A note on $p$-rational fields and the abc-conjecture

TL;DR

This paper explores the connection between the generalized abc-conjecture and p-rationality of certain number fields, proving that if the conjecture holds, then many primes induce p-rationality in these fields.

Contribution

It establishes a conditional link between the generalized abc-conjecture and the abundance of primes making specific number fields p-rational, extending recent suggestions in the field.

Findings

Conditional proof of p-rational primes in real quadratic fields

Extension of results to imaginary S_3-extensions

Quantitative lower bound on the number of p-rational primes

Abstract

In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c\,\log X$ prime numbers $p \leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by B\"ockle-Guiraud-Kalyanswamy-Khare.

Equations

$$\#\{\mathrm{prime}\mathrm{number}p,p\le X,{\alpha }^{p-1}\text{nequiv}1(\mathrm{mod}{p}^2)\}\ge c\log X,$$

$$\#\{\mathrm{prime}\mathrm{number}p\le X,r_{\psi }({\mathcal{T}}_p)<r_{\psi }(E_{\mathrm{K}})\mathrm{for}\mathrm{some}irred.{\mathbb{Q}}_p-char.\psi |\chi \}\ge c\log X,$$

$$\#\{\mathrm{prime}\mathrm{number}p\le X,\mathrm{K}\mathrm{is}p-\mathrm{rational}\}\ge c\log X,$$

$$\mathrm{Rad}(I)=\prod _{\mathfrak{p}|I}\mathrm{N}(\mathfrak{p}),$$

$$\prod _v\max \{|a{|}_v,|b{|}_v,|c{|}_v\}⩽C_{K,\epsilon }{\left(\mathrm{Rad}(abc)\right)}^{1+\epsilon }$$

$$|\mathrm{N}(u_1-u_2)|\le C_{\mathrm{K},\epsilon }\mathrm{Rad}((u_1-u_2){)}^{1+\epsilon }.$$

$$\#\{n⩽X,\phi (n)\ge \frac{1}{2}n\}\ge (\frac{6}{{\pi }^2}-\frac{1}{2})X+O(\log X).$$

$$|\mathrm{N}({\Phi }_n(u^k))|\ge \exp (cn),$$

$$|\tau (u^k)-{\zeta }^j|\ge 1-|\tau (u^k)|\ge \frac{1}{2}.$$

$$\mathrm{N}({\Phi }_n(u^k))=\prod _{i=1}^m\prod _{j\in T_n}|{\sigma }_i(u^k)-{\zeta }^j|\ge {\left((a^k-1)2^{-m+1}\right)}^{\phi (n)},$$

$$\mathrm{N}({\Phi }_n(u^k))\ge \exp (cn),$$

$$|\mathrm{N}({\Phi }_n(u))|\ge \exp (cn),$$

$$|\mathrm{N}(u^n-1)|{\ll }_{\mathrm{K},\epsilon }\mathrm{Rad}(I_n{J}_n{)}^{1+\epsilon }{\ll }_{\mathrm{K},\epsilon }{\left(\mathrm{N}(I_n)\mathrm{N}(J_n{)}^{1/2}\right)}^{1+\epsilon }.$$

$$\mathrm{N}(J_n{)}^{1/2}{\ll }_{\mathrm{K},\epsilon }\mathrm{N}(I_n{)}^{\epsilon }\mathrm{N}(J_n{)}^{\epsilon /2}{\ll }_{\mathrm{K},\epsilon }|\mathrm{N}(u^n-1){|}^{\epsilon },$$

$$\mathrm{N}(J_n){\ll }_{\mathrm{K},\epsilon }|\mathrm{N}(u^n-1){|}^{2\epsilon }.$$

$$\mathrm{N}(B_n){\ll }_{\mathrm{K},\epsilon }|\mathrm{N}(u^n-1){|}^{2\epsilon }.$$

$$|\mathrm{N}(u^n-1)|\le \prod _{i=1}^m(|{\sigma }_i(u){|}^n+1)\le 2^m({\beta }^m{)}^n,$$

$$\mathrm{N}(B_n){\ll }_{\mathrm{K},\epsilon }{2}^{2m\epsilon }({\beta }^m{)}^{2n\epsilon }.$$

$$\mathrm{N}(A_n)=\mathrm{N}({\Phi }_n(u))/\mathrm{N}(B_n){\gg }_{\mathrm{K},\epsilon }\exp (n(c-2m\epsilon \log \beta )).$$

$$\mathrm{N}(A_n){\gg }_{\mathrm{K},\epsilon }\exp (n(c-2m\epsilon \log \beta )),$$

$$\frac{1}{m}\#\{n,n_0\le n⩽n_1,\phi (n)\ge \frac{1}{2}n\}$$

$$\le \#\{p_n\le X,p_n\text{ prime }|\exists \text{leavevmode}{\mathfrak{p}}_n\in {\mathcal{O}}_k,{\mathfrak{p}}_n|p_n,u^n\equiv 1(\mathrm{mod}{\mathfrak{p}}_n)\text{ and }{u}^n\text{nequiv}1(\mathrm{mod}{\mathfrak{p}}_n^2)\}.$$

Full text

The authors thank Bruno Anglès for pointing them the work of Ichimura. They also thank Georges Gras for constructive observations, Jean-François Jaulent for encouragements, useful comments and the extremely attentive reading, Gebhard Böckle for the exchanges concerning [4] and Zakariae Bouazaoui for his interest in this work. They also want to thank the anonymous referees for their careful works and helpful remarks. The authors were partially supported by the ANR project FLAIR (ANR-17-CE40-0012). CM was also supported by the EIPHI Graduate School (ANR-17-EURE-0002)

Introduction

Let ${\rm K}$ be a number field and let $p$ be a prime number. To simplify, we assume $p$ odd. Denote by ${\rm K}_{p}$ the maximal pro-$p$-extension of ${\rm K}$ unramified outside $p$; put $G_{p}:={\rm Gal}({\rm K}_{p}/{\rm K})$.

By class field theory, the pro-$p$ group ${\rm G}_{p}$ is finitely generated and one knows, since Shafarevich and Koch, that moreover ${\rm G}_{p}$ is finitely presented (meaning that $H^{2}({\rm G}_{p},\mathbb{F}_{p})$ is finite). In fact, ${\rm G}_{p}$ may be pro-$p$ free, for example when ${\rm K}=\mathbb{Q}$, or when ${\rm K}$ is an imaginary quadratic field (when $p>3$) and $p$ doesn’t divide the class number of ${\rm K}$, or when ${\rm K}=\mathbb{Q}(\zeta_{p})$ for $p$ regular primes, etc.

A number field ${\rm K}$ for which ${\rm G}_{p}$ is pro-$p$ free is called $p$-rational ([25]). Observe that ${\rm K}$ is $p$-rational if and only if the Leopoldt conjecture holds for ${\rm K}$ at $p$ and the torsion ${\mathscr{T}}_{p}$ of the abelianization ${\rm G}_{p}^{ab}$ of ${\rm G}_{p}$ is trivial (see [28], or [27, Chapter X, §3]).

The study of ${\mathscr{T}}_{p}$ and of the $p$-rationality started in the beginning of the 80’s with Gras, Nguyen Quang Do, Movahhedi, Jaulent, and their students. Since the literature is rich: see for example [24], [26], [14], [21], [25], [22], [31], [8] etc. See also [13, Chapitre IV, §3 and §4] for a well-detailed presentation of ${\mathscr{T}}_{p}$, of the Leopoldt conjecture and of $p$-rational fields. In the spirit of our paper, let us mention here the works of Byeon [5] and Assim-Bouazzaoui [1] where they showed the infiniteness of $3$ and $5$-rational real quadratic fields.

Let us also precise at this level that a recent series of papers in different topics in number theory showed the interest of $p$-rational fields: Goren [7], Greenberg [16], Böckle-Guiraud-Kalyanswamy-Khare [4], David-Pries [6], Hajir-Maire [17], Hajir-Maire-Ramakrishna [18], etc.

Assuming Leopoldt conjecture (for ${\rm K}$ at $p$), the $p$-rationality of ${\rm K}$ is therefore equivalent to the nullity of ${\mathscr{T}}_{p}$. Observe that ${\mathscr{T}}_{p}\simeq H^{2}({\rm G}_{p},\mathbb{Z}_{p})^{*}$ for a cohomological point of view (see [29]). When the $p$-Sylow of the class group of ${\rm K}$ is trivial, the quantity ${\mathscr{T}}_{p}$ is isomorphic to the torsion of the quotient of the units of the $p$-adic completions ${\rm K}_{v}$ of ${\rm K}$ by the closure of the global units. Moreover, if we assume that no ${\rm K}_{v}$ contains the $p$-roots of the unity (which is always the case when $p>[{\rm K}:\mathbb{Q}]+1$), then the triviality of ${\mathscr{T}}_{p}$ is equivalent to the triviality of the normalized $p$-adic regulator defined by Gras [11, Definition 5.1]. Recently, Gras [9], [10], Pitoun-Varescon [30], Barbulescu-Ray [2] published a series of papers more concentrated on the computations of ${\mathscr{T}}_{p}$, and on some heuristics. In [12, Conjecture 8.11], Gras proposed the following conjecture:

Conjecture** (Gras).**

Let ${\rm K}$ be a number field. Then for large $p$, ${\rm K}$ is $p$-rational.

This conjecture is in the same spirit of the Wieferich prime numbers problem. Indeed, given an odd prime number $p$, to compute the $p$-valuation of $2^{p-1}-1$ is equivalent to compute the normalized $p$-adic regulator of the $2$-units of $\mathbb{Q}$. In particular, in this case the nontriviality of the normalized $p$-adic regulator is equivalent for $p$ to verify the congruence $2^{p-1}\equiv 1({\rm mod}\ p^{2})$.

In [32] Silverman showed how the Wieferich prime numbers are related to the $abc$-conjecture. Let us be more precise. Given an integer $\alpha\in\mathbb{Q}^{\times}\backslash\{\pm 1\}$, Silverman proved that if the $abc$-conjecture holds then as $X\rightarrow\infty$

[TABLE]

where $c>0$ is some absolute constant. See also [15], and [33] for a generalization of Wieferich primes in number fields.

Observe now that the generalized $abc$-conjecture has already been used in the context of Iwasawa theory. Indeed in [19] Ichimura gave a relationship between the Greenberg conjecture and the $abc$-conjecture. A consequence of his work is that, for example, for any real quadratic field ${\rm K}$ if the generalized $abc$-conjecture holds in ${\rm K}$, then the set of primes $p$ for which ${\rm K}$ is $p$-rational, is infinite. See also [4].

The goal of our work is to precise the quantity of such primes $p$, greatly inspired by the computations of Silverman.

Our main result involves the isotypic subspaces ${\mathscr{T}}_{p}^{\chi}$ of ${\mathscr{T}}_{p}$. Let us observe here that the authors studied previously in [23] such cutting and the arithmetic consequences of the nullity of some ${\mathscr{T}}_{p}^{\chi}$.

Let ${\rm K}/\mathbb{Q}$ be a Galois extension of Galois group ${\rm G}$. Let us fix an odd prime number $p\nmid\#{\rm G}$. For an irreducible $\mathbb{Q}_{p}$-character $\psi$ of ${\rm G}$, let $r_{\psi}(E_{\rm K})$ be the $\psi$-rank of $\mathbb{Q}_{p}\otimes E_{\rm K}$, where $E_{\rm K}$ denotes the units of the ring of integers ${\mathscr{O}}_{\rm K}$ of ${\rm K}$. Let us also cut ${\mathscr{T}}_{p}$ by its isotypic subspaces ${\mathscr{T}}_{p}^{\psi}$, and denote by $r_{\psi}({\mathscr{T}}_{p})$ the $\psi$-rank of ${\mathscr{T}}_{p}$. Observe that, assuming Leopoldt conjecture, the number field ${\rm K}$ is $p$-rational if and only if $r_{\psi}({\mathscr{T}}_{p})=0$ for all irreducible $\mathbb{Q}_{p}$-characters $\psi$. Moreover we will see that for $p\gg 0$, $r_{\psi}({\mathscr{T}}_{p})\leq r_{\psi}(E_{\rm K})$ for all $\psi$.

We will then focus on some special units $u$ of $E_{\rm K}$: we denote by $\mathbb{S}$ the set of algebraic integers $u\in\overline{\mathbb{Q}}$ having no conjugate on the unit circle.

Here we prove:

Theorem A**.**

Let ${\rm K}/\mathbb{Q}$ be a Galois extension of Galois group ${\rm G}$ and let $\chi$ be an irreducible $\mathbb{Q}$-character of ${\rm G}$ such that the $\chi$-component of $\mathbb{Q}\otimes E_{\rm K}$ contains some unit $u\in\mathbb{S}$. If the generalized $abc$-conjecture holds for ${\rm K}$, then as $X\rightarrow\infty$

[TABLE]

for some constant $c>0$ depending on ${\rm K}$.

(Of course, in Theorem A one considers only prime numbers $p\nmid\#{\rm G}$.) As consequence we obtain the following result (the real quadratic case was suggested in [4]):

Corollary**.**

Let ${\rm K}/\mathbb{Q}$ be a real quadratic field or an imaginary $S_{3}$-extension. If the generalized $abc$-conjecture holds for ${\rm K}$, then as $X\rightarrow\infty$

[TABLE]

for some constant $c>0$ depending on ${\rm K}$.

Remark 1**.**

It is well known that Leopoldt conjecture holds in the situations of Corollary, but we don’t assume Leopoldt conjecture in Theorem A.

Let us add one additionnal remark about the units in $\mathbb{S}$.

Remark 2**.**

The following observations will be useful for us:

• $-$

an unit $u\neq\pm 1$ for which all the conjugates are real is in $\mathbb{S}$;

• $-$

every cubic field contains some unit $u\in\mathbb{S}$;

• $-$

Pisot numbers are in $\mathbb{S}$.

See also [3] on the abundance of Pisot units.

Our work contains two sections. In the first one, we introduce the objects we need. In the second section, we give the proofs of our results.

1. The objects

We start with a Galois extension ${\rm K}/\mathbb{Q}$ of degree $m$ and Galois group ${\rm G}$. We denote by ${\rm N}$ the norm in ${\rm K}/\mathbb{Q}$.

Let ${\mathscr{O}}_{\rm K}$ be the ring of integers of ${\rm K}$, $E_{\rm K}$ be the units of ${\mathscr{O}}_{\rm K}$, and $\mu_{\rm K}$ be the group of the roots of the unity of ${\rm K}$.

Let $p$ be an odd prime number. In all that will follow, we suppose that:

• $(i)$

$p\nmid\#{\rm G}$,

• $(ii)$

$p$ is unramified in ${\rm K}/\mathbb{Q}$,

• $(iii)$

$p$ does not divide the class number $h_{\rm K}$ of ${\rm K}$.

One excludes this way only a finite set of prime numbers $p$. In particular, there exists an explicit prime number $p_{0}$ such that every $p>p_{0}$ satisfies $(i),(ii)$ and $(iii)$.

1.1. $p$-rational fields and isotypic components

1.1.1.

Let $S_{p}$ be the set of places of ${\rm K}$ above $p$. For $v\in S_{p}$, denote by ${\rm K}_{v}$ the completion of ${\rm K}$ at $v$, by ${\mathscr{O}}_{v}$ the ring of integers of ${\rm K}_{v}$, and by $\pi_{v}$ an uniformizer of ${\rm K}_{v}$. Then the $p$-completion ${\mathscr{E}}_{\rm K}:=\mathbb{Z}_{p}\otimes E_{K}$ of $E_{\rm K}$ embeds diagonally, via $\iota$, in ${\mathscr{U}}_{p}:=\prod_{v\in S_{p}}{\mathscr{U}}_{v}^{1}$, where ${\mathscr{U}}_{v}^{1}:=1+\pi_{v}{\mathscr{O}}_{v}$ is the group of principal units of ${\rm K}_{v}$. Observe that here ${\mathscr{U}}_{p}\simeq\mathbb{Z}_{p}^{m}$. By $p$-adic class field theory (and due to the fact that $p\nmid h_{\rm K}$), the group ${\rm G}_{p}^{ab}$ is isomorphic to ${\mathscr{U}}_{p}/\iota({\mathscr{E}}_{\rm K})$. Then, assuming Leopoldt conjecture for ${\rm K}$ at $p$ (meaning here that $\iota$ is injective), the number field ${\rm K}$ is $p$-rational if and only if ${\mathscr{U}}_{p}/{\iota({\mathscr{E}}_{\rm K})}$ is without torsion.

1.1.2.

Observe that as $p$ is unramified in ${\rm K}/\mathbb{Q}$, we also get that $p\nmid|\mu_{\rm K}|$, and as $p\nmid\#{\rm G}$, the character (as ${\rm G}$-module) of ${\mathscr{E}}_{\rm K}$ is equal to the character of $\mathbb{Q}_{p}\otimes(\mathbb{Q}\otimes E_{\rm K})\simeq{\rm Ind}_{D_{\infty}}^{\rm G}\mathds{1}$, where $D_{\infty}$ is the decomposition group of an archimedean place in ${\rm K}/\mathbb{Q}$ and where $\mathds{1}$ is the trivial character. In particular, ${\mathscr{E}}_{\rm K}$ is a submodule of the regular representation.

To be complete, ${\mathscr{U}}_{p}$ is isomorphic to the regular representation (here ${\mathscr{U}}_{v}^{1}$ has no nontrivial root of unity).

1.1.3.

Let us fix an irreducible $\mathbb{Q}$-character $\chi$ of ${\rm G}$. Let $\mathbb{Q}[{\rm G}]e_{\chi}\simeq{\rm M}_{n_{\chi}}(D)$ be the simple algebra of $\mathbb{Q}[{\rm G}]$ associated to $\chi$, where $D$ is a skew field of degree $s_{\chi}^{2}$ over its center (the integer $s_{\chi}$ is the Schur index of $\chi$). Then $\chi=s_{\chi}\sum_{\psi|\chi}\psi$, where the sum is taken over irreducible $\mathbb{Q}_{p}$-characters $\psi$ dividing $\chi$ (here $p\nmid\#{\rm G}$).

Let $E_{\rm K}^{\chi}$ be the $\chi$-component of the $\mathbb{Q}[{\rm G}]$-module $\mathbb{Q}\otimes E_{\rm K}$, then the character of $E_{\rm K}^{\chi}$ is written as $t_{\chi}\chi$ for some $t_{\chi}\in\{0,\cdots,n_{\chi}\}$. Given an irreducible $\mathbb{Q}_{p}$-character $\psi|\chi$, the integer $s_{\chi}t_{\chi}$ is then the $\psi$-rank $r_{\psi}(E_{\rm K})$ of $\mathbb{Q}_{p}\otimes E_{\rm K}$.

If $M$ is a $\mathbb{Z}_{p}[{\rm G}]$-module of finite type, the $\psi$-rank $r_{\psi}(M)$ of $M$ is defined as $\displaystyle{r_{\psi}(M):=\frac{1}{{\rm deg}(\psi)}\dim_{{\mathbb{F}}_{p}}(M^{\psi}/(M^{\psi})^{p})}$.

As seen before $r_{\psi}(E_{\rm K})=r_{\psi}({\mathscr{E}}_{\rm K})$, obviously $r_{\psi}({\mathscr{E}}_{\rm K})\geq r_{\psi}(\iota({\mathscr{E}}_{\rm K}))$, and Leopoldt conjecture is equivalent to the equality $r_{\psi}({\mathscr{E}}_{\rm K})=r_{\psi}(\iota({\mathscr{E}}_{\rm K}))$ for every $\chi$ and $\psi$. Observe that one knows that $r_{\psi}(\iota({\mathscr{E}}_{\rm K}))\geq 1$ when $r_{\psi}({\mathscr{E}}_{\rm K})\neq 0$ (see [20]).

\remaname* \the\smf@thm.*

When ${\rm G}$ is abelian, one has $r_{\psi}({\mathscr{E}}_{\rm K})\leq 1$.

As seen before, with all the assumptions, the torsion of ${\mathscr{U}}_{p}/\iota({\mathscr{E}}_{\rm K})$ is isomorphic to ${\mathscr{T}}_{p}$. Thus, $r_{\psi}({\mathscr{T}}_{p})\leq r_{\psi}({\mathscr{E}}_{\rm K})$. If for every $\psi|\chi$ the $\psi$-rank of ${\mathscr{U}}_{p}/\iota({\mathscr{E}}_{\rm K})$ is maximal, meaning $r_{\psi}({\mathscr{T}}_{p})=r_{\psi}({\mathscr{E}}_{\rm K})$, then necessarily, for every unit $x\in E_{\rm K}^{\chi}$ such that $x\equiv 1({\rm mod}\ {\mathfrak{p}})$ for all ${\mathfrak{p}}|p$, one must have $x\equiv 1({\rm mod}\ {\mathfrak{p}}^{2})$ for all ${\mathfrak{p}}|p$.

\lemmname** \the\smf@thm.**

If there exists an unit $u\in E_{\rm K}^{\chi}$ such that $u\equiv 1({\rm mod}\ {\mathfrak{p}}_{0})$ but $u\nequiv 1({\rm mod}\ {\mathfrak{p}}_{0}^{2})$ for some ${\mathfrak{p}}_{0}|p$, then $r_{\psi}({\mathscr{T}}_{p})<r_{\psi}({\mathscr{E}}_{\rm K})$ for some $\psi|\chi$.

Proof.

Put $x=u^{{\rm N}({\mathfrak{p}}_{0})-1}\in E_{\rm K}^{\chi}$, where ${\rm N}({\mathfrak{p}})=\#{\mathscr{O}}_{\rm K}/{\mathfrak{p}}$. Observe that $x\equiv 1({\rm mod}\ {\mathfrak{p}})$ for every ${\mathfrak{p}}|p$ (the extension ${\rm K}/\mathbb{Q}$ is Galois) but, easily, one also has $x\nequiv 1({\rm mod}\ {\mathfrak{p}}_{0}^{2})$. We conclude with the small discussion above. ∎

1.2. The generalized $abc$-conjecture

See [34]. If $I\subset{\mathscr{O}}_{\rm K}$ is an integral ideal, let us denote by ${\rm Rad}(I)$ the following ideal:

[TABLE]

where the product is taken over prime ideal ${\mathfrak{p}}$ dividing $I$ and where as usual ${\rm N}({\mathfrak{p}})=\#{\mathscr{O}}_{K}/{\mathfrak{p}}$ is the absolute norm of ${\mathfrak{p}}$.

The generalized $abc$-conjecture for ${\rm K}$ states that for any $\varepsilon>0$, there exists a constant $C_{K,\varepsilon}>0$ such that the inequality :

[TABLE]

holds for all nonzero $a,b,c\in{\mathscr{O}}_{K}$ verifying $a+b=c$, $(a,b)=1$, where the product is taken over all absolute values of ${\rm K}$ and where $|\cdot|_{v}$ denotes the normalized norm of ${\rm K}_{v}$ (such that $\prod_{v}|x|_{v}=1$ for all $x\in{\rm K}^{\times}$).

Here we use it in the case where $b=u_{2}$ and $c=u_{1}$ are two distinct units of ${\rm K}$ and $a=u_{1}-u_{2}$ : for every $\varepsilon>0$, there exists a constant $C_{{\rm K},\varepsilon}$ such that for all $u_{1}\neq u_{2}\in E_{\rm K}$, one has

[TABLE]

2. Proofs

2.1.

As explained in Introduction, some part of the proof is greatly inspired by [32].

Let ${\rm K}/\mathbb{Q}$ be a Galois extension of degree $m$. Consider the number field ${\rm L}:={\rm K}(\zeta)$ where $\zeta$ is a primitive $n$th-root of $1$. The extension ${\rm L}/\mathbb{Q}$ is Galois of degree $O(\varphi(n))$.

Let $T_{n}$ be the set of integers $j\in\{1,\cdots,n-1\}$ coprime to $n$. We denote by $\Phi_{n}$ the $n$th cyclotomic polynomial: $\displaystyle{\Phi_{n}(u)=\prod_{j\in T_{n}}(u-\zeta^{j})}$. The polynomial $\Phi_{n}$ is of degree $\varphi(n)$. Thereafter, we will focus on integer $n$ such that $\varphi(n)\geq\frac{1}{2}n$. Recall Lemma 6 of [32]:

[TABLE]

We start with the key lemma extending Lemma 5 of [32].

\lemmname** \the\smf@thm.**

Let $u\in E_{\rm K}\cap\mathbb{S}$. Then there exists some $k\in\mathbb{Z}_{>0}$ such that

[TABLE]

for $n$ such that $\varphi(n)\geq\frac{1}{2}n$, where $c>0$ is a constant depending on $u$ and $k$.

Proof.

As $u\in\mathbb{S}$, there exists an embedding $\sigma:{\rm K}\hookrightarrow\mathbb{C}$ such that $|\sigma(u)|\geq a>1$, for some real $a$. Hence, for $k\in\mathbb{Z}_{>0}$, we get $|\sigma(u^{k})|\geq a^{k}$, and then $|\sigma(u^{k})-\zeta^{j}|\geq a^{k}-1$.

Let us choose an another embedding $\tau$. We want to give some ”good” lower bound for $|\tau(u^{k})-\zeta^{j}|$. As $u\in\mathbb{S}$ there is only two situations.

$\bullet$ If $|\tau(u)|<1$, then clearly for sufficiently large $k$, we get

[TABLE]

$\bullet$ If $|\tau(u)|>1$, for sufficiently large $k$, we get $|\tau(u^{k})-\zeta^{j}|\geq 1$.

Putting all of this together, we obtain

[TABLE]

Consequently, by taking sufficiently large $k$, we get that for every $n$ with $\varphi(n)\geq\frac{1}{2}n$

[TABLE]

where the $\sigma_{i}$’s are the embeddings of ${\rm K}$ in $\mathbb{C}$ and where $c>0$ is some constant (depending on $u$, $k$ and $m$). ∎

Suppose now that $u\in E_{\rm K}$ is such that

[TABLE]

for every $n$ such that $\varphi(n)\geq\frac{1}{2}n$ (which is always possible by Lemma 2.1).

Let us write $(u^{n}-1)=I_{n}J_{n}$, with $I_{n}$ and $J_{n}$ relatively prime and where if ${\mathfrak{p}}|I_{n}$, then ${\mathfrak{p}}^{2}\nmid I_{n}$, and if ${\mathfrak{p}}|J_{n}$ then ${\mathfrak{p}}^{2}|J_{n}$. Then, if we write $u^{n}-1+1=u^{n}$, the generalized $abc$-conjecture implies that

[TABLE]

Hence, as $|{\rm N}(u^{n}-1)|={\rm N}(I_{n})N(J_{n})$, we get

[TABLE]

and then

[TABLE]

Now let us also write $(\Phi_{n}(u))=A_{n}B_{n}$, with $A_{n}$ and $B_{n}$ relatively prime and where if ${\mathfrak{p}}|A_{n}$, then ${\mathfrak{p}}^{2}\nmid A_{n}$, and if ${\mathfrak{p}}|B_{n}$ then ${\mathfrak{p}}^{2}|B_{n}$. Of course, $B_{n}|J_{n}$, and then

[TABLE]

Choose $\beta>1$ such that $|\sigma_{i}(u)|\leq\beta$ for all $i$. Then

[TABLE]

which implies

[TABLE]

Hence,

[TABLE]

We finally obtain:

\propname** \the\smf@thm.**

If the generalized $abc$-conjecture holds then for all $\varepsilon>0$, one has

[TABLE]

for every $n$ such that $\varphi(n)\geq\frac{1}{2}n$.

Take now $\varepsilon>0$ such that $\displaystyle{\varepsilon<\frac{c}{2m{\rm log}(\beta)}}$. Thanks to Proposition 2.1, there exists $n_{0}\in\mathbb{Z}_{>0}$ such that for all $n\geq n_{0}$, with $\varphi(n)\geq\frac{1}{2}n$, then ${\rm N}(A_{n})>n^{m}$, where we recall that $m=[{\rm K}:\mathbb{Q}]$. Then, for each such $n$, there exists a prime ideal ${\mathfrak{p}}_{n}\subset{\mathscr{O}}_{K}$, dividing $A_{n}$ but not $n$: indeed if it was not the case then as $A_{n}$ is square free, $A_{n}$ would divide $n$, which contradicts ${\rm N}(A_{n})>n^{m}$. Observe that ${\mathfrak{p}}_{n}|(u^{n}-1)$ implies ${\rm N}({\mathfrak{p}}_{n})\leq 2^{m}\beta^{mn}$.

As ${\mathfrak{p}}_{n}\nmid n$, the polynomial $X^{n}-1$ is separable over ${\mathscr{O}}_{K}/{\mathfrak{p}}_{n}$. Thus $u$ is a simple root of $X^{n}-1=\prod_{d|n}\Phi_{d}(X)$ modulo ${\mathfrak{p}}_{n}$ and, as ${\mathfrak{p}}_{n}$ divides $\Phi_{n}(u)$, its order in $\left({\mathscr{O}}_{\rm K}/{\mathfrak{p}}_{n}\right)^{\times}$ is exactly $n$. Furthermore, ${\mathfrak{p}}_{n}$ is a divisor of $A_{n}$, so ${\mathfrak{p}}_{n}^{2}$ does not divide $u^{n}-1$ (in other words ${\mathfrak{p}}_{n}|I_{n}$).

Let $p_{n}$ be the prime number such that $p_{n}\mathbb{Z}={\mathfrak{p}}_{n}\cap\mathbb{Z}$.

In conclusion, we obtain:

\propname** \the\smf@thm.**

Take $u\in E_{\rm K}$ as before. For each $n\geq n_{0}$ such that $\varphi(n)\geq\frac{1}{2}n$, there exists a prime ideal ${\mathfrak{p}}_{n}\subset{\mathscr{O}}_{\rm K}$ such that

• $(i)$

${\mathfrak{p}}_{n}|\Phi_{n}(u)$* and $u^{n}\nequiv 1({\rm mod}\ {\mathfrak{p}}_{n}^{2})$,*

• $(ii)$

$u$* is of order $n$ in $\left({\mathscr{O}}_{\rm K}/{\mathfrak{p}}_{n}\right)^{\times}$,*

• $(iii)$

${\rm N}({\mathfrak{p}}_{n})\leq\gamma^{n}$, for some $\gamma$ depending only on ${\rm K}$.

By $(ii)$ of Proposition 2.1, it follows that ${\mathfrak{p}}_{n}={\mathfrak{p}}_{n^{\prime}}$ if and only if $n=n^{\prime}$. Observe that a set of primes ${\mathfrak{p}}_{n}$ of size $Y$ gives at least $Y/m$ primes $p_{n}$.

Now given $X\geq 1$, let $n_{1}$ be the largest integer such that $\gamma^{n_{1}}\leq X$. Assume $X$ sufficiently large to ensure $n_{0}\leq n_{1}$. Then, for each $n\in[n_{0},n_{1}]$ such that $\varphi(n)\geq\frac{1}{2}n$, there exists a prime ideal ${\mathfrak{p}}_{n}\subset{\mathscr{O}}_{\rm K}$ for which $u^{n}\equiv 1({\rm mod}\ {\mathfrak{p}}_{n})$ and $u^{n}\nequiv 1({\rm mod}\ {\mathfrak{p}}_{n}^{2})$. Note that $p_{n}\leq{\rm N}({\mathfrak{p}}_{n})\leq\gamma^{n}\leq\gamma^{n_{1}}\leq X$. Thereby:

[TABLE]

In conclusion, one has found at least $c\ {\rm log}X$ prime numbers $p_{n}\leq X$ satisfying $(i)$ of Proposition 2.1 for some ${\mathfrak{p}}_{n}|p_{n}$.

2.2.

Proof of Theorem A. Let $\chi$ be an irreducible $\mathbb{Q}$-character of ${\rm G}$ such that there exists some $u\in E_{\rm K}^{\chi}\cap\mathbb{S}$. By the previous section, there exists $k\geq 1$ such that $u^{kn}\equiv 1({\rm mod}\ {\mathfrak{p}}_{n})$ and $u^{kn}\nequiv 1({\rm mod}\ {\mathfrak{p}}_{n}^{2})$ for at least $c\ {\rm log}X$ prime numbers $p_{n}\leq X$ (where ${\mathfrak{p}}_{n}|p_{n}$). We conclude with Lemma 1.1.3 (after forgetting the prime numbers smaller than $p_{0}$).

Proof of the Corollary.

Observe first that, in the two cases, the Leopoldt conjecture holds and the field ${\rm K}$ contains some unit in $\mathbb{S}$ (see Remark 2). Take $p>p_{0}$. The choice of the character is the following : if ${\rm K}$ is real quadratic, let $\chi=\psi$ be the nontrivial character of ${\rm G}$ ; if ${\rm K}/\mathbb{Q}$ is an imaginary $S_{3}$-extension, let $\chi$ be the irreducible $\mathbb{Q}$-character of ${\rm G}$ of degree $2$ (observe that $\chi=\psi$ is also $\mathbb{Q}_{p}$-irreducible). Then $\mathbb{Q}\otimes E_{\rm K}=E_{\rm K}^{\chi}$, $r_{\psi}(E_{\rm K})=1$, and ${\mathscr{T}}_{p}={\mathscr{T}}_{p}^{\psi}$. Therefore by Theorem A, ${\mathscr{T}}_{p}=\{1\}$ for at least $c\ {\rm log}X$ prime numbers $p\leq X$.