leq k<\ell^{n}/2\\ \ell\nmid k\end{subarray}}(1-\zeta_{\ell^{n}}^{k})^{c_{k}}\;\in\;\langle\pm\zeta_{\ell^{n}},\,V_{n-1}\rangle

if and only if $c_{k}=c_{m}$ whenever $k\equiv m\pmod{\ell^{n-1}}$ .

Proof.

The group $V_{n-1}$ is generated, modulo roots of unity, by $1-\zeta_{\ell^{n-1}}^{d}$ with $\ell\nmid d$ . By the proof of Lemma 8,

\begin{split}1-\zeta_{\ell^{n-1}}^{d}=1-\zeta_{\ell^{n}}^{\ell d}=\prod_{i=0}^{\ell-1}(1-\zeta_{\ell^{n}}^{d+i\ell^{n-1}}).\end{split}

The lemma follows from Lemma 8. ∎

Given $a\in\mathbb{Z}_{\ell}$ , it makes sense to reduce $a$ modulo $\ell^{n}$ and therefore it makes sense to write $\zeta_{\ell^{n}}^{a}$ . We write $\{a\}_{n}$ for the unique integer satisfying

0\leq\{a\}_{n}<\ell^{n}/2,\qquad\{a\}_{n}\equiv\pm a\pmod{\ell^{n}}.

Lemma 10.

Let $a_{1},\dotsc,a_{r}\in\mathbb{Z}_{\ell}$ and $c_{1},\dotsc,c_{r}\in\mathbb{Z}$ . Suppose

(i)

$c_{1}\neq 0$ .
(ii)

$a_{1}\not\equiv 0\pmod{\ell}$ .
(iii)

$a_{1}\neq\pm a_{2},\pm a_{3},\cdots\pm a_{r}\pmod{\ell^{n}}$ .

Write

(5)

\varepsilon_{n}\;=\;\prod_{1\leq i\leq r}(1-\zeta_{\ell^{n}}^{a_{i}})^{c_{i}}.

Then, $\varepsilon_{n}\notin\langle\pm\zeta_{\ell^{n}},V_{n-1}\rangle$ for all sufficiently large $n$ .

Proof.

If $a_{j}\equiv 0\pmod{\ell}$ then $(1-\zeta_{\ell^{n}}^{a_{j}})\in V_{n-1}$ . We may therefore suppose $a_{j}\not\equiv 0\pmod{\ell}$ for all $j$ . Write

\delta_{n}\;=\;\prod_{1\leq i\leq r}\left(1-\zeta_{\ell^{n}}^{\{a_{i}\}_{n}}\right)^{c_{i}}.

In view of the identity (4) it will be sufficient to show that $\delta_{n}\notin\langle\pm\zeta_{\ell^{n}},V_{n-1}\rangle$ for $n$ sufficiently large. Also, in view of Lemma 9, it is sufficient to show for sufficiently large $n$ that $\{a_{1}\}_{n}\not\equiv\{a_{j}\}_{n}\pmod{\ell^{n}}$ for all $2\leq j\leq n$ . This is equivalent to $a_{1}\neq\pm a_{j}$ for $2\leq j\leq n$ which is hypothesis (iii). This completes the proof. ∎

The following corollary easily follows from Lemma 10.

Corollary 11.

Let $a_{1},\dotsc,a_{r}\in\mathbb{Z}_{\ell}$ and $c_{1},\dotsc,c_{r}\in\mathbb{Z}$ . Suppose

(i)

$c_{1}\equiv 1\pmod{2}$ .
(ii)

$a_{1}\not\equiv 0\pmod{\ell}$ .
(iii)

$a_{1}\neq\pm a_{2},\pm a_{3},\cdots\pm a_{r}\pmod{\ell^{n}}$ .

Let $\varepsilon_{n}$ be as in (5). Then, $\varepsilon_{n}\notin\langle\pm\zeta_{\ell^{n}},V_{n-1},V_{n}^{2}\rangle$ for all sufficiently large $n$ .

Units and $S$ -units from cyclotomic polynomials

For $m\geq 1$ , let $\Phi_{m}(X)\in\mathbb{Z}[X]$ be the $m$ -th cyclotomic polynomial defined by

\Phi_{m}(X)=\prod_{\begin{subarray}{c}1\leq i\leq m\\ (i,m)=1\end{subarray}}(X-\zeta_{m}^{i}).

These satisfy the identity [22, Chapter 2]

(6)

X^{m}-1=\prod_{d\mid m}\Phi_{d}(X).

It follows from the Möbius inversion formula that

(7)

\Phi_{m}(X)=\prod_{d\mid m}(X^{d}-1)^{\mu(m/d)}

where $\mu$ denotes the Möbius function.

Lemma 12.

Let $\ell$ be a prime and $n\geq 1$ . Let $m\geq 1$ , and suppose $\ell^{n}\nmid m$ .

(a)

$\Phi_{m}(\zeta_{\ell^{n}})\in V_{n}\subseteq\mathcal{O}(\Omega_{n,\ell},S)^{\times}$ , where $S=\{\upsilon_{\ell}\}$ .
(b)

If $m\neq\ell^{u}$ for all $u\geq 0$ , then $\Phi_{m}(\zeta_{\ell^{n}})\in C_{n}\subseteq\mathcal{O}(\Omega_{n,\ell})^{\times}$ .

Moreover,

\upsilon_{\ell}(\Phi_{\ell^{t}}(\zeta_{\ell^{n}}))\;=\;\begin{cases}\frac{1}{\ell^{n-1}(\ell-1)}&t=0\\ \frac{1}{\ell^{n-t}}&1\leq t\leq n-1.\\ \end{cases}

Proof.

Let $t=\operatorname{ord}_{\ell}(m)<n$ . Observe that $\Phi_{m}(X)\mid(X^{m}-1)$ . Hence $\Phi_{m}(\zeta_{\ell^{n}})\cdot\mathcal{O}(\Omega_{n,\ell})$ divides $(1-\zeta_{\ell^{n}}^{m})\cdot\mathcal{O}(\Omega_{n,\ell})$ . By Lemma 7 we have $(1-\zeta_{\ell^{n}}^{m})\cdot\mathcal{O}(\Omega_{n,\ell})\,=\,\lambda_{n}^{\ell^{t}}$ , giving (a).

For (b), write $m=\ell^{t}k$ where $k>1$ . Then $\Phi_{m}(X)$ divides the polynomial $(X^{m}-1)/(X^{\ell^{t}}-1)$ . Therefore $\Phi_{m}(\zeta)\cdot\mathcal{O}(\Omega_{n,\ell})$ divides

\frac{(1-\zeta_{\ell^{n}}^{m})}{(1-\zeta_{\ell^{n}}^{\ell^{t}})}\cdot\mathcal{O}(\Omega_{n,\ell})=\frac{\lambda_{n}^{\ell^{t}}}{\lambda_{n}^{\ell^{t}}}=1\cdot\mathcal{O}(\Omega_{n,\ell}).

Thus $\Phi_{m}(\zeta)$ is a unit, giving (b).

The final part of the Lemma follows from Lemma 7, and the formulae

\Phi_{\ell^{t}}(X)=\begin{cases}X-1&t=0\\ (X^{\ell^{t}}-1)/(X^{\ell^{t-1}}-1)&t\geq 1.\end{cases}

∎

Lemma 13.

Let $n\geq 2$ if $\ell\neq 2$ and $n\geq 3$ if $\ell=2$ . Then $V_{n}/\langle\pm\zeta_{\ell^{n}}\rangle$ is free with basis

\{\Phi_{m}(\zeta_{\ell^{n}})\;:\;1\leq m<\ell^{n}/2,\;\ell\nmid m\}.

Proof.

This follows from Lemma 8 thanks to identities (6) and (7). ∎

3. The $S$ -unit equation over $\mathbb{Q}(\zeta_{\ell^{n}})^{+}$

We continue with the notation of the previous section. In particular, let $K$ be a subfield of $\overline{\mathbb{Q}}$ and $S$ be a finite set of primes of $K$ . Let $k$ be a non-zero rational integer. We shall make frequent use of the correspondence between elements of $(\mathbb{P}^{1}-\{0,k,\infty\})(\mathcal{O}(K,S))$ and the set of solutions to the $S$ -unit equation

\varepsilon+\delta=k,\qquad\varepsilon,~{}\delta\in\mathcal{O}(K,S)^{\times},

sending $\varepsilon\in(\mathbb{P}^{1}-\{0,k,\infty\})(\mathcal{O}(K,S))$ to $(\varepsilon,\delta)=(\varepsilon,k-\varepsilon)$ .

Now, as before let $\ell$ be a rational prime, $n$ is a positive integer. If $\ell=2$ suppose $n\geq 2$ . Write $\zeta=\zeta_{\ell^{n}}$ . We write $\Omega_{n,\ell}^{+}=\mathbb{Q}(\zeta+1/\zeta)$ for the index $2$ totally real subfield of $\Omega_{n,\ell}$ . We write

\Omega_{\infty,\ell}^{+}=\bigcup_{n=1}^{\infty}\Omega_{n,\ell}^{+}.

In this section, for suitable $S$ , we produce solutions to the $S$ -unit equations over $\Omega_{\infty,\ell}^{+}$ .

As before, $\Phi_{m}$ denotes the $m$ -cyclotomic polynomial. It is convenient to record the first few $\Phi_{m}$ :

	$\displaystyle\Phi_{1}=X-1,\qquad\Phi_{2}=X+1,\qquad\Phi_{3}=X^{2}+X+1,$
	$\displaystyle\Phi_{4}=X^{2}+1,\qquad\Phi_{5}=X^{4}+X^{3}+X^{2}+X+1,$
	$\displaystyle\Phi_{6}=X^{2}-X+1,\qquad\Phi_{7}=X^{6}+X^{5}+X^{4}+X^{3}+X^{2}+X+1,$
	$\displaystyle\Phi_{8}=X^{4}+1,\qquad\Phi_{9}=X^{6}+X^{3}+1,\qquad\Phi_{10}=X^{4}-X^{3}+X^{2}-X+1.$

We shall call a polynomial $F\in\mathbb{Z}[X]$ super-cyclotomic if it is of the form $X^{m}f_{1}f_{2}\cdots f_{k}$ where each $f_{i}(X)$ is a cyclotomic polynomial. We know, thanks to Lemma 12, that if $F$ is super-cyclotomic and $\ell$ is a prime, then $F(\zeta_{\ell^{n}})\in\mathcal{O}(\Omega_{n},\{\upsilon_{\ell}\})^{\times}$ for $n$ sufficiently large. We wrote a short computer program that lists all super-cyclotomic polynomials of degree at most $20$ and searches for ternary relations of the form $F-G=kH$ with $F$ , $G$ , $H$ super-cyclotomic, $\gcd(F,G,H)=1$ and $k$ is a positive integer. Note that any such relation $F-G=kH$ gives points $\varepsilon_{n}=F(\zeta_{\ell^{n}})/H(\zeta_{\ell^{n}})\in(\mathbb{P}^{1}-\{0,k,\infty\})(\mathcal{O}(\Omega_{n},\{\upsilon_{\ell}\}))$ , for $n$ sufficiently large. We found the following ternary relations between super-cyclotomic polynomials.

(8)		$\displaystyle\Phi_{2}(X)^{2}-\Phi_{3}(X)\;=\;X;$
(9)		$\displaystyle\Phi_{2}(X)^{2}-\Phi_{4}(X)\;=\;2X;$
(10)		$\displaystyle\Phi_{2}(X)^{2}-\Phi_{6}(X)\;=\;3X;$
(11)		$\displaystyle\Phi_{2}(X)^{2}-\Phi_{1}(X)^{2}\;=\;4X;$
(12)		$\displaystyle\Phi_{2}(X)^{4}-\Phi_{10}(X)\;=\;5X\Phi_{3}(X);$
(13)		$\displaystyle\Phi_{2}^{2}(X)\Phi_{3}(X)-\Phi_{1}(X)^{2}\Phi_{6}(X)\;=\;6X\Phi_{4}(X);$
(14)		$\displaystyle\Phi_{7}(X)-\Phi_{1}(X)^{6}\;=\;7X\Phi_{6}(X)^{2};$
(15)		$\displaystyle\Phi_{2}(X)^{4}-\Phi_{1}(X)^{4}\;=\;8X\Phi_{4}(X);$
(16)		$\displaystyle\Phi_{2}(X)^{4}\Phi_{5}(X)-\Phi_{1}(X)^{4}\Phi_{10}(X)\;=\;10X\Phi_{4}(X)^{3}.$

From the identities (6) and (7) one easily sees that $F(X^{k})$ is super-cyclotomic for any super-cyclotomic polynomial $F$ and any positive integer $k$ , thus each of the nine identities above in fact yields an infinite family of identities. We pose the following open problems:

•

Are there ternary linear relations between super-cyclotomic polynomials that are outside these nine families?
•

Classify all ternary linear relations between super-cyclotomic polynomials.

Lemma 14.

Let $c:\Omega_{\ell}\rightarrow\Omega_{\ell}$ denote complex conjugation. Let $n\geq 1$ and let $\zeta=\zeta_{\ell^{n}}$ be an $\ell^{n}$ -th root of $1$ . Let $m\geq 1$ and suppose $\ell^{n}\nmid m$ . Then

\frac{\Phi_{m}(\zeta)^{c}}{\Phi_{m}(\zeta)}\;=\;\begin{cases}\zeta^{-\varphi(m)}&m\geq 2\\ -\zeta^{-1}&m=1.\end{cases}

Proof.

Note that $\zeta^{c}=\zeta^{-1}$ . So

\frac{\Phi_{1}(\zeta)^{c}}{\Phi_{1}(\zeta)}=\frac{\zeta^{-1}-1}{\zeta-1}=-\zeta^{-1},\qquad\frac{\Phi_{2}(\zeta)^{c}}{\Phi_{2}(\zeta)}=\frac{\zeta^{-1}+1}{\zeta+1}=\zeta^{-1}.

Let $m\geq 3$ . The polynomial $\Phi_{m}$ is monic of degree $\varphi(m)$ , and its roots are the primitive $m$ -th roots of $1$ which come in distinct pairs $\eta$ , $\eta^{-1}$ . Thus the trailing coefficient is $1$ . It follows that $X^{\varphi(m)}\Phi_{m}(X^{-1})$ is monic and has the same roots as $\Phi_{m}$ , therefore

\Phi_{m}(X)=X^{\varphi(m)}\Phi_{m}(X^{-1}).

Hence

\frac{\Phi_{m}(\zeta)^{c}}{\Phi_{m}(\zeta)}=\frac{\Phi_{m}(\zeta^{-1})}{\Phi_{m}(\zeta)}=\zeta^{-\varphi(m)}.

∎

Lemma 15.

Let $S=\{\upsilon_{\ell}\}$ . Let

k\in\{1,2,3,4,5,6,7,8,10\}.

Then $(\mathbb{P}^{1}-\{0,k,\infty\})(\mathcal{O}(\Omega_{\infty,\ell}^{+},S))$ is infinite.

Proof.

The proof makes use of identities (8)–(16). We prove the lemma for $k=10$ using the identity (16); the other cases are similar. Let $n\geq 3$ and let $\zeta=\zeta_{\ell^{n}}$ . Let

\varepsilon=\frac{\Phi_{2}(\zeta)^{4}\Phi_{5}(\zeta)}{\zeta\Phi_{4}(\zeta)^{3}},\qquad\delta=\frac{-\Phi_{1}(\zeta)^{4}\Phi_{10}(\zeta)}{\zeta\Phi_{4}(\zeta)^{3}}.

By the identity, $\varepsilon+\delta=10$ . By Lemma 12 we know that $\varepsilon$ , $\delta$ are $S$ -units. A priori, $\varepsilon$ , $\delta$ belong to $\Omega_{\infty,\ell}$ . However, an easy application of Lemma 14 shows that $\varepsilon^{c}=\varepsilon$ and $\delta^{c}=\delta$ , so $\varepsilon$ , $\delta\in\Omega_{\infty,\ell}^{+}$ . It follows that $\varepsilon$ is an $\mathcal{O}(\Omega_{\infty,\ell}^{+},S)$ -point on $\mathbb{P}^{1}-\{0,10,\infty\}$ . This point depends on $\zeta=\zeta_{\ell^{n}}$ . Let us make sure that we really obtain infinitely many such points as we vary $n$ . Write

\varepsilon_{n}\;=\;\frac{\Phi_{2}(\zeta_{\ell^{n}})^{4}\Phi_{5}(\zeta_{\ell^{n}})}{\zeta_{\ell^{n}}\Phi_{4}(\zeta_{\ell^{n}})^{3}}\;=\;\frac{(1-\zeta_{\ell^{n}}^{2})^{7}(1-\zeta_{\ell^{n}}^{5})}{\zeta_{\ell^{n}}(1-\zeta_{\ell^{n}})^{5}(1-\zeta_{\ell^{n}}^{4})^{3}}\in V_{n}.

To show that we obtain infinitely many distinct $\varepsilon_{n}$ it is enough to show that $\varepsilon_{n}\notin V_{n-1}$ for $n$ sufficiently large. This follows by an easy application of Lemma 9; to illustrate this let $\ell=5$ and suppose $\varepsilon_{n}\in V_{n-1}$ . Note that $1-\zeta_{5^{n}}^{5}\in V_{n-1}$ . It follows that

(1-\zeta_{5^{n}})^{-5}(1-\zeta_{5^{n}}^{2})^{7}(1-\zeta_{5^{n}}^{4})^{-3}\;\in\;\langle\pm\zeta_{\ell^{n}},V_{n-1}\rangle.

Now in the product on the left the exponent of $1-\zeta_{5^{n}}$ is $-5$ whereas the exponent of $1-\zeta_{5^{n}}^{1+5^{n-1}}$ is $0$ , contradicting Lemma 9. The proof is similar for $\ell=2$ , and for $\ell\neq 2$ , $5$ . It follows that we have infinitely many $\mathcal{O}(\Omega_{\infty,\ell}^{+},S)$ -points on $\mathbb{P}^{1}-\{0,10,\infty\}$ . ∎

Proof of Theorem 2 for $\ell=2$ and $3$

For $\ell=2$ , $3$ , we have $\Omega_{\infty,\ell}^{+}=\mathbb{Q}_{\infty,\ell}$ . Indeed, if $\ell=2$ then $\mathbb{Q}_{n,2}=\Omega_{n+2,2}^{+}$ and if $\ell=3$ then $\mathbb{Q}_{n,3}=\Omega_{n+1,3}^{+}$ . Therefore Theorem 2 with $\ell=2$ and $3$ follows immediately from Lemma 15 for $k\in\{1,2,3,4,5,6,7,8,10\}$ .

Also, if $\ell=2$ , then the infinitely many solutions $\varepsilon+\delta=6$ yields infinitely many solutions for $2\varepsilon+2\delta=12$ and $4\varepsilon+4\delta=24$ . And if $\ell=3$ , then the infinitely many solutions $\varepsilon+\delta=4$ yields infinitely many solutions $3\varepsilon+3\delta=12$ , and similarly infinitely many solutions $\varepsilon+\delta=8$ yields infinitely many solutions $3\varepsilon+3\delta=24$ . This proves Theorem 2 for $\ell=2$ , $3$ and $k\in\{12,24\}$ . ∎

Proof of Theorem 1 for $\ell=2$

Theorem 1 for $\ell=2$ is simply a special case of Theorem 2.∎

4. The unit equation over $\mathbb{Q}(\zeta_{\ell^{n}})^{+}$

For roots of unity $\alpha$ , $\beta$ , we let

	$\displaystyle E(\alpha,\beta)$	$\displaystyle=\frac{\alpha^{2}+\alpha^{-2}}{\left(\alpha\beta^{-1}+\alpha^{-1}\beta\right)(\alpha\beta+\alpha^{-1}\beta^{-1})}=\frac{\Phi_{8}(\alpha)}{\Phi_{4}(\alpha\beta)\Phi_{4}(\alpha/\beta)},$
	$\displaystyle F(\alpha,\beta)$	$\displaystyle=\frac{\beta^{2}+\beta^{-2}}{\left(\alpha\beta^{-1}+\alpha^{-1}\beta\right)(\alpha\beta+\alpha^{-1}\beta^{-1})}=\frac{\Phi_{8}(\beta)}{\Phi_{4}(\alpha\beta)\Phi_{4}(\beta/\alpha)}.$

We easily check that

(17)

E(\alpha,\beta)+F(\alpha,\beta)=1.

Lemma 16.

Suppose $\ell$ is odd and $n\geq 1$ . Let $\zeta=\zeta_{\ell^{n}}$ . Let $i$ , $j$ be integers satisfying $i$ , $j$ , $i+j$ , $i-j\not\equiv 0\pmod{\ell^{n}}$ . Then $E(\zeta^{i},\zeta^{j})$ , $F(\zeta^{i},\zeta^{j})\in\mathcal{O}(\Omega_{n,\ell}^{+})^{\times}$ , and satisfy the unit equation

(18)

\varepsilon+\delta=1,\qquad\varepsilon,~{}\delta\in\mathcal{O}(\Omega_{n,\ell}^{+})^{\times}.

Moreover,

(19)

\upsilon_{\ell}(E(\zeta^{i},\zeta^{j})-F(\zeta^{i},\zeta^{j}))=\frac{\ell^{\operatorname{ord}_{\ell}(i+j)}+\ell^{\operatorname{ord}_{\ell}(i-j)}}{\ell^{n-1}(\ell-1)}

Proof.

It is clear that $E(\zeta^{i},\zeta^{j})$ , $F(\zeta^{i},\zeta^{j})$ are fixed by complex conjugation $\zeta\mapsto\zeta^{-1}$ and so belong to $\Omega_{n,\ell}^{+}$ . By Lemma 12, $E(\zeta^{i},\zeta^{j})$ and $F(\zeta^{i},\zeta^{j})$ are units. It remains to check (19). We observe

E(\zeta^{i},\zeta^{j})-F(\zeta^{i},\zeta^{j})=\frac{(\zeta^{i-j}-\zeta^{j-i})(\zeta^{i+j}-\zeta^{-i-j})}{(\zeta^{i-j}+\zeta^{j-i})(\zeta^{i+j}+\zeta^{-i-j})}=\frac{(\zeta^{2(i-j)}-1)(\zeta^{2(i+j)}-1)}{\Phi_{4}(\zeta^{i-j})\Phi_{4}(\zeta^{i+j})}.

The denominator is a unit by Lemma 12. Now (19) follows from Lemma 7. ∎

Lemma 17.

Let $\ell$ be an odd prime. Then $(\mathbb{P}^{1}-\{0,1,\infty\})(\mathcal{O}(\Omega_{\infty,\ell}^{+}))$ is infinite.

Proof.

We deduce this from Lemma 16. Let us take for example $i=2$ and $j=1$ . Let $n\geq 2$ and let

\varepsilon_{n}=E(\zeta_{\ell^{n}}^{2},\zeta_{\ell^{n}}),\qquad\delta_{n}=F(\zeta_{\ell^{n}}^{2},\zeta_{\ell^{n}}).

By Lemma 16, $\varepsilon_{n}$ , $\delta_{n}\in\mathcal{O}(\Omega_{\infty,\ell}^{+})^{\times}$ and satisfy $\varepsilon_{n}+\delta_{n}=1$ . Thus $\varepsilon_{n}\in(\mathbb{P}^{1}-\{0,1,\infty\})(\mathcal{O}(\Omega_{\infty,\ell}^{+}))$ . Moreover,

\upsilon_{\ell}(2\varepsilon_{n}-1)=\upsilon_{\ell}(\varepsilon_{n}-\delta_{n})=\begin{cases}\frac{2}{\ell^{n-1}(\ell-1)}&\ell>3\\ \frac{2}{3^{n-1}}&\ell=3,\end{cases}

by (19). Thus $\varepsilon_{n}\neq\varepsilon_{m}$ whenever $n\neq m$ . Hence $(\mathbb{P}^{1}-\{0,1,\infty\})(\mathcal{O}(\Omega_{\infty,\ell}^{+}))$ is infinite. ∎

Remark. Lemma 17 applies only for $\ell$ odd; for $\ell=2$ it is easy to show that the statement is false. Indeed, and let $\eta_{n}$ be the prime ideal of $\mathcal{O}(\Omega_{n,2}^{+})$ above $2$ . Then $\mathcal{O}(\Omega_{n,2}^{+})/\eta_{n}\cong\mathbb{F}_{2}$ , and a solution to $\varepsilon+\delta=1$ with $\varepsilon$ , $\delta\in\mathcal{O}(\Omega_{n,2}^{\plus})^{\times}$ reduced modulo $\eta_{n}$ gives $1+1\equiv 1\pmod{2}$ which is impossible.

Proof of Theorem 1 for $\ell=3$

We recall that $\mathbb{Q}_{\infty,3}=\Omega_{\infty,3}^{+}$ . Therefore Theorem 1 for $\ell=3$ follows immediately from Lemma 17. ∎

5. The $S$ -unit equation over $\mathbb{Q}_{\infty,5}$

The purpose of the is section is to prove Theorems 1 and 2 for $\ell=5$ . These in fact follow immediately from the following lemma.

Lemma 18.

Let $\upsilon_{5}$ be the unique prime of $\mathbb{Q}_{\infty,5}$ above $5$ , and write $S=\{\upsilon_{5}\}$ . Then

(i)

$(\mathbb{P}^{1}-\{0,k,\infty\})(\mathcal{O}(\mathbb{Q}_{\infty,5},S))$ is infinite for $k=1$ , $4$ ;
(ii)

$(\mathbb{P}^{1}-\{0,2,\infty\})(\mathcal{O}(\mathbb{Q}_{\infty,5}))$ is infinite.

Proof.

Let $a\in\mathbb{Z}_{5}^{\times}$ be the element satisfying

a^{2}=-1,\qquad a\equiv 2\pmod{5};

such an element exists and is unique by Hensel’s Lemma. Let $\sigma:\Omega_{\infty,5}\rightarrow\Omega_{\infty,5}$ be the field automorphism satisfying

\sigma(\zeta_{5^{n}})\;=\;\zeta_{5^{n}}^{a}

for $n\geq 1$ . Note that $\sigma$ is an automorphism of order $4$ , and fixes a subfield of $\Omega_{\infty,5}$ of index $4$ . This subfield is precisely $\mathbb{Q}_{\infty,5}$ .

Let

\begin{gathered}F=(x_{1}x_{2}^{2}+x_{3}x_{4}^{2})(x_{1}^{2}x_{4}+x_{2}x_{3}^{2}),\\ G=(x_{1}^{2}x_{2}+x_{3}^{2}x_{4})(x_{1}x_{4}^{2}+x_{2}^{2}x_{3}),\\ H=(x_{1}-x_{3})(x_{2}-x_{4})(x_{1}x_{2}-x_{3}x_{4})(x_{1}x_{4}-x_{2}x_{3}).\end{gathered}

Observe $F$ , $G$ , $H$ are invariant under the $4$ -cycle $(x_{1},x_{2},x_{3},x_{4})$ . One can check that $F-G=H$ . Let $n\geq 2$ and write $\zeta=\zeta_{5^{n}}$ . Let

\varepsilon_{n}=\frac{F(\zeta,\zeta^{a},\zeta^{a^{2}},\zeta^{a^{3}})}{H(\zeta,\zeta^{a},\zeta^{a^{2}},\zeta^{a^{3}})},\qquad\delta_{n}=-\frac{G(\zeta,\zeta^{a},\zeta^{a^{2}},\zeta^{a^{3}})}{H(\zeta,\zeta^{a},\zeta^{a^{2}},\zeta^{a^{3}})}.

From the identity $F-G=H$ we have $\varepsilon_{n}+\delta_{n}=1$ . We shall show that $\varepsilon_{n}$ , $\delta_{n}\in\mathcal{O}(\mathbb{Q}_{\infty,5},S)^{\times}$ .

Since $\sigma$ cyclically permutes $\zeta,\zeta^{a},\zeta^{-1},\zeta^{-a}$ we conclude that $f(\zeta,\zeta^{a},\zeta^{-1},\zeta^{-a})\in\mathbb{Q}_{\infty,5}$ for $f=F$ , $G$ , $H$ . Thus $\varepsilon_{n}$ , $\delta_{n}\in\mathbb{Q}_{\infty,5}$ . Moreover,

	$\displaystyle F\;=\;x_{2}x_{3}^{3}x_{4}^{2}\cdot\Phi_{2}(x_{1}x_{2}^{2}/x_{3}x_{4}^{2})\Phi_{2}(x_{1}^{2}x_{4}/x_{2}x_{3}^{2}),$
	$\displaystyle G\;=\;x_{2}^{2}x_{3}^{3}x_{4}\cdot\Phi_{2}(x_{1}^{2}x_{2}/x_{3}^{2}x_{4})\Phi_{2}(x_{1}x_{4}^{2}/x_{2}^{2}x_{3}),$
	$\displaystyle H\;=\;x_{2}x_{3}^{3}x_{4}^{2}\cdot\Phi_{1}(x_{1}/x_{3})\cdot\Phi_{1}(x_{2}/x_{4})\cdot\Phi_{1}(x_{1}x_{2}/x_{3}x_{4})\cdot\Phi_{1}(x_{1}x_{4}/x_{2}x_{3}).$

Hence

\begin{split}\varepsilon_{n}&=\frac{\Phi_{2}(\zeta^{2+4a})\Phi_{2}(\zeta^{4-2a})}{\Phi_{1}(\zeta^{2})\Phi_{1}(\zeta^{2a})\Phi_{1}(\zeta^{2+2a})\Phi_{1}(\zeta^{2-2a})}\\ &=\frac{(1-\zeta^{4+8a})(1-\zeta^{8-4a})}{(1-\zeta^{2})(1-\zeta^{2a})(1-\zeta^{2+2a})(1-\zeta^{2-2a})(1-\zeta^{2+4a})(1-\zeta^{4-2a})}.\end{split}

and

\begin{split}\delta_{n}&=\frac{-\zeta^{2a}\Phi_{2}(\zeta^{4+2a})\Phi_{2}(\zeta^{2-4a})}{\Phi_{1}(\zeta^{2})\Phi_{1}(\zeta^{2a})\Phi_{1}(\zeta^{2+2a})\Phi_{1}(\zeta^{2-2a})}\\ &=\frac{-\zeta^{2a}(1-\zeta^{8+4a})(1-\zeta^{4-8a})}{(1-\zeta^{2})(1-\zeta^{2a})(1-\zeta^{2+2a})(1-\zeta^{2-2a})(1-\zeta^{4+2a})(1-\zeta^{2-4a})}.\end{split}

We checked, using the fact that $a\equiv 7\pmod{25}$ , that the exponents of $\zeta$ in the above expressions for $\varepsilon_{n}$ and $\delta_{n}$ all have $5$ -adic valuation $0$ or $1$ . It follows from this that $\varepsilon_{n}$ , $\delta_{n}\in V_{n}\subseteq\mathcal{O}(\Omega_{n},S)^{\times}$ for $n\geq 2$ . Hence $\varepsilon_{n}$ , $\delta_{n}\in\mathbb{Q}_{\infty,5}\cap\mathcal{O}(\Omega_{n},S)^{\times}=\mathcal{O}(\mathbb{Q}_{\infty,5},S)^{\times}$ for $n\geq 2$ . To complete the proof of the lemma for $k=1$ it is enough to show that $\varepsilon_{n}\neq\varepsilon_{m}$ for $n>m$ , and for this it is enough to show that $\varepsilon_{n}\notin\langle\pm\zeta_{5^{n}},V_{n-1}\rangle$ for $n\geq 2$ . Since $a\equiv 7\pmod{25}$ we see that

4+8a\equiv 10,\qquad 8-4a\equiv 5,\qquad 2+4a\equiv 5,\qquad 4-2a\equiv 15\pmod{25}.

Thus the factors

1-\zeta^{4+8a},\qquad 1-\zeta^{8-4a},\qquad 1-\zeta^{2+4a},\qquad 1-\zeta^{4-2a}

all belong to $V_{n-1}$ . Hence it is enough to show that

(20)

(1-\zeta^{2})(1-\zeta^{2a})(1-\zeta^{2+2a})(1-\zeta^{2-2a})

does not belong to $\langle\pm\zeta_{5^{n}},V_{n-1}\rangle$ . However, the exponents $2$ , $2a$ , $2+2a$ , $2-2a$ are respectively $2$ , $4$ , $1$ , $3$ modulo $5$ , and hence certainly distinct modulo $5^{n-1}$ . It follows from Lemma 9 that the product (20) does not belong to $\langle\pm\zeta_{5^{n}},V_{n-1}\rangle$ completing the proof for $k=1$ .

The proof for $k=2$ is similar, and is based on the identity $F-G=2H$ where

\begin{gathered}F\;=\;(x_{1}^{2}+x_{1}x_{3}+x_{3}^{2})(x_{2}^{2}+x_{2}x_{4}+x_{4}^{2})\;=\;x_{3}^{2}x_{4}^{2}\cdot\Phi_{3}(x_{1}/x_{3})\cdot\Phi_{3}(x_{2}/x_{4}),\\ G\;=\;(x_{1}^{2}-x_{1}x_{3}+x_{3}^{2})(x_{2}^{2}-x_{2}x_{4}+x_{4}^{2})\;=\;x_{3}^{2}x_{4}^{2}\cdot\Phi_{6}(x_{1}/x_{3})\cdot\Phi_{6}(x_{2}/x_{4}),\\ H\;=\;(x_{1}x_{4}+x_{2}x_{3})(x_{1}x_{2}+x_{3}x_{4})\;=\;x_{2}x_{3}^{2}x_{4}\cdot\Phi_{2}(x_{1}x_{4}/x_{2}x_{3})\cdot\Phi_{2}(x_{1}x_{2}/x_{3}x_{4}),\end{gathered}

and likewise the proof for $k=4$ is based on the identity $F-G=4H$ where

\begin{gathered}F\;=\;(x_{1}+x_{3})^{2}(x_{2}+x_{4})^{2}=x_{3}^{2}x_{4}^{2}\cdot\Phi_{2}(x_{1}/x_{3})^{2}\Phi_{2}(x_{2}/x_{4})^{2},\\ G\;=\;(x_{1}-x_{3})^{2}(x_{2}-x_{4})^{2}=x_{3}^{2}x_{4}^{2}\cdot\Phi_{1}(x_{1}/x_{3})^{2}\Phi_{1}(x_{2}/x_{4})^{2},\\ H\;=\;(x_{1}x_{2}+x_{3}x_{4})(x_{1}x_{4}+x_{2}x_{3})=x_{2}x_{3}^{2}x_{4}\cdot\Phi_{2}(x_{1}x_{2}/x_{3}x_{4})\Phi_{2}(x_{1}x_{4}/x_{2}x_{3}).\end{gathered}

∎

Remark. It is appropriate to remark on how the identities in the above proof were found. Write

{\Psi}_{m}(X,Y)=Y^{\varphi(m)}\Phi_{m}(X/Y)

for the homogenization of the $m$ -th cyclotomic polynomial. Now consider

f(x_{1},x_{2},x_{3},x_{4})=\Psi_{m}(u,v)

where $u$ , $v$ are monomials in variables $x_{1},x_{2},x_{3},x_{4}$ . Let $\ell$ be a prime. We see that evaluating any such $f$ at $(\zeta^{\alpha},\zeta^{\beta},\zeta^{\gamma},\zeta^{\delta})$ gives an element of $V_{n}$ (provided that it does not vanish). We considered products of such $f$ of total degree up to $20$ and picked out ones that are invariant under the $4$ -cycle $(x_{1},x_{2},x_{3},x_{4})$ , and searched for ternary relations between them. This yielded the identities used in the above proof.

Proof of Theorems 1 and 2 for $\ell=5$ .

Theorems 1 and 2 for $\ell=5$ follow immediately from Lemma 18. ∎

6. The $S$ -unit equation over $\mathbb{Q}_{\infty,7}$

Lemma 19.

Let $\upsilon_{7}$ be the unique prime of $\mathbb{Q}_{\infty,7}$ above $7$ , and write $S=\{\upsilon_{7}\}$ . Then $(\mathbb{P}^{1}-\{0,1,\infty\})(\mathcal{O}(\mathbb{Q}_{\infty,7},S))$ is infinite.

Proof.

In view of the proof of Lemma 18, it would be natural to seek polynomials $F$ , $G$ , $H$ in variables $x_{1},\dotsc,x_{6}$ satisfying the following properties

•

$F\pm G=H$ ;
•

$F$ , $G$ , $H$ are invariant under the $6$ -cycle $(x_{1},x_{2},\dotsc,x_{6})$ ;
•

each is a product of polynomials

$f(x_{1},x_{2},\dotsc,x_{6})=\Psi_{m}(u,v)$

with $u$ , $v$ monomials in $x_{1},\dotsc,x_{6}$ .

Unfortunately, an extensive search has failed to produce any such triple of polynomials. We therefore need to proceed a little differently.

Let $a\in\mathbb{Z}_{7}$ be the element satisfying

a^{2}+a+1=0,\qquad a\equiv 2\pmod{7};

such an element exists and is unique by Hensel’s Lemma. Let $\sigma$ , $c:\Omega_{\infty,7}\rightarrow\Omega_{\infty,7}$ be the field automorphisms satisfying

\sigma(\zeta_{7^{n}})\;=\;\zeta_{7^{n}}^{a},\qquad c(\zeta_{7^{n}})\;=\;\zeta_{7^{n}}^{-1}

for $n\geq 1$ . Then $\mathbb{Q}_{\infty,7}$ is the field fixed by the subgroup of $\operatorname{Gal}(\Omega_{\infty,7}/\mathbb{Q})$ generated by $\sigma$ and $c$ . We work with polynomials in variables $x_{1}$ , $x_{2}$ , $x_{3}$ . Let

\begin{gathered}F\;=\;(x_{1}x_{2}^{2}+x_{3}^{3})(x_{2}x_{3}^{2}+x_{1}^{3})(x_{3}x_{1}^{2}+x_{2}^{3})\\ G\;=\;(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{1})(x_{1}x_{2}-x_{3}^{2})(x_{2}x_{3}-x_{1}^{2})(x_{3}x_{1}-x_{2}^{2})\\ H\;=\;(x_{1}^{2}x_{2}+x_{3}^{3})(x_{2}^{2}x_{3}+x_{1}^{3})(x_{3}^{2}x_{1}+x_{2}^{3}).\\ \end{gathered}

These satisfy the identity $F-G=H$ . Moreover, they are invariant under the $3$ -cycle $(x_{1},x_{2},x_{3})$ and all the factors are of the form $\Psi_{m}(u,v)$ where $m=1$ or $2$ , and where $u$ , $v$ are suitable monomials in $x_{1}$ , $x_{2}$ , $x_{3}$ . Evaluating any of $F$ , $G$ , $H$ at $(\zeta,\zeta^{a},\zeta^{a^{2}})$ yields an $S$ -unit belonging to $\Omega_{n,7}^{\langle\sigma\rangle}$ . Now we let

F^{\prime}=\frac{F(x_{1}^{2},x_{2}^{2},x_{3}^{2})}{x_{1}^{6}x_{2}^{6}x_{3}^{6}},\qquad G^{\prime}=\frac{G(x_{1}^{2},x_{2}^{2},x_{3}^{2})}{x_{1}^{6}x_{2}^{6}x_{3}^{6}},\qquad H^{\prime}=\frac{H(x_{1}^{2},x_{2}^{2},x_{3}^{2})}{x_{1}^{6}x_{2}^{6}x_{3}^{6}}.

Observe that the rational functions $F^{\prime}$ , $G^{\prime}$ , $H^{\prime}$ satisfy $F^{\prime}-G^{\prime}=H^{\prime}$ and are moreover invariant under the $3$ -cycle $(x_{1},x_{2},x_{3})$ . Moreover, $F^{\prime}$ , $G^{\prime}$ , $H^{\prime}$ evaluated at $(\zeta,\zeta^{a},\zeta^{a^{2}})$ yield $S$ -units belonging to $\Omega_{n,7}^{\langle\sigma\rangle}$ . We need to check that these in fact belong to $\mathbb{Q}_{n-1,7}=\Omega_{n,7}^{\langle\sigma,c\rangle}$ and so we need to check that these expressions are invariant under $c$ . This follows immediately on observing that $F^{\prime}$ , $G^{\prime}$ , $H^{\prime}$ may be rewritten as

\begin{gathered}F^{\prime}\;=\;\left(\frac{x_{1}x_{2}^{2}}{x_{3}^{3}}+\frac{x_{3}^{3}}{x_{1}x_{2}^{2}}\right)\left(\frac{x_{2}x_{3}^{2}}{x_{1}^{3}}+\frac{x_{1}^{3}}{x_{2}x_{3}^{2}}\right)\left(\frac{x_{3}x_{1}^{2}}{x_{2}^{3}}+\frac{x_{2}^{3}}{x_{3}x_{1}^{2}}\right)\\ G^{\prime}\;=\;\left(\frac{x_{1}}{x_{2}}-\frac{x_{2}}{x_{1}}\right)\left(\frac{x_{2}}{x_{3}}-\frac{x_{3}}{x_{2}}\right)\left(\frac{x_{3}}{x_{1}}-\frac{x_{1}}{x_{3}}\right)\left(\frac{x_{1}x_{2}}{x_{3}^{2}}-\frac{x_{3}^{2}}{x_{1}x_{2}}\right)\left(\frac{x_{2}x_{3}}{x_{1}^{2}}-\frac{x_{1}^{2}}{x_{2}x_{3}}\right)\left(\frac{x_{3}x_{1}}{x_{2}^{2}}-\frac{x_{2}^{2}}{x_{3}x_{1}}\right)\\ H^{\prime}\;=\;\left(\frac{x_{1}^{2}x_{2}}{x_{3}^{3}}+\frac{x_{3}^{3}}{x_{1}^{2}x_{2}}\right)\left(\frac{x_{2}^{2}x_{3}}{x_{1}^{3}}+\frac{x_{1}^{3}}{x_{2}^{2}x_{3}}\right)\left(\frac{x_{3}^{2}x_{1}}{x_{2}^{3}}+\frac{x_{2}^{3}}{x_{3}^{3}x_{1}}\right).\\ \end{gathered}

Thus $F^{\prime}$ , $G^{\prime}$ , $H^{\prime}$ evaluated at $(\zeta,\zeta^{a},\zeta^{a^{2}})$ yield elements of $\mathcal{O}(\mathbb{Q}_{\infty,7},S)^{\times}$ . We write

\varepsilon_{n}=\frac{F^{\prime}(\zeta,\zeta^{a},\zeta^{a^{2}})}{H^{\prime}(\zeta,\zeta^{a},\zeta^{a^{2}})},\qquad\delta_{n}=-\frac{G^{\prime}(\zeta,\zeta^{a},\zeta^{a^{2}})}{H^{\prime}(\zeta,\zeta^{a},\zeta^{a^{2}})}.

Then $\varepsilon_{n}$ , $\delta_{n}$ belong to $\mathcal{O}(\mathbb{Q}_{\infty,7},S)^{\times}$ and satisfy $\varepsilon_{n}+\delta_{n}=1$ . In fact it is straightforward to check that $\varepsilon_{n}\notin\langle\pm\zeta_{7^{n}},V_{n-1}\rangle$ , from which it follows that $\varepsilon_{n}\neq\varepsilon_{m}$ for $n>m$ . The details are similar to those of the proof of Lemma 18 and we omit them. ∎

7. Isogeny classes of elliptic curves over $\mathbb{Q}_{\infty,\ell}$

The purpose of this section is to prove Theorem 4. Since isogenous elliptic curves share the same set of bad primes, the corresponding theorem over number fields is an immediate consequence of Shafarevich’s theorem. However, as we intend to show in the following section, Shafarevich’s theorem does not generalize to elliptic curves over $\mathbb{Q}_{\infty,\ell}$ . We shall instead rely on a theorem of Kato to control $\mathbb{Q}_{\infty,\ell}$ -points on certain modular Jacobians.

Our first lemma shows that there are only finitely many primes that can divide the degree of a cyclic isogeny of $E$ .

Lemma 20.

Let $\ell$ be a prime and let $E/\mathbb{Q}_{\infty,\ell}$ be an elliptic curve without potential complex multiplication. Then there is a constant $B$ , depending on $E$ , such that for primes $p\geq B$ , the elliptic curve $E$ has no $p$ -isogenies defined over $\mathbb{Q}_{\infty,\ell}$ .

Proof.

Let $n$ be the least positive integer such that $E$ admits a model defined over $\mathbb{Q}_{n,\ell}$ . By a famous theorem of Serre [14], there is a constant $B$ , depending on $E$ , such that for $p\geq B$ the mod $p$ representation

\overline{\rho}_{E,p}\;:\;\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{n,\ell})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})

is surjective. We may suppose that $B\geq 5$ . Thus, for $p\geq B$ , the Galois group $\operatorname{Gal}(\mathbb{Q}_{n,\ell}(E[p])/\mathbb{Q}_{n,\ell})$ is isomorphic to $\operatorname{GL}_{2}(\mathbb{F}_{p})$ which is non-solvable. We will show that $E$ has no $p$ -isogeny defined over $\mathbb{Q}_{\infty,\ell}$ . Suppose otherwise. Then such an isogeny is in fact defined over $\mathbb{Q}_{m,\ell}$ for some $m\geq n$ . It follows that the extension $\mathbb{Q}_{m,\ell}(E[p])/\mathbb{Q}_{m,\ell}$ has Galois group isomorphic to a subgroup of a Borel subgroup of $\operatorname{GL}_{2}(\mathbb{F}_{p})$ , with is solvable. As the extension $\mathbb{Q}_{m,\ell}/\mathbb{Q}_{n,\ell}$ is cyclic, we conclude that $\mathbb{Q}_{m,\ell}(E[p])/\mathbb{Q}_{n,\ell}$ is solvable. However, this contains the non-solvable subextension $\mathbb{Q}_{n,\ell}(E[p])/\mathbb{Q}_{n,\ell}$ , giving a contradiction. ∎

We shall make use of the following theorem of Kato [9, Theorem 14.4] building on work of Rohrlich [11].

Theorem 21 (Kato).

Let $\ell$ be a prime. Let $A$ be an abelian variety defined over $\mathbb{Q}$ and admitting a surjective map $J_{1}(N)\rightarrow A$ for some $N\geq 1$ . Then $A(\mathbb{Q}_{\infty,\ell})$ is finitely generated.

Lemma 22.

Let $p$ , $\ell$ be primes. Let $E$ be an elliptic curve defined over $\mathbb{Q}_{\infty,\ell}$ without potential complex multiplication. Then, for $m$ sufficiently large, $E$ has no $p^{m}$ -isogenies defined over $\mathbb{Q}_{\infty,\ell}$ .

Proof.

Let $r$ be the least positive integer such that the modular curve $X=X_{0}(p^{r})$ has genus at least $2$ , and write $J=J_{0}(p^{r})$ for the corresponding modular Jacobian. It follows from Kato’s theorem that $J(\mathbb{Q}_{\infty,\ell})$ is finitely generated, and therefore that $J(\mathbb{Q}_{\infty,\ell})=J(\mathbb{Q}_{n,\ell})$ for some $n\geq 1$ . Consider the Abel-Jacobi map

X\hookrightarrow J,\qquad P\mapsto[P-\infty]

where $\infty\in X(\mathbb{Q})$ denotes the infinity cusp. It follows from this embedding that $X(\mathbb{Q}_{\infty,\ell})=X(\mathbb{Q}_{n,\ell})$ . By Faltings’ theorem, this set is finite.

Let $k=\#X(\mathbb{Q}_{\infty,\ell})$ and let $s=kr$ . To prove the lemma we in fact show that $E$ has no cyclic isogenies of degree $p^{s}$ defined over $\mathbb{Q}_{\infty,\ell}$ . Suppose otherwise, and let $\psi:E\rightarrow E^{\prime}$ be a cyclic isogeny of degree $p^{s}$ defined over $\mathbb{Q}_{\infty,\ell}$ . Then, we may factor $\psi$ into a sequence of cyclic isogenies defined over $\mathbb{Q}_{\infty,\ell}$

E=E_{0}\;\xrightarrow{\psi_{1}}\;E_{1}\;\xrightarrow{\psi_{2}}\;E_{2}\;\cdots\xrightarrow{\psi_{k}}\;E_{k}=E^{\prime}

where $\psi_{i}$ is degree $p^{r}$ . Note that $E_{i}$ and $E_{j}$ are non-isomorphic over $\overline{\mathbb{Q}}$ for $i\neq j$ ; indeed they are related by a cyclic isogeny and $E$ does not have potential complex multiplication. Thus the elliptic curves $E_{0},E_{1},\dotsc,E_{k}$ support distinct $\mathbb{Q}_{\infty,\ell}$ -points on $X=X_{0}(p^{r})$ . This contradicts the fact that $\#X(\mathbb{Q}_{\infty,\ell})=k$ . ∎

Remark. A famous theorem of Serre [13, Section 2.1] asserts that the $p$ -adic Tate module of a non-CM elliptic curve defined over a number field is irreducible. It is in fact possible to deduce Lemma 22 from Serre’s theorem for $\ell\neq p$ , but we have been unable to do this for $\ell=p$ .

Proof of Theorem 4.

Let $E^{\prime}$ belong to the $\mathbb{Q}_{\infty,\ell}$ -isogeny class of $E$ . Let $\psi:E\rightarrow E^{\prime}$ be an isogeny defined over $\mathbb{Q}_{\infty,\ell}$ . This has kernel of the form $\mathbb{Z}/a\times\mathbb{Z}/ab$ where $a$ , $b$ are positive integers, and so it can be factored into a composition

E\rightarrow E/E[a]\cong E\rightarrow E^{\prime}

where the final morphism is cyclic of degree $b$ . Thus to prove the proposition, it is enough to show that $E$ has finitely many cyclic isogenies defined over $\mathbb{Q}_{\infty,\ell}$ . The degree of any such isogeny is divisible by primes $p<B$ where $B$ is as in Lemma 20. Also, for any $p<B$ , we know the exponent of $p$ in the degree of a cyclic isogeny $E\rightarrow E^{\prime}$ is bounded by Lemma 22. Thus there are finitely many cyclic isogenies of $E$ defined over $\mathbb{Q}_{\infty,\ell}$ . ∎

8. From $S$ -unit equations to elliptic curves

The aim of this section is to prove Theorem 3. We start by recalling a few facts about Legendre elliptic curves (Proposition III.1.7 of [17] and its proof). Let $K$ be a field of characteristic $\neq 2$ and let $\lambda\in(\mathbb{P}^{1}-\{0,1,\infty\})(K)$ . Associated to $\lambda$ is the Legendre elliptic curve

E_{\lambda}\;:\;Y^{2}=X(X-1)(X-\lambda).

This model respectively has discriminant and $j$ -invariant

(21)

\Delta=16\lambda^{2}(1-\lambda)^{2},\qquad j=\frac{64(\lambda^{2}-\lambda+1)^{3}}{\lambda^{2}(1-\lambda)^{2}}.

Moreover, for $\lambda$ , $\mu\in(\mathbb{P}^{1}-\{0,1,\infty\})(K)$ , the Legendre elliptic curves $E_{\lambda}$ and $E_{\mu}$ are isomorphic over $K$ (or over $\overline{K}$ ) if and only if

\mu\;\in\;\left\{\lambda,\,\frac{1}{\lambda},\,1-\lambda,\,\frac{1}{1-\lambda},\,\frac{\lambda}{\lambda-1},\,\frac{\lambda-1}{\lambda}\right\}.

Now let $K$ be a number field and $S$ a finite set of non-archimedean places. We let $S^{\prime}$ be the set of non-archimedean places which are either in $S$ or above $2$ . We let $\lambda\in(\mathbb{P}^{1}-\{0,1,\infty\})(\mathcal{O}(K,S))$ . Then $\lambda$ , $1-\lambda\in\mathcal{O}(K,S)^{\times}$ . It follows from the expression for the discriminant that $E_{\lambda}$ has good reduction away for $S^{\prime}$ .

Proof of Theorem 3

Let $\ell=2$ , $3$ , $5$ or $7$ . Let $S$ be given by (1) and let $S^{\prime}=S\cup\{\upsilon_{2}\}$ as in the statement of Theorem 3. In proving Theorem 1 we constructed, for each positive integer $n$ , elements $\varepsilon_{n}$ , $\delta_{n}=1-\varepsilon_{n}$ , belonging $\mathbb{Q}_{\infty,\ell}\cap V_{n}\subseteq\mathcal{O}(\mathbb{Q}_{\infty,\ell},S)^{\times}$ , and moreover verified, for $n\geq 2$ , that $\varepsilon_{n}\notin\langle\zeta_{\ell^{n}},V_{n-1}\rangle$ . We let

E_{n}\;:\;Y^{2}=X(X-1)(X-\varepsilon_{n}).

Then $E_{n}$ is defined over $\mathbb{Q}_{\infty,\ell}$ and has good reduction away from $S^{\prime}$ . We claim, for $n>m$ , that $E_{n}$ and $E_{m}$ are not isomorphic, even over $\overline{\mathbb{Q}}$ . To see this, suppose $E_{n}$ and $E_{m}$ are isomorphic. Then $\varepsilon_{n}$ equals one of $\varepsilon_{m}^{\pm 1}$ , $\delta_{m}^{\pm 1}$ , $(-\varepsilon_{m}\delta_{m})^{\pm 1}$ . This gives a contradiction as all of these belong to $\langle\pm\zeta_{\ell^{n}},V_{n-1}\rangle$ . This proves the claim.

It remains to show that the $E_{n}$ form infinitely many isogeny classes over $\mathbb{Q}_{\infty,\ell}$ . However, this immediately follows from Theorem 4 and the following lemma. ∎

Lemma 23.

For $n$ sufficiently large, $E_{n}$ does not have potential complex multiplication.

Proof.

Suppose $E_{n}$ has potential complex multiplication by an order $R$ in an imaginary quadratic field $K$ . Write $j=j(E_{n})$ . By standard CM theory [16, Theorem 5.7], we know that $\operatorname{Gal}(K(j)/K)\cong\operatorname{Pic}(R)$ and $[\mathbb{Q}(j):\mathbb{Q}]=[K(j):K]$ . Since in our case $\mathbb{Q}(j)/\mathbb{Q}$ is Galois, $\operatorname{Gal}(\mathbb{Q}(j)/\mathbb{Q})\cong\operatorname{Gal}(K(j)/K)\cong\operatorname{Pic}(R)$ . However, $\mathbb{Q}(j)\subset\mathbb{Q}_{\infty,\ell}$ is totally real. It follows [16, page 124] that $\operatorname{Pic}(R)$ is an elementary abelian $2$ -group. Since $\mathbb{Q}(j)\subset\mathbb{Q}_{\infty,\ell}$ , the Galois group of $\mathbb{Q}(j)/\mathbb{Q}$ is cyclic of order $\ell^{n}$ for some $n$ . Thus, $j\in\mathbb{Q}$ if $\ell\neq 2$ , and $j\in\mathbb{Q}_{1,2}=\mathbb{Q}(\sqrt{2})$ if $\ell=2$ . However, from the expression for $j$ in (21) we know that $[\mathbb{Q}(\varepsilon_{n}):\mathbb{Q}(j)]\leq 6$ . Thus $\varepsilon_{n}$ belongs to a subfield of $\mathbb{Q}_{\infty,\ell}$ of degree at most $12$ . The lemma follows since, by Siegel’s theorem, the $S$ -unit equation has only finitely many solutions in any number field. ∎

9. Hyperelliptic curves over $\mathbb{Q}_{\infty,\ell}$ with few bad primes

Let $\ell$ be an odd prime. Let $g\geq 2$ be an integer satisfying

(22)

\begin{cases}\text{$g\equiv(\ell-3)/4\,$ or $\,-1\pmod{(\ell-1)/2}$}&\text{ if $\ell\equiv 3\pmod{4}$}\\ \text{$g\equiv-1\pmod{(\ell-1)/4}$}&\text{ if $\ell\equiv 1\pmod{4}$}.\end{cases}

Then there is a positive integer $k$ such that

(23)

k\cdot\left(\frac{\ell-1}{2}\right)\;=\;\begin{cases}\text{$2g+1$ or $2g+2$}&\text{if $\ell\equiv 3\pmod{4}$}\\ 2g+2&\text{if $\ell\equiv 1\pmod{4}$}.\end{cases}

Let $n\geq 2$ be a positive integer satisfying

(24)

\ell^{n-1}\;\geq\;k.

In this section we construct a hyperelliptic $D_{n}$ curve of genus $g$ defined over $\mathbb{Q}_{n-1,\ell}$ with good reduction away from the primes above $2$ , $\ell$ .

Write

\mathcal{Z}_{n}=\{\zeta\in\Omega_{n,\ell}\quad:\quad\zeta^{\ell^{n}}=1,\quad\zeta^{\ell^{i}}\neq 1\textrm{ if }i<n\}

for the set of primitive $\ell^{n}$ -th roots of $1$ . Write

\mathcal{Z}_{n}^{+}\;=\;\{\zeta+\zeta^{-1}\;:\;\zeta\in\mathcal{Z}_{n}\}\;\subset\;\Omega_{n,\ell}^{+}.

We note that any element of $\mathcal{Z}_{n}^{+}$ generates $\Omega_{n,\ell}^{+}$ .

Lemma 24.

$\displaystyle\#\mathcal{Z}_{n}^{+}=\ell^{n-1}(\ell-1)/2$ .

Proof.

We note that $\#\mathcal{Z}_{n}=\varphi(\ell^{n})=\ell^{n-1}(\ell-1)$ . Suppose $\alpha$ , $\beta\in\mathcal{Z}_{n}$ . Then

(25)

(\alpha+\alpha^{-1})-(\beta+\beta^{-1})\;=\;\alpha^{-1}\cdot(1-\alpha\beta)\cdot(1-\alpha\beta^{-1}).

Thus $\alpha+\alpha^{-1}=\beta+\beta^{-1}$ if and only if $\alpha=\beta$ or $\alpha=\beta^{-1}$ . The lemma follows. ∎

Write

G_{n}=\operatorname{Gal}(\Omega_{n,\ell}^{+}/\mathbb{Q}_{n-1,\ell}),\qquad H_{n}=\operatorname{Gal}(\Omega_{n,\ell}^{+}/\Omega_{n-1,\ell}^{+}).

We note that these are both cyclic subgroups of $\operatorname{Gal}(\Omega_{n,\ell}^{+}/\mathbb{Q})$ having orders

\#G_{n}=(\ell-1)/2,\qquad\#H_{n}=\ell.

Lemma 25.

Fix $\zeta\in\mathcal{Z}_{n}$ . Let

(26)

\eta_{i}\;=\;\zeta^{1+\ell^{n-1}(i-1)}+\zeta^{-1-\ell^{n-1}(i-1)},\qquad 1\leq i\leq\ell.

Then $\eta_{1},\dotsc,\eta_{\ell}\in\mathcal{Z}_{n}^{+}$ form a single orbit under the action of $H_{n}$ , but have pairwise disjoint orbits under the action of $G_{n}$ .

Proof.

Let $\kappa\in\operatorname{Gal}(\Omega_{n,\ell}/\mathbb{Q})$ be given by $\kappa(\zeta)=\zeta^{1+\ell^{n-1}}$ . We note that $\kappa$ has order $\ell$ and fixes $\Omega_{n-1,\ell}$ . We denote the restriction of $\kappa$ to $\Omega_{n,\ell}^{+}$ by $\tau$ ; this is a cyclic generator of $H_{n}$ . Note that

\eta_{i}=\tau^{i-1}(\zeta+\zeta^{-1}),\qquad 1\leq i\leq\ell.

Let $\sigma_{1}$ , $\sigma_{2}\in G_{n}$ . Let $1\leq i<j\leq\ell$ and suppose $\sigma_{1}(\eta_{i})=\sigma_{2}(\eta_{j})$ . Thus $\sigma_{1}\tau^{i-1}(\eta_{1})=\sigma_{2}\tau^{j-1}(\eta_{1})$ , so $\tau^{1-j}\sigma_{2}^{-1}\sigma_{1}\tau^{i-1}$ fixes $\eta_{1}$ . As $\eta_{1}$ generates $\Omega_{n,\ell}^{+}$ , we have $\tau^{1-j}\sigma_{2}^{-1}\sigma_{1}\tau^{i-1}=1$ is the identity element in $\operatorname{Gal}(\Omega_{n,\ell}^{+}/\mathbb{Q})$ . However, $\operatorname{Gal}(\Omega_{n,\ell}^{+}/\mathbb{Q})$ is abelian, so

\tau^{i-j}=\sigma_{1}^{-1}\sigma_{2}\in G_{n}\cap H_{n}=\{1\}.

Since $1\leq i\leq j\leq\ell$ and $\tau$ has order $\ell$ we have $i=j$ . ∎

The Galois group $G_{n}$ acts faithfully on $\mathcal{Z}_{n}^{+}$ . This action has $\ell^{n-1}$ orbits. Assumption (24) ensures that the number of orbits is at least $k$ . If $k>\ell$ , then we extend the list $\eta_{1},\dotsc,\eta_{\ell}\in\mathcal{Z}_{n}^{+}$ to $\eta_{1},\dotsc,\eta_{k}\in\mathcal{Z}_{n}^{+}$ , so that the $\eta_{i}$ continue to have disjoint orbits under the action of $G_{n}$ ; if $\ell=3$ the choice of $\eta_{4}$ will be important later, and we choose $\eta_{4}=\zeta^{2}+\zeta^{-2}$ . Consider the curve

(27)

D_{n}\;:\;Y^{2}=\prod_{j=1}^{k}\prod_{\sigma\in G_{n}}(X-\eta_{j}^{\sigma}).

Lemma 26.

The curve $D_{n}$ is hyperelliptic of genus $g$ , is defined over $\mathbb{Q}_{n-1,\ell}$ , and has good reduction away from the primes above $2$ and $\ell$ .

Proof.

Our assumption on the orbits ensures that the polynomial on the right hand-side of (27) is separable. By (23), the degree of the polynomial is either $2g+1$ or $2g+2$ . Thus $D_{n}$ is a hyperelliptic curve of genus $g$ . A priori, $D_{n}$ is defined over $\Omega_{n,\ell}^{+}$ . However, the roots of the hyperelliptic polynomial are permuted by the action of $G_{n}=\operatorname{Gal}(\Omega_{n,\ell}^{+}/\mathbb{Q}_{n-1,\ell})$ and so the polynomial belongs to $\mathbb{Q}_{n-1,\ell}[X]$ . Hence $D_{n}$ is defined over $\mathbb{Q}_{n-1,\ell}$ .

Let $u_{1},\dotsc,u_{d}$ be the roots of the hyperelliptic polynomial. Then the discriminant of hyperelliptic polynomial is

\prod_{1\leq i<j\leq d}(u_{i}-u_{j})^{2}.

However, $u_{i}$ , $u_{j}$ are distinct elements of $\mathcal{Z}_{n}^{+}$ . Thus there are $\alpha$ , $\beta\in\mathcal{Z}_{n}$ with $\alpha\neq\beta$ , $\beta^{-1}$ such that $u_{i}=\alpha+\alpha^{-1}$ , $u_{j}=\beta+\beta^{-1}$ . From the identity (25),

u_{i}-u_{j}\;=\;\alpha^{-1}(1-\alpha\beta^{-1})(1-\alpha\beta).

Since $\alpha\beta$ and $\alpha\beta^{-1}$ are non-trivial $\ell$ -power roots of $1$ , we see that $u_{i}-u_{j}$ is a $\{\upsilon_{\ell}\}$ -unit, and hence the discriminant of the hyperelliptic polynomial of $D_{n}$ is a $\{\upsilon_{\ell}\}$ -unit. ∎

Given four pairwise distinct elements $z_{1}$ , $z_{2}$ , $z_{3}$ , $z_{4}$ of a field $K$ , we shall employ the notation $(z_{1},z_{2}\,;\,z_{3},z_{4})$ to denote the cross ratio

(z_{1},z_{2}\,;\,z_{3},z_{4})\;=\;\frac{(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{1}-z_{4})(z_{2}-z_{3})}.

We extend the cross ratio to four distinct elements $z_{1},z_{2},z_{3},z_{4}$ of $\mathbb{P}^{1}(K)$ in the usual way. We let $\operatorname{GL}_{2}(K)$ act on $\mathbb{P}^{1}(K)$ via fractional linear transformations

\gamma(z)=\frac{az+b}{cz+d},\qquad\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}.

It is well-known and easy to check that these fractional linear transformations leave the cross ratio unchanged:

(\gamma(z_{1}),\gamma(z_{2})\,;\,\gamma(z_{3}),\gamma(z_{4}))\;=\;(z_{1},z_{2}\,;\,z_{3},z_{4}).

Lemma 27.

Let $\overline{K}$ be an algebraically closed field of characteristic $0$ . Let

D\;:\;Y^{2}=\prod_{i=1}^{d}(X-a_{i}),\qquad D^{\prime}\;:\;Y^{2}=\prod_{i=1}^{d}(X-b_{i}),

be genus $g$ curves defined over $\overline{K}$ where the polynomials on the right are separable. If $D$ , $D^{\prime}$ are isomorphic then there is some permutation $\mu\in S_{d}$ such that for all quadruples of pairwise distinct indices $1\leq r,s,t,u\leq d$

(a_{r},a_{s}\,;\,a_{t},a_{u})\;=\;(b_{\mu(r)},b_{\mu(s)}\,;\,b_{\mu(t)},b_{\mu(u)}).

Proof.

We shall make use of the following standard description (e.g. [2, Proposition 6.11]) of isomorphisms of hyperelliptic curves: every isomorphism $\pi\;:\;D\rightarrow D^{\prime}$ is of the form

\pi(X,Y)\;=\;\left(\frac{aX+b}{cX+d},\frac{eY}{(cX+d)^{g+1}}\right)

for some

\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\operatorname{GL}_{2}(\overline{K}),\qquad e\in\overline{K}^{\times}.

Observe that $\pi(a_{i},0)$ has $Y$ -coordinate $0$ ; thus

\{\gamma(a_{1}),\dotsc,\gamma(a_{d})\}\;=\;\{b_{1},\dotsc,b_{d}\}.

Hence there is a permutation $\mu\in S_{d}$ such that $\gamma(a_{i})=b_{\mu(i)}$ . The lemma follows from the invariance of the cross ratio under the action of $\operatorname{GL}_{2}(\overline{K})$ . ∎

Lemma 28.

Let $\ell\geq 11$ be prime. Then there is some $a\in\mathbb{Z}_{\ell}^{\times}$ of order $\ell-1$ such that

(28)

1+a^{2}\;\not\equiv\;0,\;\pm(1-a^{2}),\,\pm(a+a^{3}),\,\pm(a-a^{3}),\\ \pm(1+a^{3}),\,\pm(1-a^{3}),\,\pm(a+a^{2}),\,\pm(a-a^{2})\pmod{\ell}.

Proof.

Making use of the fact that a polynomial of degree $n$ has at most $n$ roots, we see that the number of $a\in\mathbb{F}_{\ell}$ that do not satisfy (28) is (very crudely) bounded by $37$ . An element $a\in\mathbb{Z}_{\ell}^{\times}$ of order $\ell-1$ is the unique Hensel lift of an element $a\in\mathbb{F}_{\ell}^{\times}$ of order $\ell-1$ . There are precisely $\varphi(\ell-1)$ elements of order $\ell-1$ in $\mathbb{F}_{\ell}^{\times}$ . A theorem of Shapiro [15, page 23], asserts that $\varphi(n)>n^{\log{2}/\log{3}}$ for $n\geq 30$ . We note that if $\ell\geq 317$ then $\varphi(\ell-1)\geq 316^{\log{2}/\log{3}}\approx 37.8$ , and so the lemma holds for $\ell\geq 317$ . For the range $11\leq\ell\leq 317$ we checked the lemma by brute force computer enumeration. ∎

Lemma 29.

Let $n>m$ be sufficiently large. Then $D_{n}$ and $D_{m}$ are non-isomorphic, even over $\overline{\mathbb{Q}}$ .

Proof.

Note that all roots of the hyperelliptic polynomial for $D_{n}$ in (27) belong to $\mathcal{Z}_{n}^{+}$ . It follows from (25) that the cross ratio of any four of them belongs to $V_{n}$ . Suppose $D_{n}$ and $D_{m}$ are isomorphic. Let $u_{1},u_{2},u_{3},u_{4}$ be any distinct roots of the hyperelliptic polynomial for $D_{n}$ given in (27). Then, by Lemma 27,

(u_{1},u_{2}\,;\,u_{3},u_{4})\;\in V_{m}\subseteq V_{n-1}.

We shall obtain a contradiction through a careful choice of the four roots $u_{1},\dotsc,u_{4}$ .

We first suppose that $k\geq 2$ and $\ell\geq 5$ . Let $\zeta=\zeta_{\ell^{n}}$ and $b=1+\ell^{n-1}$ . Then, by Lemma 25, $\eta_{1}=\zeta+\zeta^{-1}$ and $\eta_{2}=\zeta^{b}+\zeta^{-b}$ . Let $a\in\mathbb{Z}_{\ell}^{\times}$ have order $\ell-1$ . Let $\kappa\in\operatorname{Gal}(\Omega_{n,\ell}/\mathbb{Q}_{n-1,\ell})$ be given by $\kappa(\zeta)=\zeta^{a}$ . Then $\kappa$ is a cyclic generator for $\operatorname{Gal}(\Omega_{n,\ell}/\mathbb{Q}_{n-1,\ell})$ . We shall denote the restriction of $\kappa$ to $\Omega_{n,\ell}^{+}$ by $\mu$ . Then $\mu$ is a cyclic generator for $G_{n}=\operatorname{Gal}(\Omega_{n,\ell}^{+}/\mathbb{Q}_{n-1,\ell})$ having order $(\ell-1)/2$ . We shall take

	$\displaystyle u_{1}$	$\displaystyle=\eta_{1}=\zeta+\zeta^{-1},$	$\displaystyle\quad u_{2}$	$\displaystyle=\mu(\eta_{1})=\zeta^{a}+\zeta^{-a},$
	$\displaystyle\quad u_{3}$	$\displaystyle=\eta_{2}=\zeta^{b}+\zeta^{-b},$	$\displaystyle\quad u_{4}$	$\displaystyle=\mu(\eta_{2})=\zeta^{ab}+\zeta^{-ab}.$

We compute the cross ratio with the help of identity (25), finding

(u_{1},u_{2}\,;\,u_{3},u_{4})\;=\;\frac{(1-\zeta^{1+b})(1-\zeta^{1-b})(1-\zeta^{a+ab})(1-\zeta^{a-ab})}{(1-\zeta^{1+ab})(1-\zeta^{1-ab})(1-\zeta^{a+b})(1-\zeta^{a-b})}.

As $b\equiv 1\pmod{\ell}$ , and clearly $a\not\equiv\pm 1\pmod{\ell}$ , it is easy to check that $1+b$ is the only one out of the eight exponents of $\zeta$ above that is $\pm 2\pmod{\ell}$ . Therefore by Lemma 10, the cross ratio is not an element of $\langle\pm\zeta_{\ell^{n}},V_{n-1}\rangle$ for $n$ sufficiently large, giving a contradiction for the case $k\geq 2$ and $\ell\geq 5$ .

Next we suppose that $k=1$ . It follows from (23) that $\ell\geq 11$ . We choose $a\in\mathbb{Z}_{\ell}^{\times}$ as in Lemma 28, and, as above, take $\mu$ to be the corresponding generator of $G_{n}$ of order $(\ell-1)/2\geq 5$ . We take

u_{i}=\mu^{i-1}(\eta_{1})=\zeta^{a^{i-1}}+\zeta^{-a^{i-1}},\qquad 1\leq i\leq 4;

observe that these are four roots of the hyperelliptic polynomial of $D_{n}$ given in (27). The assumption that $\ell\geq 11$ ensures that $a$ has order $\geq 10$ and so $u_{1},u_{2},u_{3},u_{4}$ are indeed pairwise distinct. We compute the cross ratio with the help of identity (25), finding

(u_{1},u_{2}\,;\,u_{3},u_{4})\;=\;\frac{(1-\zeta^{1+a^{2}})(1-\zeta^{1-a^{2}})(1-\zeta^{a+a^{3}})(1-\zeta^{a-a^{3}})}{(1-\zeta^{1+a^{3}})(1-\zeta^{1-a^{3}})(1-\zeta^{a+a^{2}})(1-\zeta^{a-a^{2}})}.

Using Lemma 9 and our choice of $a$ given by Lemma 28 we conclude that this cross ratio does not belong to $\langle\pm\zeta_{\ell^{n}},V_{n-1}\rangle$ for $n$ sufficiently large. This gives a contradiction for the case $k=1$ .

Finally, we consider $\ell=3$ . It follows from (23) that $k\geq 5$ . Recall our choices of $\eta_{1}$ , $\eta_{2}$ , $\eta_{3}$ in Lemma 25, and our choice of $\eta_{4}=\zeta^{2}+\zeta^{-2}$ in the particular case $\ell=3$ . We choose the four roots $u_{i}=\eta_{i}$ for $i=1,\dotsc,4$ , and obtain,

(u_{1},u_{2}\,;\,u_{3},u_{4})\;=\;\frac{(1-\zeta^{2+2\times 3^{n-1}})(1-\zeta^{-2\times 3^{n-1}})(1-\zeta^{3+3^{n-1}})(1-\zeta^{-1+3^{n-1}})}{(1-\zeta^{3})(1-\zeta^{-1})(1-\zeta^{2})(1-\zeta^{-3^{n-1}})}.

As before, with the help of Lemma 10, we easily verify that the cross ratio is not an element of $\langle\pm\zeta_{\ell^{n}},V_{n-1}\rangle$ for $n$ sufficiently large. This completes the proof. ∎

Proof of Theorem 5

If $\ell=3$ or $5$ then (22) does not impose any restriction on the genus. Therefore we obtain, as above, for every genus $g\geq 2$ , infinitely many $\overline{\mathbb{Q}}$ -isomorphism classes of genus $g$ hyperelliptic curves, defined over $\mathbb{Q}_{\infty,\ell}$ , with good reduction away from $\{\upsilon_{2},\upsilon_{\ell}\}$ .

It remains to deal with $\ell=7$ , $11$ and $13$ . Here, (22) imposes the restriction

g\equiv\begin{cases}\text{$1$ or $2\bmod{3}$}&\text{if $\ell=7$}\\ \text{$2$ or $4\bmod{5}$}&\text{if $\ell=11$}\\ \text{$2\bmod{3}$}&\text{if $\ell=13$}.\end{cases}

We very briefly sketch how to remove the restriction. Instead of $D_{n}$ defined as in (27), we consider the more general

D_{n}\;:\;Y^{2}=h(X)\cdot\prod_{j=1}^{k}\prod_{\sigma\in G_{n}}(X-\eta_{j}^{\sigma})

where

•

$h$ is a monic divisor of $X(X-1)(X+1)$ ;
•

$k$ and $h$ are chosen to obtain the desired genus;
•

$\eta_{j}\in\mathcal{Z}_{n}^{+}$ are chosen as before.

These $D_{n}$ are clearly defined over $\mathbb{Q}_{n-1,\ell}$ . To check that they have good reduction away from $S^{\prime}=\{\upsilon_{2},\upsilon_{\ell}\}$ , we need to verify that the difference of any two distinct roots $u$ , $v$ of the hyperelliptic polynomial belongs to $\mathcal{O}(\Omega_{n},S^{\prime})^{\times}$ . The proof of Lemma 26 shows this if $u$ , $v\in\mathcal{Z}_{n}^{+}$ . For the remaining possible differences it is enough to note that

\alpha+\alpha^{-1}=\alpha^{-1}\Phi_{4}(\alpha),\qquad\alpha+\alpha^{-1}+1=\alpha^{-1}\Phi_{3}(\alpha),\qquad\alpha+\alpha^{-1}-1=\alpha^{-1}\Phi_{6}(\alpha)

which are all units by Lemma 12. We omit the remaining details. ∎

10. Isogeny classes of hyperelliptic curves over $\mathbb{Q}_{\infty,\ell}$

A beautiful theorem of Kummer asserts that the index of the cyclotomic units $C_{n}$ in the full unit group $\mathcal{O}(\Omega_{n,\ell})^{\times}$ equals the class number $h_{n}^{+}$ of $\Omega_{n,\ell}^{+}$ . In this section, with the help of Kummer’s theorem, we prove for certain primes $\ell$ the existence of infinitely many isogeny classes of hyperelliptic Jacobians over $\mathbb{Q}_{\infty,\ell}$ with good reduction away from $\ell$ . We first prove a few elementary lemmas.

Lemma 30.

Let $K$ be a field of characteristic not $2$ , and let $L=K(\sqrt{\alpha_{1}},\dots,\sqrt{\alpha_{r}})$ where $\alpha_{i}\in K^{\times}$ . Then for any $x\in K$ such that $\sqrt{x}\in L$ , we have

x\;=\;\alpha_{1}^{e_{1}}\cdots\alpha_{r}^{e_{r}}q^{2}

for some integers $e_{i}\in\mathbb{Z}$ and $q\in K$ .

Proof.

Let $M$ be a field of characteristic not $2$ , and let $d\in M$ be a non-square. Let $x\in M$ and suppose $\sqrt{x}\in M(\sqrt{d})$ . Then $\sqrt{x}=y+z\sqrt{d}$ for some $y$ , $z\in M$ . Squaring, we deduce that $yz=0$ . Thus $x=y^{2}$ or $x=dz^{2}$ .

We now prove the lemma by induction on $r$ . The above establishes the case $r=1$ . Let $r\geq 2$ , and let $x\in K$ satisfy $\sqrt{x}\in L$ . Letting $M=K(\sqrt{\alpha_{1}},\dotsc,\sqrt{\alpha_{r-1}})$ we see that $x\in M$ and $\sqrt{x}\in M(\sqrt{\alpha_{r}})$ . Thus, by the above, $\sqrt{x}\in M$ or $\sqrt{x\alpha_{r}}\in M$ . In other words,

\sqrt{x\cdot\alpha_{r}^{e}}\;\in\;M=K(\sqrt{\alpha_{1}},\dotsc,\sqrt{\alpha_{r-1}})

for some $e\in\{0,1\}$ . By the inductive hypothesis, there are $e_{1},\dotsc,e_{r-1}\in\mathbb{Z}$ and $q\in K$ such that

x\cdot\alpha_{r}^{e}\;=\;\alpha_{1}^{e_{1}}\cdots\alpha_{r-1}^{e_{r-1}}q^{2}.

The proof is complete on taking $e_{r}=-e$ . ∎

Lemma 31.

Let $\ell$ be an odd prime. Let $q\in\Omega_{\infty,\ell}$ satisfy $q^{2}\in V_{n}$ . If the class number $h_{n}^{+}$ of $\Omega_{n,\ell}^{+}$ is odd, then $q\in V_{n}$ .

Proof.

Let $q\in\Omega_{\infty,\ell}$ satisfy $q^{2}\in V_{n}\subset\Omega_{n,\ell}$ . As the extension $\Omega_{\infty,\ell}/\Omega_{n,\ell}$ is pro- $\ell$ , we conclude that $q\in\Omega_{n,\ell}$ . However, $V_{n}\subseteq\mathcal{O}(\Omega_{n,\ell},\{\upsilon_{\ell}\})^{\times}$ , where, as usual, $\upsilon_{\ell}$ denotes the prime above $\ell$ . Thus $q\in\mathcal{O}(\Omega_{n,\ell},\{\upsilon_{\ell}\})^{\times}$ . We claim that

[\mathcal{O}(\Omega_{n,\ell},\{\upsilon_{\ell}\})^{\times}:V_{n}]\;=\;h_{n}^{\plus}.

The lemma follows immediately from the claim. To prove the claim, consider the commutative diagram with exact rows

where $\kappa(\alpha)=\operatorname{ord}_{(1-\zeta)}(\alpha)$ . By the snake lemma,

\mathcal{O}(\Omega_{n,\ell},\{\upsilon_{\ell}\})^{\times}/V_{n}\;\cong\;\mathcal{O}(\Omega_{n,\ell})^{\times}/C_{n}.

Write $C_{n}^{+}=C_{n}\cap\Omega_{n,\ell}^{+}$ . The aforementioned theorem of Kummer asserts that

[\mathcal{O}(\Omega_{n,\ell})^{\times}:C_{n}]\;=\;[\mathcal{O}(\Omega_{n,\ell}^{\plus})^{\times}:C_{n}^{\plus}]\;=\;h_{n}^{\plus};

see, for example, [22, Exercise 8.5] for the first equality, and [22, Theorem 8.2] for the second. This proves the claim. ∎

Lemma 32.

Let $K$ be a field of characteristic $\neq 2$ . Let $f\in K[X]$ be a monic separable polynomial of odd degree $d\geq 5$ . Write $f=\prod_{i=1}^{d}(X-\alpha_{i})$ with $\alpha_{i}\in\overline{K}$ . Let $C/K$ be a hyperelliptic curve given by $Y^{2}=f(X)$ with Jacobian $J$ . Then

K(J[2])=K(\alpha_{1},\dotsc,\alpha_{d}),\qquad K(J[4])=K(J[2])\left(\big{\{}\sqrt{\alpha_{i}-\alpha_{j}}\big{\}}_{1\leq i,j\leq d}\right).

Proof.

Write $\infty$ for the point at infinity on the given model for $C$ . The expression given for $K(J[2])$ is well-known; it may be seen by observing (see, for example [12]) that the classes of the classes of degree $0$ divisors $[(\alpha_{i},0)-\infty]$ with $i=1,\dotsc,d$ generate $J[2]$ .

Yelton [23, Theorem 1.2.2] gives a high-powered proof of the given expression for $K(J[4])$ . For the convenience of the reader we give a more elementary argument. Let $L=K(J[2])$ . The theory of $2$ -descent on hyperelliptic Jacobians furnishes, for any field $M\supseteq L$ , an injective homomorphism [12], [19]

J(M)/2J(M)\;\hookrightarrow\;\prod_{i=1}^{d}M^{*}/(M^{*})^{2}

known as the $X-\Theta$ -map. This in particular sends the $2$ -torsion point $[(\alpha_{i},0)-\infty]$ to

\left((\alpha_{i}-\alpha_{1})\,,\dotsc\,,\,(\alpha_{i}-\alpha_{i-1})\,,\,\prod_{j\neq i}(\alpha_{i}-\alpha_{j})\,,\,(\alpha_{i}-\alpha_{i+1})\,,\,\dotsc\,,\,(\alpha_{i}-\alpha_{d})\right).

The field $K(J[4])$ is the smallest extension of $M$ of $L$ such that all the images of the $2$ -torsion generators $[(\alpha_{i},0)-\infty]$ are trivial in $\prod_{i=1}^{d}M^{*}/(M^{*})^{2}$ . This is plainly the extension

M=L\left(\big{\{}\sqrt{\alpha_{i}-\alpha_{j}}\big{\}}_{1\leq i,j\leq d}\right).

∎

Lemma 33.

Let $p$ be a prime for which $2$ is a primitive root (i.e. $2$ is a generator for $\mathbb{F}_{p}^{\times}$ ). Let $G$ be a cyclic group of order $p$ , and let $V$ be an $\mathbb{F}_{2}[G]$ -module with $\dim_{\mathbb{F}_{2}}(V)=p-1$ . Suppose that the action of $G$ on $V-\{0\}$ is free. Then $V$ is irreducible.

Proof.

Let $W$ be a $\mathbb{F}_{2}[G]$ -submodule of $V$ , and write $d=\dim_{\mathbb{F}_{2}}(W)$ . Since the action of $G$ on $V-\{0\}$ is free, the set $W-\{0\}$ consists of $G$ -orbits, all having size $p$ . However, $\#(W-\{0\})=2^{d}-1$ , and so $p\mid(2^{d}-1)$ . By assumption, $2$ is a primitive root modulo $p$ , therefore $(p-1)\mid d$ . Since $W$ is an $\mathbb{F}_{2}$ -subspace of $V$ which has dimension $p-1$ , we see that $W=0$ or $W=V$ . ∎

Lemma 34.

Let $\ell=2p+1$ , where $\ell$ and $p$ are odd primes. Suppose $2$ is a primitive root modulo $p$ . Let $g=(\ell-3)/4$ . Let $n\geq 2$ and let $D_{n}/\mathbb{Q}_{n-1,\ell}$ be the hyperelliptic curve defined in Section 9. Let $A/\mathbb{Q}_{\infty,\ell}$ be an abelian variety and let $\phi:J(D_{n})\rightarrow A$ be an isogeny defined over $\mathbb{Q}_{\infty,\ell}$ . Then $\phi=2^{r}\phi_{\mathrm{odd}}$ where $\phi_{\mathrm{odd}}:J(D_{n})\rightarrow A$ is an isogeny of odd degree.

We remark if $\ell$ and $p$ are primes with $\ell=2p+1$ then $p$ is called a Sophie-Germain prime, and $\ell$ is called as safe prime.

Proof of Lemma 34.

Note that, in the notation of Section 9, $k=1$ , and the hyperelliptic polynomial for $D_{n}$ has odd degree $2g+1=(\ell-1)/2=p$ , and consists of a single orbit under action of $G_{n}=\operatorname{Gal}(\Omega_{n}^{+}/\mathbb{Q}_{n-1,\ell})$ :

D_{n}\;:\;y^{2}\;=\;\prod_{\sigma\in G_{n}}(X-\eta_{1}^{\sigma}),\qquad\eta_{1}=\zeta_{\ell^{n}}+\zeta_{\ell^{n}}^{-1}.

In particular, the hyperelliptic polynomial is irreducible over $\mathbb{Q}_{\infty,\ell}$ . It follows from this (e.g. [19, Lemma 4.3]) that $J(\mathbb{Q}_{\infty,\ell})[2]=0$ , where $J$ denotes $J(D_{n})$ for convenience. We note, by Lemma 32, that $\mathbb{Q}_{\infty,\ell}(J[2])=\mathbb{Q}_{\infty,\ell}(\eta_{1})=\Omega_{\infty,\ell}^{+}$ . We consider the action of $G_{\infty}:=\operatorname{Gal}(\Omega_{\infty,\ell}^{+}/\mathbb{Q}_{\infty,\ell})$ on $J[2]$ . The group $G_{\infty}$ is cyclic of order $(\ell-1)/2=p$ . Any element fixed by this action belongs to $J(\mathbb{Q}_{\infty,\ell})[2]=0$ . Thus $G_{\infty}$ acts freely on $V-\{0\}$ , where $V:=J[2]$ .

Now let $\phi:J\rightarrow A$ be an isogeny defined over $\mathbb{Q}_{\infty,\ell}$ . Then $W:=\ker(\phi)\cap J[2]$ is a subgroup of $V$ stable under the action of $G_{\infty}$ , and therefore an $\mathbb{F}_{2}[G_{\infty}]$ -submodule of the $\mathbb{F}_{2}[G_{\infty}]$ -module $V$ . Observe that $\dim_{\mathbb{F}_{2}}(V)=2g=p-1$ . By hypothesis, $2$ is a primitive root modulo $p$ . We apply Lemma 33 to deduce that $W=0$ or $W=V$ . Therefore, either $\phi$ already has odd degree, or $J[2]\subseteq\ker(\phi)$ . In the latter case, observe that $\phi=2\phi^{\prime}$ where $\phi^{\prime}:J\rightarrow A$ is an isogeny defined over $\mathbb{Q}_{\infty,\ell}$ of degree $\deg(\phi)/2^{2g}$ . As $\phi$ has finite degree, by repeating the argument we eventually arrive at $\phi=2^{r}\phi_{\mathrm{odd}}$ . ∎

Lemma 35.

Let $\ell=2p+1$ , where $\ell$ and $p$ are odd primes. Suppose $2$ is a primitive root modulo $p$ . Suppose that the class number $h_{n}^{+}$ of $\Omega_{n,\ell}^{+}$ is odd for all $n$ . Let $g=(\ell-3)/4$ . For $n\geq 2$ let $D_{n}/\mathbb{Q}_{n-1,\ell}$ be the genus $g$ hyperelliptic curve defined in Section 9. Let $n>m$ be sufficiently large. Then there are no isogenies $J(D_{n})\to J(D_{m})$ defined over $\mathbb{Q}_{\infty,\ell}$ .

The assumption that $h_{n}^{+}$ is odd for all $n$ may seem at first sight very restrictive. However, it is conjectured [3] that $h_{n+1}^{+}=h_{n}^{+}$ for all but finitely many pairs $(\ell,n)$ . Moreover, Washington [21] has shown that $\operatorname{ord}_{p}(h_{n})$ remains bounded as $n\rightarrow\infty$ , for any fixed prime $p$ .

Proof of Lemma 35.

Write $J_{n}$ for $J(D_{n})$ . Suppose there is an isogeny $\phi:J_{n}\rightarrow J_{m}$ defined over $\mathbb{Q}_{\infty,\ell}$ . By Lemma 34 we may suppose that $\phi$ has odd degree, and so $\ker(\phi)\cap J_{n}[4]=0$ . Thus $\phi$ restricted to $J_{n}[4]$ induces an isomorphism of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{\infty,\ell})$ -modules $J_{n}[4]\cong J_{m}[4]$ . In particular, $\mathbb{Q}_{\infty,\ell}(J_{n}[4])=\mathbb{Q}_{\infty,\ell}(J_{m}[4])$ . As in the proof of Lemma 34 we have $\mathbb{Q}_{\infty,\ell}(J_{n}[2])=\mathbb{Q}_{\infty,\ell}(J_{m}[2])=\Omega_{\infty,\ell}^{+}$ . Thus, by Lemma 32, the equality $\mathbb{Q}_{\infty,\ell}(J_{n}[4])=\mathbb{Q}_{\infty,\ell}(J_{m}[4])$ may be rewritten as

\Omega_{\infty,\ell}^{+}\Big{(}\big{\{}\sqrt{\vartheta_{n,i}-\vartheta_{n,j}}\big{\}}_{1\leq i,j\leq(\ell-1)/2}\Big{)}\;=\;\Omega_{\infty,\ell}^{+}\Big{(}\big{\{}\sqrt{\vartheta_{m,i}-\vartheta_{m,j}}\big{\}}_{1\leq i,j\leq(\ell-1)/2}\Big{)}

where $\vartheta_{r,i}:=\mu_{r}^{i-1}(\zeta_{\ell^{r}}+\zeta_{\ell^{r}}^{-1})$ where $\mu_{r}$ is a cyclic generator of $G_{r}$ . This, in particular, implies that

\sqrt{\vartheta_{n,2}-\vartheta_{n,1}}\;\in\;\Omega_{\infty,\ell}^{+}\Big{(}\big{\{}\sqrt{\vartheta_{m,i}-\vartheta_{m,j}}\big{\}}_{1\leq i,j\leq(\ell-1)/2}\Big{)}

We apply Lemma 30 to obtain

\displaystyle\vartheta_{n,2}-\vartheta_{n,1}=\pm

\displaystyle\prod_{1\leq i<j\leq\frac{\ell-1}{2}}(\vartheta_{m,i}-\vartheta_{m,j})^{e_{i,j}}\cdot q^{2}

for some integers $e_{i,j}\in\mathbb{Z}$ and $q\in\Omega_{\infty,\ell}^{+}$ . By Lemma 31, we have $q\in V_{n}$ . The generator $\mu_{n}$ of $G_{n}$ is given by $\mu_{n}(\zeta_{\ell^{n}}+\zeta_{\ell^{n}}^{-1})=\zeta_{\ell^{n}}^{a}+\zeta_{\ell^{n}}^{-a}$ where $a\in\mathbb{Z}_{\ell}^{\times}$ has order $(\ell-1)$ . Note

\vartheta_{n,2}-\vartheta_{n,1}\;=\;\zeta_{\ell^{n}}^{a}+\zeta_{\ell^{n}}^{-a}-\zeta_{\ell^{n}}-\zeta_{\ell^{n}}^{-1}\;=\;\zeta_{\ell^{n}}^{-a}(1-\zeta_{\ell^{n}}^{a+1})(1-\zeta_{\ell^{n}}^{a-1}).

Thus,

(1-\zeta_{\ell^{n}}^{a+1})(1-\zeta_{\ell^{n}}^{a-1})\;\in\;\langle\pm\zeta_{\ell^{n}},V_{m},V_{n}^{2}\rangle.

However, $(a+1)\not\equiv\pm(a-1)\pmod{\ell}$ . Now Corollary 11 gives a contradiction. ∎

Proof of Theorem 6

Let $\ell\geq 11$ . Let

g\;=\;\lfloor(\ell-3)/4\rfloor\;=\;\begin{cases}(\ell-3)/4&\text{$\ell\equiv 3\pmod{4}$}\\ (\ell-5)/4&\text{$\ell\equiv 1\pmod{4}$}.\end{cases}

Thus $g$ satisfies (22). Let $D_{n}$ be as in Section 9. By Lemma 26, the hyperelliptic curve $D_{n}/\mathbb{Q}_{n-1,\ell}$ has genus $g$ , and good reduction away from $\{\upsilon_{2},\upsilon_{\ell}\}$ . Moreover, by Lemma 29, we have $D_{n}$ and $D_{m}$ are non-isomorphic, even over $\overline{\mathbb{Q}}$ , for $n>m$ sufficiently large.

Now suppose

(i)

$\ell=2p+1$ where $p$ is also an odd prime;
(ii)

$2$ as a primitive root modulo $p$ .

It then follows from Lemma 35 that $J(D_{n})$ and $J(D_{m})$ are non-isogenous over $\mathbb{Q}_{\infty,\ell}$ provided $h_{n}^{+}$ is odd for all $n$ , where $h_{n}^{+}$ denotes the class number of $\Omega_{n,\ell}^{+}$ . Write $h_{n}$ for the class number of $\Omega_{n,\ell}$ . It is known thanks to the work of Estes [4] that $h_{1}$ is odd for all primes $\ell$ satisfying (i) and (ii) (a simplified proof of this result is given Stevenhagen [18, Corollary 2.3]). Moreover, Ichimura and Nakajima [8] show, for primes $\ell\leq 509$ , that the ratio $h_{n}/h_{1}$ is odd for all $n$ . The primes $11\leq\ell\leq 509$ satisfying both (i) and (ii) are $11$ , $23$ , $59$ , $107$ , $167$ , $263$ , $347$ , $359$ . Thus for these primes $h_{n}$ is odd for all $n$ . As $h_{n}^{+}\mid h_{n}$ (see for example [22, Theorem 4.10]), we know for these primes that $h_{n}^{+}$ is odd for all $n$ . This completes the proof. ∎

Remark.

•

A key step in our proof of Theorem 6 is showing that $J(D_{n})[2]$ is irreducible as an $\mathbb{F}_{2}[G_{\infty}]$ -module whenever $\ell=2p+1$ where $p$ is a prime having $2$ as a primitive root. It can be shown for all other $\ell$ that the $\mathbb{F}_{2}[G_{\infty}]$ -module $J(D_{n})[2]$ is in fact reducible.
•

Another key step is the argument in the proof of Lemma 35 showing that for $n>m$ sufficiently large, the Jacobians $J(D_{n})$ and $J(D_{m})$ are not related via odd degree isogenies defined over $\mathbb{Q}_{\infty,\ell}$ . This step can be made to work, with very minor modifications to the argument, for all $\ell\geq 11$ , and all choices of genus $g$ given in (22).

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Curves with few bad primes over cyclotomic ℤℓ\mathbb{Z}_{\ell}-extensions

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Conjecture (Mazur).

Conjecture (Parshin and Zarhin).

Theorem (Zarhin).

Theorem 1.

Theorem 2.

Theorem 3.

Remarks

Theorem 4.

Theorem 5.

Theorem 6.

2. Units and SS-units of ℚ​(ζ)\mathbb{Q}(\zeta)

Lemma 7.

Proof.

Cyclotomic units and SS-units

Lemma 8.

Proof.

Lemma 9.

Proof.

Lemma 10.

Proof.

Corollary 11.

Units and SS-units from cyclotomic polynomials

Lemma 12.

Proof.

Lemma 13.

Proof.

3. The SS-unit equation over ℚ​(ζℓn)+\mathbb{Q}(\zeta_{\ell^{n}})^{+}

Lemma 14.

Proof.

Lemma 15.

Proof.

Proof of Theorem 2 for ℓ=2\ell=2 and 33

Proof of Theorem 1 for ℓ=2\ell=2

4. The unit equation over ℚ​(ζℓn)+\mathbb{Q}(\zeta_{\ell^{n}})^{+}

Lemma 16.

Proof.

Lemma 17.

Proof.

Proof of Theorem 1 for ℓ=3\ell=3

5. The SS-unit equation over ℚ∞,5\mathbb{Q}_{\infty,5}

Lemma 18.

Proof.

Proof of Theorems 1 and 2 for ℓ=5\ell=5.

6. The SS-unit equation over ℚ∞,7\mathbb{Q}_{\infty,7}

Lemma 19.

Proof.

7. Isogeny classes of elliptic curves over ℚ∞,ℓ\mathbb{Q}_{\infty,\ell}

Lemma 20.

Proof.

Theorem 21 (Kato).

Lemma 22.

Proof.

Proof of Theorem 4.

8. From SS-unit equations to elliptic curves

Proof of Theorem 3

Lemma 23.

Proof.

9. Hyperelliptic curves over ℚ∞,ℓ\mathbb{Q}_{\infty,\ell} with few bad primes

Lemma 24.

Proof.

Lemma 25.

Proof.

Lemma 26.

Proof.

Lemma 27.

Proof.

Lemma 28.

Proof.

Lemma 29.

Proof.

Proof of Theorem 5

10. Isogeny classes of hyperelliptic curves over ℚ∞,ℓ\mathbb{Q}_{\infty,\ell}

Lemma 30.

Proof.

Lemma 31.

Curves with few bad primes
over cyclotomic $\mathbb{Z}_{\ell}$ -extensions

2. Units and $S$ -units of $\mathbb{Q}(\zeta)$

Cyclotomic units and $S$ -units

Units and $S$ -units from cyclotomic polynomials

3. The $S$ -unit equation over $\mathbb{Q}(\zeta_{\ell^{n}})^{+}$

Proof of Theorem 2 for $\ell=2$ and $3$

Proof of Theorem 1 for $\ell=2$

4. The unit equation over $\mathbb{Q}(\zeta_{\ell^{n}})^{+}$

Proof of Theorem 1 for $\ell=3$

5. The $S$ -unit equation over $\mathbb{Q}_{\infty,5}$

Proof of Theorems 1 and 2 for $\ell=5$ .

6. The $S$ -unit equation over $\mathbb{Q}_{\infty,7}$

7. Isogeny classes of elliptic curves over $\mathbb{Q}_{\infty,\ell}$

8. From $S$ -unit equations to elliptic curves

9. Hyperelliptic curves over $\mathbb{Q}_{\infty,\ell}$ with few bad primes

10. Isogeny classes of hyperelliptic curves over $\mathbb{Q}_{\infty,\ell}$