Genus theory and $\epsilon$-conjectures on p-class groups

TL;DR

This paper investigates the impact of genus theory on the p-rank epsilon-conjecture for p-class groups, proving the conjecture for degree p cyclic fields and analyzing potential obstructions in the proof.

Contribution

It proves the p-rank epsilon-conjecture for degree p cyclic fields and analyzes the role of genus theory and exceptional classes as obstructions.

Findings

Proved the p-rank epsilon-conjecture for degree p cyclic extensions.

Identified the role of exceptional p-classes as potential obstructions.

Compared epsilon-conjectures with p-adic Brauer-Siegel type conjectures.

Abstract

We suspect that the ``genus part'' of the class number of a number field K may be an obstruction for an ``easy proof'' of the classical p-rank epsilon-conjecture for p-class groups and, a fortiori, for a proof of the ``strong epsilon-conjecture'': \# (Cl\_K \otimes \Z\_p) <<\_(d,p,epsilon) ($\sqrt$D\_K)^epsilon for all K of degree d. We analyze the weight of genus theory in this inequality by means of an infinite family of degree p cyclic fields with many ramified primes, then we prove the p-rank epsilon-conjecture: \# (Cl\_K \otimes \F\_p) <<\_(d,p,epsilon) ($\sqrt$D\_K)^epsilon, for d=p and the family of degree p cyclic extensions (Theorem 2.5) then sketch the case of arbitrary base fields. The possible obstruction for the strong form, in the degree p cyclic case, is the order of magnitude of the set of ``exceptional'' p-classes given by a well-known non-predictible algorithm, but controled thanks to recent density results due to Koymans--Pagano. Then we compare the epsilon-conjectures with some p-adic conjectures, of Brauer- Siegel type, about the torsion group T\_K of the Galois group of the maximal abelian p-ramified pro-p-extension of totally real number fields K. We give numerical computations with the corresponding PARI/GP programs.

Equations

$$\text{raise}1.5pt\#(\mathcal{C}{\ell }_K\otimes {\mathbb{F}}_p)\le C_{d,p,\epsilon }\cdot (\sqrt{D_K}{)}^{\epsilon },$$

$$\text{raise}1.5pt\#(\mathcal{C}{\ell }_K\otimes {\mathbb{Z}}_p)\le C_{d,p,\epsilon }\cdot (\sqrt{D_K}{)}^{\epsilon }.$$

$$1\to (\mathcal{C}{\ell }_K\otimes {\mathbb{Z}}_p{)}^G\text{relbar}\to \mathcal{C}{\ell }_K\otimes {\mathbb{Z}}_p\text{relbar}\to (\mathcal{C}{\ell }_K\otimes {\mathbb{Z}}_p{)}^{1-\sigma }\to 1$$

$$\raise 1.5pt\hbox{${\scriptscriptstyle\#}$}\big{(}{\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}\otimes\mathbb{Z}_{p}\big{)}^{G_{N,p}}=p^{N-1}.$$

$$\displaystyle\lim_{X\to\infty}\bigg{[}\frac{\raise 1.5pt\hbox{${\scriptscriptstyle\#}$}\big{\{}K:\ \sqrt{D_{K}}<X\ \ \&\ \ \raise 1.5pt\hbox{${\scriptscriptstyle\#}$}\big{(}{\mathcal{C}}\hskip-2.0pt{\ell}(K)\otimes\mathbb{Z}_{p}\big{)}^{1-\sigma}>h(\sqrt{D_{K}}\,)\,\big{\}}}{\raise 1.5pt\hbox{${\scriptscriptstyle\#}$}\{K:\ \sqrt{D_{K}}<X\}}\bigg{]}=0.$$

$$\raise 1.5pt\hbox{${\scriptscriptstyle\#}$}\big{(}{\mathcal{C}}\hskip-2.0pt{\ell}(K)\otimes\mathbb{Z}_{p}\big{)}^{1-\sigma}>\displaystyle C\ \frac{\big{(}\sqrt{D_{K}}\big{)}^{\varepsilon}}{p^{N-1}}=:C\,h_{\varepsilon}(\sqrt{D_{K}}),$$

$$M_{i+1}/M_i:=(M/M_i{)}^G,for\; 0\le i\le m-1,$$

$$\raise 1.5pt\hbox{${\scriptscriptstyle\#}$}\big{(}M_{i+1}/M_{i}\big{)}=\frac{p^{N-1}}{(\Lambda_{i}:\Lambda_{i}\cap{\rm N}_{F/\mathbb{Q}}(F^{\times}))}\ \,\&\ \,\Lambda_{i}:=\{x\in\mathbb{Q}^{\times},\ (x)\in{\rm N}_{F/\mathbb{Q}}({\mathcal{I}}_{i})\},$$

$$(y)=\mathfrak{A}\cdot {\mathfrak{B}}^{1-\sigma },$$

$$t_0^N=0\le t_1^N\le \cdots \le t_m^N=N-1.$$

$$\displaystyle\mathop{\raise 2.0pt\hbox{$\sum$}}\limits_{i=(r-1)\,(p-1)}^{r\,(p-1)-1}\big{(}N-1-t_{i}^{N}\big{)}=(p-1)\cdot(N-1)-\displaystyle\mathop{\raise 2.0pt\hbox{$\sum$}}\limits_{i=(r-1)\,(p-1)}^{r\,(p-1)-1}t_{i}^{N}.$$

$${\mathcal{C}}\hskip-2.0pt{\ell}_{F}\otimes\mathbb{Z}_{q}\simeq\mathbb{Z}[\mu_{p}]\big{/}{\rm Phys.~{}Rev.~{}D}_{{\mathfrak{q}}\mid q}{\mathfrak{q}}^{n_{\mathfrak{q}}},\ \hbox{$n_{\mathfrak{q}}\geq 0$},$$

$$\ell_{k}>\hbox{$\frac{p-1}{2}$}\cdot k\cdot{\rm log}\big{(}\hbox{$\frac{\ell_{k}}{p}$}\big{)},\ \hbox{for all $k\geq 1$}.$$

$$\pi(x;1,p)\leq\displaystyle\frac{2\,x}{(p-1)\,{\rm log}\big{(}\frac{x}{p}\big{)}};$$

$$\text{raise}1.5pt\#(\mathcal{C}{\ell }_{F_{N,p}}\otimes {\mathbb{Z}}_p)=:p^{N-1+\Delta (N)}\le C_{p,p,\epsilon }\cdot (\sqrt{D_{N,p}}{)}^{\epsilon },$$

$$p^{N-1+\delta (N)}=p^{(p-1)(N-1)-{\sum }_{i=1}^{p-2}{t}_i^N}\le C_{p,p,\epsilon }\cdot (\sqrt{D_{N,p}}{)}^{\epsilon },$$

$$X(N):=$$

$$=$$

$$X(N)=$$

$$=$$

$$X(N)=$$

$$\displaystyle\hskip 56.9055pt-\varepsilon\cdot\hbox{$\frac{p-1}{2}$}\big{[}N{\rm log}(N)+N\cdot\gamma_{p}+\hbox{$\frac{1}{2}$}{\rm log}(N)+O(1)\big{]}.$$

$$X(N)$$

$$-\log (p)-\epsilon \cdot \frac{p-1}{2}N\cdot {\gamma }_p-\epsilon \cdot \frac{p-1}{4}\log (N)-\epsilon \cdot O(1)$$

$$\displaystyle=N\cdot\Big{[}-\varepsilon\cdot\hbox{$\frac{p-1}{2}$}\,{\rm log}(N)+\hbox{$\frac{\Delta(N)}{N}$}\,{\rm log}(p)$$

$$\displaystyle\hskip 91.04872pt+{\rm log}(p)-\hbox{$\frac{{\rm log}(p)}{N}$}-\varepsilon\cdot O(1)\Big{]}$$

$$\displaystyle=N\cdot\Big{[}-\varepsilon\cdot\hbox{$\frac{p-1}{2}$}\,{\rm log}(N)+\hbox{$\frac{\Delta(N)}{N}$}\,{\rm log}(p)$$

$$\displaystyle\hskip 128.0374pt+(1-N^{-1})\,{\rm log}(p)-\varepsilon\cdot O(1)\Big{]}.$$

$$X_{0}(N)=-\varepsilon\,\hbox{$\frac{p-1}{2}$}\,N\,{\rm log}(N)+N\,\big{[}(p-1)\,{\rm log}(p)-\varepsilon\,O(1)-o(1)\big{]}.$$

$$\begin{aligned} &(N-1+\delta(N))\,{\rm log}(p)\ \leq\ {\rm log}(C_{\varepsilon})+\varepsilon\,\hbox{$\frac{p-1}{2}$}\hbox{$\sum_{k=1}^{N}$}{\rm log}(\ell_{k})\\ &\leq\ {\rm log}(C_{\varepsilon})+\varepsilon\,\hbox{$\frac{p-1}{2}$}\hbox{$\sum_{k=1}^{N}$}\big{[}{\rm log}(c_{p})+2\,{\rm log}(p-1)+2\,{\rm log}(k)\big{]}\\ &\leq\ {\rm log}(C_{\varepsilon})+\varepsilon\,\hbox{$\frac{p-1}{2}$}\cdot\big{[}N\,\gamma^{\prime}_{p}+2\,{\rm log}(N!)\big{]}\end{aligned},$$

$$(N-1+\delta(N)){\rm log}(p)\leq{\rm log}(C_{\varepsilon})+\varepsilon\hbox{$\frac{p-1}{2}$}\big{[}2N{\rm log}(N)+N\,\gamma^{\prime\prime}_{p}+{\rm log}(N)+O(1)\big{]},$$

$$\log (C_{\epsilon })$$

$$-\epsilon \frac{p-1}{2}N{\gamma }_p^{\prime \prime }-\epsilon \frac{p-1}{2}\log (N)-\epsilon O(1)$$

$$\displaystyle\geq N\cdot\Big{[}-\varepsilon\,(p-1)\,{\rm log}(N)+\hbox{$\frac{\delta(N)}{N}$}\,{\rm log}(p)+O(1)\Big{]}.$$

$$\log (C_{p,p,c,\epsilon })\le$$

Full text

1. 1 Introduction

2. 2 A possible obstruction due to exceptional $p$-classes

3. 2.1 Definition of the family $(F_{N,p})_{N\geq 1}$

4. 2.2 About the exceptional $p$-classes

5. 2.2.1 The algorithm

6. 2.2.2 Examples of maximal rank

7. 2.3 Estimation of $C_{p,p,\varepsilon}$ for the fields $F_{N,p}$

8. 2.4 A lower bound for $C_{p,p,\varepsilon}$

9. 3 PARI/GP programs computing ${\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}$

10. 3.1 Structure of some ${\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}$

11. 3.2 Examples of exeptional $p$-classes

12. 4 Genus theory – Abelian $p$-ramification theory

13. 4.1 Reminders on Genus theory

14. 4.1.1 Genus field and genus number $g_{K/k}$

15. 4.1.2 Variants of the strong $\varepsilon$-conjecture

16. 4.1.3 Computation of some successive local maxima

17. 4.1.4 Reciprocal study

18. 4.2 Reminders on $p$-ramification theory

19. 4.2.1 Structure of the $p$-torsion group ${\mathcal{T}}_{K}$

20. 4.2.2 The $p$-adic Brauer–Siegel conjecture for ${\mathcal{T}}_{K}$

21. 4.2.3 Estimation of $\widetilde{\mathcal{C}}_{F_{N,p},p}$

22. 4.3 $p$-adic Brauer–Siegel conjecture versus $\varepsilon$-conjectures

23. 4.3.1 Analysis by means of CM-fields

24. 4.3.2 Computation of $\widetilde{\mathcal{C}}_{K,p}$ for imaginary quadratic fields

25. 4.3.3 Conclusion

26. 4.4 Reflection theorem and $p$-rank $\varepsilon$-conjectures

1. Introduction

For any number field $K$, we denote by ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$ the class group of $K$ in the restricted sense and by ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p}$ its $p$-class group, for any prime number $p$; to avoid any ambiguity, we shall write ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{F}_{p}$ for the “$p$-torsion group”, often denoted ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}[p]$ in most papers, only giving the $p$-rank ${\rm rk}_{p}({\mathcal{C}}\hskip-2.0pt{\ell}_{K})$ of ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$. One knows the following classical result (weak form of theorems of Brauer, Brauer–Siegel, Tsfasman–Vladŭţ–Zykin [41, 44]):

For all $\varepsilon>0$ there exists a constant $C_{d,\varepsilon}$ such that \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}\leq C_{d,\varepsilon}\cdot(\sqrt{D_{K}}\,)^{1+\varepsilon}, where $d$ is the degree of $K$ and $D_{K}$ the absolute value of its discriminant.

But it is clear that, arithmetically, the behaviour of the $p$-Sylow subgroups of ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$ depends, in conflicting manners (regarding $p$), on many parameters (signature, ramification, prime divisors of $d$, action of Galois groups, etc.).

After the Cohen–Lenstra–Martinet, Adam–Malle, Delaunay–Jouhet, Gerth, Koymans–Pagano,… heuristics, conjectures, or density statements, on the order and structure of ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p}$ [4, 5, 1, 30, 8, 15, 28], many authors study and prove inequalities of the form \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{F}_{p})\leq C_{d,p,\varepsilon}\cdot(\sqrt{D_{K}}\,)^{\,c+\varepsilon}, with positive constant $c<1$ as small as possible (e.g., under GRH, the inequality \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{F}_{p})\leq C_{d,p,\varepsilon}\cdot(\sqrt{D_{K}}\,)^{\,1-\frac{1}{p\,(d-1)}+\varepsilon} [10, Proposition 1]; see also [14, § 1.1] for more examples and comments). The various links between this $\varepsilon$-conjecture and the above classical heuristics (or results) are described in [33, § 1.1, Theorem 1.2, Remark 3.3].

For a general history upon today about such inequalities, we refer to some recent papers of the bibliography (e.g., [9, 10, 14, 33, 43]) in which the reader can have a more complete list of recent contributions.

For short, we shall call “$p$-rank $\varepsilon$-conjecture” the case:

[TABLE]

and “strong $\varepsilon$-conjecture” the case:

[TABLE]

These kind of results do not separate the case of totally real fields, for which we know that the class number is rather small, regarding the case of non totally real fields (see the tables in Washington’s book [42] and papers by Schoof as [36, 38], among many contributions; however, some real fields may have exceptional large class numbers regarding $D_{K}$ [6, 7]). Moreover, by nature, the results that we have quoted deal with upper bounds of the $p$-rank ${\rm rk}_{p}({\mathcal{C}}\hskip-2.0pt{\ell}_{K})$ and precisely we shall see that genus theory gives, when it applies, large and maximal $p$-ranks and possibly unbounded orders, which probably makes harder proofs by classical complex analytic way.

Remark 1.1*.*

Nevertheless, many arguments and computations are in favor of the strong $\varepsilon$-conjecture \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p})\leq C_{d,p,\varepsilon}\cdot(\sqrt{D_{K}}\,)^{\varepsilon}, except possibly for subfamilies of density zero, even if such a conjecture is “a conjecture at infinity for fixed $p$” because, as we shall see in numerical calculations, the constants $C_{d,p,\varepsilon}$ are enormous, especially for $d=p$, so that, for “usual fields”, the inequalities are trivial up to some huge values of the discriminant. So it will be difficult to give convincing computations “at infinity”.

In this context, we shall only prove the case of the $p$-rank $\varepsilon$-conjecture for degree $p$ cyclic fields $K$ (Theorem 2.5):

Main Theorem. Denote by ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$ the class group of any number field $K$. Let $p\geq 2$ be a prime number; then the $p$-rank $\varepsilon$-conjecture saying that: \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{F}_{p})\ll_{d,p,\varepsilon}(\sqrt{D_{K}}\,)^{\varepsilon} for all $K$ of degree $d$, is true for the subfamily of all cyclic extension $F/\mathbb{Q}$ of degree $d=p$.

For these fields, with Galois group $G=:\langle\sigma\rangle$, we shall use the exact sequence:

[TABLE]

and consider the “non-genus part” $({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p})^{1-\sigma}$ (or set of “exceptional $p$-classes”) as a random object for which densities results are known (see Remark 2.2).

The case $d=p=2$ is known from Gauss genera theory, but its generalization is not obvious since for $p=2$ the $p$-rank is canonical (given by Chevalley’s formula (1)) while for $p>2$ it depends on the algorithm which determines the complete structure of ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p}$; indeed, for $d=p=2$ and any class $\gamma$ such that $\gamma^{2}=1$, we can write $\gamma^{\sigma-1}=\gamma^{\sigma+1-2}={\rm N}_{K/\mathbb{Q}}(\gamma)\,\gamma^{-2}=1$, which does not work for $d=p>2$.

To our knowledge, for $d=p>2$, only the case of cyclic cubic fields is proved [10, Corollary 1, case (3)] for the $3$-rank $\varepsilon$-conjecture.

For CM fields, we shall try to put the “minus part” of ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p}$ in “duality” with the “plus part” of the torsion group ${\mathcal{T}}_{K}$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $K$ for which, on the contrary, we know that there is no complete strong $\varepsilon$-conjecture because of some explicit families of density zero.

This suggests that, for any given $p$, the strong $\varepsilon$-conjecture may be true “for almost all field” in a sense to be specified.

2. A possible obstruction due to exceptional $p$-classes

We shall illustrate the above comments with a family of fields with optimal $p$-ranks, and give numerical illustrations with PARI/GP [32] programs.

2.1. Definition of the family $(F_{N,p})_{N\geq 1}$

We consider a fixed prime $p\geq 2$ and the sequence of all odd prime numbers $\ell_{k}$, totally split in $\mathbb{Q}(\mu_{p})$, whence such that $\ell_{k}\equiv 1\pmod{p}$. For $p>2$, let $F_{N,p}$ be any cyclic extension of degree $p$, of global conductor $f_{N,p}=\prod_{k=1}^{N}\ell_{i}$, for any $N\geq 1$. For $p=2$ we shall consider $F_{N,2}:=\mathbb{Q}(\sqrt{\pm\,\ell_{1}\cdots\ell_{N}})$ $=\mathbb{Q}(\sqrt{\pm\,3\cdot 5\cdots\ell_{N}})$ with $\pm\,$ such that $2$ be unramified, and consider the restricted sense for class groups, which is more canonical for our purpose. We may consider degree $p$ cyclic extensions ramified at $p$ without any modification of the forthcoming reasonings, except an useless complexity of redaction (this modifies the above conductors up to the constant factor $p^{2}$ or $8$).

The discriminant $D_{N,p}$ of $F_{N,p}$ is the product of the conductors associated to each nontrivial character of $G_{N,p}:={\rm Gal}(F_{N,p}/\mathbb{Q})$, thus $D_{N,p}=f_{N,p}^{p-1}$. There are $(p-1)^{N-1}$ such fields, all contained in $\mathbb{Q}(\mu_{\ell_{1}\cdots\ell_{N}})$.

The Chevalley formula [3] gives the number of ambiguous classes (i.e., invariant by $G_{N,p}$) which is equal to the genus number for cyclic fields:

[TABLE]

It is obvious that the group of ambiguous classes is $p$-elementary of $p$-rank $N-1\leq{\rm rk}_{p}({\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}})$ and generated by the ramified primes ${\mathfrak{l}}_{i}\mid\ell_{i}$, $1\leq i\leq N$, with a (non-trivial) relation $\prod_{i=1}^{N}{\mathfrak{l}}_{i}^{n_{i}}=(a)$, $a\in F_{N,p}^{\times}$, due to the classical Kummer theory over $\mathbb{Q}(\mu_{p})$ giving $F_{N,p}(\mu_{p})=\mathbb{Q}(\mu_{p})(\sqrt[p]{\alpha})$, $\alpha\in\mathbb{Q}(\mu_{p})^{\times}$ depending of canonical Gauss sums.

2.2. About the exceptional $p$-classes

We know that \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}\otimes\mathbb{F}_{p}) and \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}\otimes\mathbb{Z}_{p}), both depend of a non-predictible algorithm that we recall below; this shows that the so-called Brumer–Silverman–Duke–Zhang–Ellenberg–Venkatesh $\varepsilon$-conjecture on the $p$-ranks is not so different from the strong $\varepsilon$-conjecture, in a logical point of view, except that we shall see that the $p$-rank is an $O(N)$ contrary to \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}\otimes\mathbb{Z}_{p}) whose order of magnitude is unknown.

Definition 2.1**.**

Let $\delta(N)\geq 0$ be such that \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}\otimes\mathbb{F}_{p})=p^{N-1+\delta(N)} and let $\Delta(N)\geq\delta(N)$ be such that \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}\otimes\mathbb{Z}_{p})=p^{N-1+\Delta(N)}.

In other words, p^{\Delta(N)}=\raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}\otimes\mathbb{Z}_{p})^{1-\sigma} measures what we shall call the set of “exceptional $p$-classes” (i.e., non-invariant) obtained via the classical filtration of ${\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}\otimes\mathbb{Z}_{p}$ (a general theoretical and numerical approach was given in [19, 20] and [39] after the historical papers of Inaba [25], Redei–Reichardt [35], Fröhlich [11, 12] and others; for a wide generalization to ray class groups and a survey, see [18]). Then $N-1+\delta(N)$ is the $p$-rank.

Remark 2.2*.*

During the writing of this paper we have been informed, by Peter Koymans and Carlo Pagano, of their work [28] proving that the $p$-class groups of cyclic degree $p$ fields have, under GRH, the (explicit) distribution conjectured by Frank Gerth III [15] and generalizing results of Jack Klys by means of methods developed by Etienne Fouvry–Jürgen Klüners, Alexander Smith and others.

These results prove that the strong $\varepsilon$-conjecture for degree $p$ cyclic fields $K$ is true (in the meaning that this occurs with probability $1$), except possibly for very sparse families of fields of density zero. One deduces this from [28, Theorem 1.1] by proving that for any map $h:\mathbb{R}\to\mathbb{R}$, such that $h(X)\to\infty$ as $X\to\infty$, we have the following density property about the sets of exceptional $p$-classes of the $p$-class groups:

[TABLE]

The possible infinite “bad families” of degree $p$ cyclic fields $K$ are such that for some fixed $\varepsilon$ and for any $C>0$:

[TABLE]

where $N$ is the number of ramified primes of $K$; we shall verify Section 2.3 that these functions $h_{\varepsilon}$ fulfill the condition at infinity.

But the Theorem 2.5 shall prove (unconditionally) the $p$-rank $\varepsilon$-conjecture.

2.2.1. The algorithm

Recall briefly the computation of $\delta(N)$ and $\Delta(N)$ in the purpose of some probabilistic considerations.

Let $F$ be any degree $p$ cyclic extension of $\mathbb{Q}$ with $N\geq 1$ ramified primes, put $M:={\mathcal{C}}\hskip-2.0pt{\ell}_{F}\otimes\mathbb{Z}_{p}$, and let $\sigma$ be a generator of $G:={\rm Gal}(F/\mathbb{Q})\simeq\mathbb{Z}/p\mathbb{Z}$.

Let $I_{F}$ be the group of ideals of $F$, prime to $p$, and let $M_{i}=:c\hskip-1.0pt{\ell}_{F}({\mathcal{I}}_{i})$, $i\geq 0$, ${\mathcal{I}}_{i}\subset I_{F}$, defined inductively by $M_{0}:=1$, and:

[TABLE]

where $m\geq 1$ is the least integer $i$ such that $M_{i}=M$ (i.e., such that $M_{i+1}=M_{i}$). Then $M_{i}=\{c\in M,\,c^{(1-\sigma)^{i}}=1\}$ for all $i\geq 0$ and $M_{1}=M^{G}$.

The sequence \raise 1.5pt\hbox{{\scriptscriptstyle#}}(M_{i+1}/M_{i}), $0\leq i\leq m$, decreases to $1$ and is bounded by \raise 1.5pt\hbox{{\scriptscriptstyle#}}M_{1} due to the injective maps $M_{i+1}/M_{i}\hookrightarrow M_{i}/M_{i-1}\hookrightarrow\cdots\hookrightarrow M_{2}/M_{1}\hookrightarrow M_{1}$ defined by the operation of $1-\sigma$.

We have, for the fields $F$, the formulas [18, Corollary 3.7], for all $i\geq 0$:

[TABLE]

with ${\mathcal{I}}_{0}=1$ and $\Lambda_{0}=1$.

The progression of the algorithm depends on the $x\in\Lambda_{i}$, $(x)={\rm N}_{F/\mathbb{Q}}({\mathfrak{A}})$, ${\mathfrak{A}}\in{\mathcal{I}}_{i}$, such that $x={\rm N}_{F/\mathbb{Q}}(y)$, $y\in F^{\times}$, giving the equation:

[TABLE]

in which the solutions ${\mathfrak{B}}$ (non-predictible) become new elements to be added to ${\mathcal{I}}_{i}$ to built ${\mathcal{I}}_{i+1}\supseteq{\mathcal{I}}_{i}$, then $\Lambda_{i+1}\supseteq\Lambda_{i}$, and so on.

Put, for all $i\geq 0$, \raise 1.5pt\hbox{{\scriptscriptstyle#}}\big{(}M_{i+1}/M_{i}\big{)}:=p^{N-1-t_{i}^{N}}, $t_{i}^{N}\geq 0$, where $p^{t_{i}^{N}}\mid p^{N-1}$ is the index $(\Lambda_{i}:\Lambda_{i}\cap{\rm N}_{F/\mathbb{Q}}(F^{\times}))$ (since $x\in\Lambda_{i}$ is norm of an ideal, it is locally a norm everywhere, except perhaps at the $N$ ramified primes, but the product formula for Hasse’s normic symbols gives the above divisibility). We have:

[TABLE]

Then \raise 1.5pt\hbox{{\scriptscriptstyle#}}M=\raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{F}\otimes\mathbb{Z}_{p})={\rm Phys.~{}Rev.~{}D}_{i\geq 0}\raise 1.5pt\hbox{{\scriptscriptstyle#}}\big{(}M_{i+1}/M_{i}\big{)}=p^{m\,(N-1)-\sum_{i?=1}^{m}\,t_{i}^{N}}.

We have the following result that we shall use to test the $\varepsilon$-conjecture.

Lemma 2.3** ([18, Lemma 4.2]).**

Let $F/\mathbb{Q}$ be a degree $p$ cyclic extension with $N\geq 1$ ramified primes. Then ${\rm rk}_{p}({\mathcal{C}}\hskip-2.0pt{\ell}_{F})=(p-1)\cdot(N-1)-\sum_{i=1}^{p-2}t_{i}^{N}$. Whence this yields $\delta(N)=(p-2)\cdot(N-1)-\sum_{i=1}^{p-2}t_{i}^{N}$.

We see that the $p$-rank may vary in the interval $[N-1,(p-1)\,(N-1)]$ and is always $O(N)$ as $N\to\infty$ which shall be of a great importance; the $p$-rank is equal to $N-1$, for all $F/\mathbb{Q}$ of degree $p$, if and only if $p=2$ as we have seen. In the same manner, the $p^{r}$-ranks are given by the expressions:

[TABLE]

It is clear that the normic indices depend on the $\mathbb{F}_{p}$-ranks of $N\times N$-matrices of suitable Hilbert symbols [20, Ch. VI, § 2] generalizing, for instance, Rédei’s matrices for the computation of the $4$-ranks of a quadratic field.

But we emphasize the fact that if some heuristics on the $\mathbb{F}_{p}$-rank of these matrices are natural, the number of steps of the algorithm (i.e., $m$) only depends on distribution results (see Remark 2.2).

2.2.2. Examples of maximal rank

In fact, for $p>2$, the maximal rank, equal to $(p-1)\cdot(N-1)$, is very rare. We remembered that in our thesis [20, p. 39], we gave the first example of cyclic cubic field, for which ${\rm rk}_{3}({\mathcal{C}}\hskip-2.0pt{\ell})=2\,(N-1)$; this example has conductor $f=9\cdot 577\cdot 757\cdot 991$ (the program below shows that the property is fulfilled for the eight fields of conductor $f$):

f=3895721091 N=4 P=x^3-1298573697x+18010351461592 Cl=[9,3,3,3,3,3] f=3895721091 N=4 P=x^3-1298573697x-16034355152657 Cl=[3,3,3,3,3,3] f=3895721091 N=4 P=x^3-1298573697x+15707980296811 Cl=[3,3,3,3,3,3] f=3895721091 N=4 P=x^3-1298573697x-10132337699792 Cl=[3,3,3,3,3,3] f=3895721091 N=4 P=x^3-1298573697x-8835062576489 Cl=[3,3,3,3,3,3] f=3895721091 N=4 P=x^3-1298573697x-7829966535011 Cl=[9,3,3,3,3,3] f=3895721091 N=4 P=x^3-1298573697x+912031593193 Cl=[3,3,3,3,3,3] f=3895721091 N=4 P=x^3-1298573697x-221623244288 Cl=[9,3,3,3,3,3]

We publish, for any further examples, a program computing all the cyclic cubic fields such that there are exceptional $3$-classes; so in most cases the $3$-rank is $N$ (instead of $N-1$) and many examples have a non-trivial $9$-rank.

{bf=7;Bf=10^6;for(f=bf,Bf,e=valuation(f,3);if(e!=0 & e!=2,next);F=f/3^e; if(Mod(F,3)!=1||core(F)!=F,next);F=factor(F);Div=component(F,1); d=component(matsize(F),1);for(j=1,d-1,D=component(Div,j); if(Mod(D,3)!=1,break));for(b=1,sqrt(4f/27),if(e==2 & Mod(b,3)==0,next); A=4f-27b^2;if(issquare(A,&a)==1,if(e==0,if(Mod(a,3)==1,a=-a); P=x^3+x^2+(1-f)/3x+(f*(a-3)+1)/27);if(e==2,if(Mod(a,9)==3,a=-a); P=x^3-f/3x-fa/27);N=omega(f);L=List;K=bnfinit(P,1);C8=component(K,8); h=component(component(C8,1),1);Cl=component(component(C8,1),2); dim=component(matsize(Cl),2);rho=0;for(i=1,dim,z=component(Cl,i); v=valuation(z,3);if(v>=1,rho=rho+1;listput(L,3^v)));if(rho>=N, print(" f=",f," N=",N," P=",P," Cl=",L)))))}

f=657 N=2 P=x^3-219x-1241 Cl=[3,3] f=657 N=2 P=x^3-219x+730 Cl=[3,3] (...) f=3913 N=3 P=x^3+x^2-1304x+17681 Cl=[3,3,3] f=3913 N=3 P=x^3+x^2-1304x-17536 Cl=[3,3,3] (...) f=4711 N=2 P=x^3+x^2-1570x+19193 Cl=[9,3] f=4711 N=2 P=x^3+x^2-1570x-13784 Cl=[9,3] f=4921 N=3 P=x^3+x^2-1640x+23876 Cl=[3,3,3] f=4921 N=3 P=x^3+x^2-1640x-729 Cl=[3,3,3] (...) f=5383 N=2 P=x^3+x^2-1794x+17744 Cl=[9,3] f=5383 N=2 P=x^3+x^2-1794x-9171 Cl=[9,3] (..) f=15561 N=4 P=x^3-5187*x+141778 Cl=[9,3,3,3]

Remark 2.4*.*

The Galois action of $G={\rm Gal}(F/\mathbb{Q})\simeq\mathbb{Z}/p\mathbb{Z}$ on the non-$p$-part of the class group gives rise, for all prime q\mid\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{F}, to:

[TABLE]

whose $\mathbb{F}_{q}$-dimension is a multiple of the residue degree of $q$ in $\mathbb{Q}(\mu_{p})/\mathbb{Q}$, which explains the rarity of such divisibilities for large residue degrees.

2.3. Estimation of $C_{p,p,\varepsilon}$

for the fields $F_{N,p}$

As we have explained, we do not consider degree $p$ cyclic extensions $F/\mathbb{Q}$ ramified at $p$. This shall modify the forthcoming computations by some $O(1)$ without any consequence on the statements since $p$ is fixed in all the sequel.

We need a lower bound of the $k$th prime number $\ell_{k}\equiv 1\pmod{p}$ to get an estimation of $f_{N,p}$.

We thank Gérald Tenenbaum for valuable indications for the good formula, from [40, Notes on Chapitre I, § 4.6], due to a result of Montgomery–Vaughan giving, for the $k$th prime number $\ell_{k}\equiv 1\pmod{p}$:

[TABLE]

Indeed, if \pi(x;1,p):=\raise 1.5pt\hbox{{\scriptscriptstyle#}}\{\ell\leq x;\ \ell\equiv 1\pmod{p}\} then:

[TABLE]

whence the result taking $x=\ell_{k}$ since $\pi(\ell_{k};1,p)=k$.

We intend to test, for the family $(F_{N,p})_{N\geq 1}$ of discriminants $D_{N,p}:=D_{F_{N,p}}$, the strong $\varepsilon$-conjecture, that is to say:

[TABLE]

where $\Delta(N)=(m-1)\cdot(N-1)-\sum_{i=1}^{m-1}t_{i}^{N}\geq 0$, related to the set of exceptional $p$-classes, has only a probabilistic value depending on $m$ and the $t_{i}^{N}$.

It will be easy, from the forthcoming calculations, to test the $p$-rank $\varepsilon$-conjecture, \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}\otimes\mathbb{F}_{p})\leq C_{p,p,\varepsilon}\cdot(\sqrt{D_{N,p}}\,)^{\varepsilon}, but considering instead the weaker inequality:

[TABLE]

even in the less favorable case $t_{i}^{N}=0$, for $1\leq i\leq p-2$ and replacing for all $N$, $D_{N,p}$ by a lower bound $D^{\prime}_{N,p}$ (in other words the existence of an inequality $p^{(p-1)\,(N-1)}\leq C^{\prime}_{p,p,\varepsilon}\cdot(\sqrt{{D^{\prime}}_{\!N,p}}\,)^{\varepsilon}$ proves the $p$-rank $\varepsilon$-conjecture for all degree $p$ cyclic fields). We fix $p$ and put $D_{N,p}=:D_{N}=:f_{N}^{p-1}$. The strong form is equivalent to prove that $\displaystyle\frac{\,p^{N-1+\Delta(N)}\,}{(\sqrt{D_{N}}\,)^{\varepsilon}}$ is bounded as $N\to\infty$, whence (N-1+\Delta(N))\,{\rm log}(p)-\varepsilon\cdot\frac{p-1}{2}\sum_{k=1}^{N}{\rm log}(\ell_{k})<\infty,\ \hbox{as N\to\infty}.

We then have, replacing $\ell_{k}$ by a lower bound $\ell^{\prime}_{k}$, to compute, using (1), (2), (3), the following quantity:

[TABLE]

We verify that \displaystyle\mathop{\raise 2.0pt\hbox{\sum}}\limits_{k=1}^{N}{\rm log}_{2}\big{(}\hbox{\frac{\ell_{k}}{p}}\big{)} can be neglected, subject to adding $-1$ to the sum, and consider, instead:

[TABLE]

The expression of $N!$ leads to {\rm log}(N!)=N{\rm log}(N)-N+\hbox{\frac{1}{2}}{\rm log}(N)+O(1), whence, with \gamma_{p}:={\rm log}\big{(}\frac{p-1}{2}\big{)}-1:

[TABLE]

Now we write $X(N)$ under the form:

[TABLE]

Replacing $\Delta(N)$ by the maximal value $(p-2)\,(N-1)$ of $\delta(N)$, the dominant term $-\varepsilon\cdot\frac{p-1}{2}{\rm log}(N)$ ensures the existence of a positive constant $C_{\varepsilon}$ since $\frac{(p-1)\,(N-1)}{N}\,{\rm log}(p)=O(1)$ giving:

[TABLE]

It is easy to verify that $X_{0}(N)$, as function of $N$, admits, for an $N_{0}\gg 0$, a computable maximum, only depending on $p$ and $\varepsilon$ (e.g., for $p=7$, $\varepsilon=0.1$, $N_{0}\approx 2935394\cdot 10^{10}$, $X_{0}(N_{0})\approx 88\cdot 10^{14}$).

This proves the $p$-rank $\varepsilon$-conjecture for the family $(F_{N,p})_{N\geq 1}$, even assuming always a maximal $p$-rank $(p-1)\,(N-1)$. Whence we can state:

Theorem 2.5**.**

Let $p\geq 2$ be a given prime number.

(i) The $p$-rank $\varepsilon$-conjecture on the existence, for all $\varepsilon>0$, of a constant $C_{d,p,\varepsilon}$ such that \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{F}_{p})\leq C_{d,p,\varepsilon}\cdot(\sqrt{D_{K}}\,)^{\varepsilon} for all $K$ of degree $d$, is true for the subfamily of all cyclic extension $F/\mathbb{Q}$ of degree $d=p$.

(ii) For any cyclic extension $F/\mathbb{Q}$ of degree $p$, let $\Delta(N)\geq 0$ be defined by \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{F}\otimes\mathbb{Z}_{p})=p^{N-1+\Delta(N)}, where $N$ is the number of ramified primes. Then the strong $\varepsilon$-conjecture, \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p})\leq C_{d,p,\varepsilon}\cdot(\sqrt{D_{K}}\,)^{\varepsilon}, is true for the family of degree $p$ cyclic extensions under the condition $\Delta(N)\ll N\,{\rm log}(N)$.

Proof.

Consider arbitrary prime numbers $\ell_{n_{i}}\equiv 1\pmod{p}$, $1\leq i\leq N$, and a field $K$, cyclic of degree $p$, of conductor $\prod_{i=1}^{N}\ell_{n_{i}}$. Thus $D_{K}\geq D_{F_{N}}$ and the maximal $p$-rank of ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$ is still $(p-1)\,(N-1)$. We have seen that if we put $\delta(N)=(p-2)\,(N-1)$ in (4), giving (5), the main term ensures the existence of the constant $C_{p,p,\varepsilon}$. We omit the details when $p^{2}$ (or $8$) divides the conductor. ∎

Very probably, considering the Remark 2.2, $\Delta(N)$ is almost all the time of order much less than $N\,{\rm log}(N)$ since $p^{r}$-ranks are in practice very rare for $r\geq 2$ as well as a maximal $p$-rank equal to $(p-1)\,(N-1)$ (see § 2.2.2); but the dominant terms in (4) being $\Delta(N)\,{\rm log}(p)-\varepsilon\cdot\frac{p-1}{2}\,N\,{\rm log}(N)$, this may give some trouble for the proof of an universal strong $\varepsilon$-conjecture assuming for instance $\Delta(N)=O(1)\,{\rm log}(N)$, or more, for infinite families of degree $p$ cyclic fields, even of density zero.

2.4. A lower bound for $C_{p,p,\varepsilon}$

This part is not essential for our purpose, but it will show that a lower bound of $C_{p,p,\varepsilon}$ is of the same order of magnitude as for the upper bound deduced from (5). We shall use the following property which may be justified from results given in [31]. For all $k\geq 1$, the $k$th prime $\ell_{k}\equiv 1\pmod{p}$ fulfills, for some constants $c_{p}$ only depending on $p$, the inequality $\ell_{k}<c_{p}\,((p-1)\,k)\cdot{\rm log}((p-1)\,k)<c_{p}\,((p-1)\,k)^{2}$. Then:

[TABLE]

where $\gamma^{\prime}_{p}:={\rm log}(c_{p})+2\,{\rm log}(p-1)>0$. Whence, from the value of ${\rm log}(N!)$:

[TABLE]

with $\gamma^{\prime\prime}_{p}=\gamma^{\prime}_{p}-2>0$. Thus:

[TABLE]

Remarks 2.6*.*

(i) As soon as we replace $\varepsilon$ by $c+\varepsilon$, $0<c<1$, as it is done in much papers giving general proofs for \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{F}_{p})\leq C_{d,p,c,\varepsilon}\cdot(\sqrt{D_{K}}\,)^{c+\varepsilon}, the above computations becomes, replacing $\Delta(N)$ by $\delta(N)$:

[TABLE]

For $\varepsilon\to 0$, the formulas (5) and (6) give a limit value $C_{p,c}$ of $C_{p,p,c,\varepsilon}$ such that ${\rm log}(C_{p,c})\approx(N-1+\delta(N))\,{\rm log}(p)-c\,\frac{p-1}{2}\,N\,{\rm log}(N)-N\,O(1)$, rapidely negative as $N$ increases, giving for degree $p$ cyclic fields, an obvious proof of the “$p$-rank $(c+\varepsilon)$-property”.

(ii) If we replace $\mathbb{Q}$ by a number field $k$ and the fields $F_{N,p}$ by the degree $p$ cyclic extensions $F_{k,N,p}$ of $k$ with $N$ ramified prime ideals of $k$, the details of computations are more complicate, but the results and comments are similar since the $p$-rank of ${\mathcal{C}}\hskip-2.0pt{\ell}_{F_{k,N,p}}$ is still equivalent to $O(N)$ because of the exact sequence of genus theory and the inequalities [16, Theorem IV.4.5.1]:

[TABLE]

where the constants $c_{k,p}$, $c^{\prime}_{k,p}$ depend on the $p$-classes and units of $k$. Then, the general algorithm computing the $p$-rank via the filtration $(M_{i})_{i\geq 0}$ is identical up to similar modifications (see [18, § 4.4]) and shall give the $p$-rank $\varepsilon$-conjecture for the relative degree $p$ cyclic case without too much difficulties. The case of abelian $p$-extensions may be accessible from the $p^{n}$-cyclic cases, with some effort…

The proof of the strong $\varepsilon$-conjecture, for the degree $p$ cyclic extensions of $k$, remains, theoretically, open, but is clearly not a folk conjecture for such very particular real fields because of the possible generalization of the density results of [28] and the conclusion, about the possible existence of pathological families of density zero, is still relevant. Meanwhile a particular study of the algorithm giving $\Delta(N)$, independently of any density results, should be a crucial step for many questions in number theory.

3. PARI/GP programs computing ${\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}$

3.1. Structure of some ${\mathcal{C}}\hskip-2.0pt{\ell}_{F_{N,p}}$

The following numerical results show that exceptional $p$-classes (indicated by ∗) are not excessively frequent for these fields. We examine the cases $p=3,5,7$, then $p=2$ (one must precise $p$ and the conductor $f=\ell_{1}\cdots\ell_{N}$, $\ell_{i}\equiv 1\pmod{p}$, in the following program giving all the $(p-1)^{N-1}$ fields of conductor $f$):

{p=3;f=713193137;V=polsubcyclo(f,p);d=matsize(V);d=component(d,2); for(k=1,d,P=component(V,k);if(nfdisc(P)!=f^(p-1),next);K=bnfinit(P,1); C8=component(K,8);C81=component(C8,1);h=component(C81,1); Cl=component(C81,2);print("p=",p," f=",f," P=",P," Cl=",Cl))}

p=3 f=1983163=713193137 P=x^3+x^2-661054x+ 49725976 Cl=[6,6,3,3] P=x^3+x^2-661054x+198463201 Cl=[3,3,3,3] P=x^3+x^2-661054x+97321888 Cl=[39,3,3,3,3] P=x^3+x^2-661054x-186270421 Cl=[3,3,3,3] P=x^3+x^2-661054x-79179619 Cl=[6,6,3,3] P=x^3+x^2-661054x-188253584 Cl=[3,3,3,3,3] P=x^3+x^2-661054x+138968311 Cl=[3,3,3,3] P=x^3+x^2-661054x-146607161 Cl=[3,3,3,3,3]* P=x^3+x^2-661054x-158506139 Cl=[3,3,3,3] P=x^3+x^2-661054x-140657672 Cl=[6,6,3,3] P=x^3+x^2-661054x+186564223 Cl=[3,3,3,3] P=x^3+x^2-661054x+81456584 Cl=[3,3,3,3] P=x^3+x^2-661054x+206395853 Cl=[3,3,3,3] P=x^3+x^2-661054x-206102051 Cl=[12,12,3,3] P=x^3+x^2-661054x+2130064 Cl=[3,3,3,3,3] P=x^3+x^2-661054x+27911183 Cl=[3,3,3,3,3]

p=3 f=85276009=71319313743 P=x^3-x^2-28425336x+58104545836 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+56995957719 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-7472705085 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+10264704787 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+51623569152 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+30901498965 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-50622365639 Cl=[6,6,3,3,3] P=x^3-x^2-28425336x-29132811371 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-46614393216 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-33226059803 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-56506410260 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-12248161589 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+23994142236 Cl=[6,6,3,3,3] P=x^3-x^2-28425336x+52220501215 Cl=[3,3,3,3,3,3] P=x^3-x^2-28425336x+11799672949 Cl=[6,6,3,3,3] P=x^3-x^2-28425336x-57529722368 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-21287418543 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-47040773261 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+36273887532 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-53436473936 Cl=[6,6,3,3,3,3]* P=x^3-x^2-28425336x+40964068027 Cl=[21,3,3,3,3] P=x^3-x^2-28425336x+45824800540 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+56313749647 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+13078813084 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+39940755919 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-37830964289 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+57848717809 Cl=[6,6,3,3,3] P=x^3-x^2-28425336x-9775157328 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-2526696563 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-58126654431 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x+28428494704 Cl=[3,3,3,3,3] P=x^3-x^2-28425336x-51986781783 Cl=[3,3,3,3,3]

p=5 f=13981=113141 P=x^5+x^4-5592x^3+46417x^2+4301003x-26664769 Cl=[5,5] P=x^5+x^4-5592x^3+46417x^2+2623283x-26664769 Cl=[5,5] P=x^5+x^4-5592x^3-261165x^2-4479065x-26832541 Cl=[55,5,5] P=x^5+x^4-5592x^3+32436x^2+1518784x+1814528 Cl=[5,5] P=x^5+x^4-5592x^3-205241x^2-2074333x-6028813 Cl=[5,5,5]* P=x^5+x^4-5592x^3-135336x^2+847696x+1143440 Cl=[5,5] P=x^5+x^4-5592x^3-205241x^2-1878599x+234675 Cl=[5,5] P=x^5+x^4-5592x^3+74379x^2+3993421x-7035445 Cl=[5,5] P=x^5+x^4-5592x^3+158265x^2-424575x-12851541 Cl=[5,5] P=x^5+x^4-5592x^3+130303x^2+442247x+346523 Cl=[5,5] P=x^5+x^4-5592x^3+46417x^2+1644613x-16878069 Cl=[5,5] P=x^5+x^4-5592x^3+214189x^2-2493763x+5715227 Cl=[5,5] P=x^5+x^4-5592x^3+74379x^2+3238447x-28174717 Cl=[5,5] P=x^5+x^4-5592x^3+32436x^2+5992704x-2659392 Cl=[5,5,5] P=x^5+x^4-5592x^3-37469x^2+4384889x-11397517 Cl=[5,5] P=x^5+x^4-5592x^3-9507x^2+7432747x+1185383 Cl=[5,5]

3.2. Examples of exeptional $p$-classes

If one drops the invariant classes from the class group, this gives a very small component (often equal to $1$); for instance we list some fields of the above tables with exceptional $p$-classes and/or nontrivial non-$p$-parts of the class group (for $p=3,5,7,11$):

p=3 f=1983163 P=x^3+x^2-661054x+97321888 Cl=[39,3,3,3,3]=[13]x[3,3,3,3,3] f=1983163 P=x^3+x^2-661054x-206102051Cl=[12,12,3,3]=[4,4]x[3,3,3,3] f=85276009 P=x^3-x^2-28425336x-53436473936 Cl=[6,6,3,3,3,3]=[2,2]x[3,3,3,3,3,3]* p=5 f=13981 P=x^5+x^4-5592x^3-261165x^2-4479065x-26832541 Cl=[55,5,5]=[11]x[5,5,5] f=13981 P=x^5+x^4-5592x^3-205241x^2-2074333x-6028813 Cl=[5, 5, 5] f=13981 P=x^5+x^4-5592x^3+32436x^2+5992704x-2659392 Cl=[5, 5, 5] p=7 f=88537 P=x^7+x^6-37944x^5-1134719x^4+324123632x^3 +13095064100x^2-393352790753x-8536744545007 Cl=[301,7]=[43]x[7,7] f=10004681 P=x^7-x^6-4287720x^5-1266715121x^4+5127549957760x^3 +2650344024068794x^2-951078919604894529x-488606218944681147667 Cl=[791,7,7]=[113]x[7,7,7] f=10004681 P=x^7-x^6-4287720x^5+3455494311x^4+176273349584x^3 -471685834336278x^2-36087798097778993x+6487901368894795147 Cl=[7,7,7,7] f=10004681 P=x^7-x^6-4287720x^5+2985274304x^4+1830007100160x^3 -1788009977372784x^2+233770322355404864x-5599630780142239232 Cl=[14,14,14]=[2,2,2]x[7,7,7] f=10004681 P=x^7-x^6-4287720x^5+6276814353x^4-4047542893720x^3 +1360785664233294x^2-232635292693132049x+15953699891990750023 Cl=[7,7,7,7]* f=10004681 P=x^7-x^6-4287720x^5-6169008811x^4-3865457699520x^3 -1247559831026016x^2-202130636944756129x-12969698603184144677 Cl=[7,7,7,7] f=10004681 P=x^7-x^6-4287720x^5+1224450448x^4+5079527488960x^3 -2161274540764336x^2-1677161713287529664x+812143879436422435328 Cl=[7,7,7,7] f=10004681 P=x^7-x^6-4287720x^5-1306733845x^4+4849019638720x^3 +2420349994235592x^2-659732951886568641x-207964718993797238079 Cl=[7,7,7,7] f=10004681 P=x^7-x^6-4287720x^5+5756570941x^4-3107102879720x^3 +760318637129246x^2-74394888056758073x+1594979915105904419 Cl=[7,7,7,7] f=10004681 P=x^7-x^6-4287720x^5-1746939809x^4+2392910471944x^3 +1648023037138232x^2+241022177190387487x-9669121620934915453 Cl=[7,7,7,7]* f=10004681 P=x^7-x^6-4287720x^5-4188081973x^4-284662313448x^3 +706702291060440x^2+90226184532822239x-3998693323498243787 Cl=[7,7,7,7] f=10004681 P=x^7-x^6-4287720x^5-2227164497x^4+1569805356712x^3 +369664737597974x^2-166416491455189217x+3813962316737479895 Cl=[7,7,7,7] f=10004681 P=x^7-x^6-4287720x^5-3307670045x^4+1296557509240x^3 +1587033521303494x^2+232482552302284071x-10617355312468435915 Cl=[203,7,7]=[203]x[7,7,7] f=10004681 P=x^7-x^6-4287720x^5+1654651731x^4+4187910318240x^3 -1965195178959414x^2-1012216801097102473x+451605062388713519719 Cl=[14,14,14]=[2,2,2]x[7,7,7] f=10004681 P=x^7-x^6-4287720x^5+1544600240x^4+4818925558272x^3 -2343209264562096x^2-1609606088655340480x+886992887186768275456 Cl=[203,7,7]=[203]x[7,7,7] f=10004681 P=x^7-x^6-4287720x^5-1306733845x^4+4427622475000x^3 +1798213808544282x^2-1313378138040048441x-613859706212867719587 Cl=[301,7,7]=[301]x[7,7,7] p=11 f=137149 P=x^11+x^10-62340x^9-2099173x^8+998038116x^7+30321726924x^6 -5078707527329x^5+3275334221180x^4+8066546096404148x^3 -214723422858644515x^2+1902599837479513519x-4121588229203611219 Cl=[253,11]=[23]x[11,11] f=137149 P=x^11+x^10-62340x^9-727683x^8+1217887963x^7+33409088063x^6 -7760886906947x^5-350751766601032x^4+3398347545513222x^3 +236507399684756272x^2+2593585988882665302x+8529384350363670191 Cl=[979,11]=[89]x[11,11] f=137149 P=x^11+x^10-62340x^9-1550577x^8+1265615815x^7+66889353347x^6 -7866931610939x^5-712972865698216x^4-12900860936489076x^3 +143276981594922336x^2+1596818122984871186x+3178173588229813309 Cl=[737,11]=[67]x[11,11] f=137149 P=x^11+x^10-62340x^9-7310835x^8+70636578x^7+43296296622x^6 +1378934348258x^5-28471672310749x^4-944763467217249x^3 +12433374265353417x^2+31266667418235948x-324164722946199831 Cl=[253,11]=[23]x[11,11] f=137149 P=x^11+x^10-62340x^9-727683x^8+1153290784x^7+18055120364x^6 -7130815486262x^5-134781314432095x^4+13777568483843493x^3 +310799275320778321x^2+883774494827373474*x-5728549445587601897 Cl=[253,11]=[23]x[11,11]

For $p=2$, $F_{N,2}$ may be real or complex; as we know, the $2$-rank is always $N-1$ and any exceptional classes give non-trivial $4$-ranks. The PARI/GP instruction ${\sf bnfnarrow}$ allows the $2$-structure in the restricted sense:

{m=1;for(N=2,100,el=prime(N);m=(-1)^((el-1)/2)elm;P=x^2-m; K=bnfinit(P,1);L=bnfnarrow(K);Cl=component(L,2);print("m=",m," Cl=",Cl))} m=-15 Cl=[2] m=+105 Cl=[2,2] m=-1155 Cl=[2,2,2] m=-15015 Cl=[12,2,2,2] m=-255255 Cl=[16,2,2,2,2]* m=+4849845 Cl=[4,2,2,2,2,2]* m=-111546435 Cl=[42,2,2,2,2,2,2] m=-3234846615 Cl=[308,2,2,2,2,2,2,2]* m=+100280245065 Cl=[2,2,2,2,2,2,2,2,2] m=+3710369067405 Cl=[34,2,2,2,2,2,2,2,2,2] m=+152125131763605 Cl=[2,2,2,2,2,2,2,2,2,2,2] m=-6541380665835015 Cl=[28284,2,2,2,2,2,2,2,2,2,2,2]* m=+307444891294245705 Cl=[14,2,2,2,2,2,2,2,2,2,2,2,2] m=+16294579238595022365 Cl=[2,2,2,2,2,2,2,2,2,2,2,2,2,2] m=-961380175077106319535 Cl=[1210734,2,2,2,2,2,2,2,2,2,2,2,2,2,2] m=-58644190679703485491635 Cl=[1622526,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] m=+3929160775540133527939545 Cl=[2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] m=-278970415063349480483707695 Cl=[83911452,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]* m=-20364840299624512075310661735 Cl=[362626834,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] m=+1608822383670336453949542277065 Cl=[4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]* m=-133532257844637925677812008996395 Cl=[2322692420,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]*

4. Genus theory – Abelian $p$-ramification theory

4.1. Reminders on Genus theory

For wide information on genus theory, see for example [12, 13, 16, 27, 29, 34].

4.1.1. Genus field and genus number $g_{K/k}$

Introduce the genus field according to an extension $K/k$ (we shall take $k=\mathbb{Q}$ in the sequel) which is the maximal subextension $H_{K/k}$ of $H_{K}$ (in the restricted sense) equal to the compositum of $K$ with an abelian extension of $k$:

[TABLE]

Thus $H_{K/k}$ is for instance equal to the compositum of $K$ with $H_{K}^{\rm ab}$ (the maximal abelian subextension of $H_{K}/k$), according to the diagram above, where $K^{\rm ab}$ is the maximal abelian subextension of $K/k$. The genus number is $g_{K/k}:=[H_{K/k}:KH_{k}]$. When $K/k$ is cyclic, the genus number $g_{K/k}$ is equal to the number of invariant classes by ${\rm Gal}(K/k)$ given by Chevalley’s formula (1). In the general Galois case, we have the similar expression g_{K/k}=\displaystyle\frac{\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{k}\cdot\prod_{\mathfrak{l}}\,e^{\rm ab}_{\mathfrak{l}}}{[K^{\rm ab}:k]\cdot(E_{k}^{\rm pos}:E_{k}^{\rm pos}\cap{\mathcal{N}}_{K/k})}, where $E_{k}^{\rm pos}$ is the group of totally positive units of $k$, $e^{\rm ab}_{\mathfrak{l}}$ the index of ramification of ${\mathfrak{l}}$ in $K^{\rm ab}/k$ and ${\mathcal{N}}_{K/k}$ the group of local norms in $K/k$. A general formula does exist for non-Galois fields (e.g.,[16, Theorem IV.4.2 & Corollaries]).

4.1.2. Variants of the strong $\varepsilon$-conjecture

Since the $p$-genus group of $K$ may be an obstruction to the strong $\varepsilon$-conjecture, we may consider, in the exact sequence $1\to{\mathcal{C}}\hskip-2.0pt{\ell}^{\prime}_{K}\otimes\mathbb{Z}_{p}\to{\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p}\to{\rm Gal}(H_{K/\mathbb{Q}}/K)\otimes\mathbb{Z}_{p}\to 1$, the number \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}^{\prime}_{K}\otimes\mathbb{Z}_{p}) (giving the number of exceptional $p$-classes instead of the whole $p$-class group) and propose the following form of the $\varepsilon$-conjecture:

Conjecture 4.1**.**

For a number field $K$, let $H_{K}$ be its Hilbert’s class field, $H_{K/\mathbb{Q}}$ its genus field and ${\mathcal{C}}\hskip-2.0pt{\ell}^{\prime}_{K}:={\rm Gal}(H_{K}/H_{K/\mathbb{Q}})$. Let $p$ be a prime number. For all $\varepsilon>0$ there exists $C^{\prime}_{d,p,\varepsilon}$ such that \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}^{\prime}_{K}\otimes\mathbb{Z}_{p})\leq C^{\prime}_{d,p,\varepsilon}\cdot(\sqrt{D_{K}}\,)^{\varepsilon} holds for all $K$ of degree $d$, except possibly for sparse families of density zero.

One may ask what happens for a “global” strong $\varepsilon$-conjecture on the form:

[TABLE]

The paper [6] of Ryan Daileda recalls some results by Littlewood showing that (under GRH) there exist imaginary quadratic fields with arbitrary large discriminant for which \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}\geq c\cdot\sqrt{D_{K}}\,{\rm log}_{2}(D_{K}), where $c$ is an absolute constant. For real quadratic fields a result of Montgomery and Weinberger is that there exist real quadratic fields with arbitrary large discriminant whose class numbers satisfy \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}\geq c\cdot\sqrt{D_{K}}\,\frac{{\rm log}_{2}(D_{K})}{{\rm log}(D_{K})}. Analogous results are known for cyclic cubic fields and Daileda proves that there exists an absolute constant $c>0$ so that there are totally real non-abelian cubic fields, with arbitrary large discriminant satisfying \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}\geq c\cdot\sqrt{D_{K}}\,\Big{(}\frac{{\rm log}_{2}(D_{K})}{{\rm log}(D_{K})}\Big{)}^{2}. All this has been generalized to CM number fields in [7].

So, if we consider an inequality of “strong global $\varepsilon$-conjecture” type:

[TABLE]

any infinite family ${\mathcal{K}}$ of fiields $K$ such that \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}\geq c\cdot\sqrt{D_{K}}, for a constant $c>0$, independent of $K\in{\mathcal{K}}$, yields:

[TABLE]

which is absurd. In other words, there is in general no strong global $\varepsilon$-conjecture; nevertheless the question of the strong form we have considered (for $p$ fixed and for ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$ or ${\mathcal{C}}\hskip-2.0pt{\ell}^{\prime}_{K}$):

[TABLE]

depends on the finiteness of fields $K$ such that \raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p})\geq c_{p}\cdot\sqrt{D_{K}}. For instance, in the quadratic case and $p>2$, this should give some $p$-class groups for which the $p$-rank and/or the exponent tend to infinity with $D_{K}$; even in the imaginary case, this may occur only for very sparse families.

4.1.3. Computation of some successive local maxima

The following program, for imaginary quadratic fields, allows a study of this question by computing the successive local maxima of $C^{\prime}_{2,\varepsilon}$, in $\sf{C}$, the discriminant and the class number obtained for each local maximum, in $\sf{D}$ and $\sf{h}$, respectively:

{eps=0.05;Cm=0;bD=2;BD=10^9;for(D=bD,BD,e=valuation(D,2);M=D/2^e; if(core(M)!=M,next);if((e==1||e>3)||(e==0&Mod(M,4)!=-1)||(e==2&Mod(M,4)!=1), next);h=qfbclassno(-D);N=omega(D);C=h/(2^(N-1)*(sqrt(D)^eps)); if(C>Cm,Cm=C;print("D=",-D," h=",h," C=",C)))}

As far as the program has run, it slows down between $\sf{C=6\cdot 10^{4}}$ and $\sf{C=7\cdot 10^{4}}$, but no conclusion is possible as expected from the above. But the most spectacular fact, that we have discover, is that, for each local maximum $\sf{C}$, the corresponding discriminant is prime (whence $\sf{h}$ odd) whatever $\varepsilon$, as shown by this short excerpt:

D h C -3 1 0.972908434869468710702241668941166407 -23 3 2.773818617890694606606085132125197163 -47 5 4.541167885124564220325740509229014479 -71 7 6.292403751297605635733619062872115785 -167 11 9.678872599268429560299054329160821597 -191 13 11.400332501352005304200415816510168367 -239 15 13.080709822134822456679612679136456819 -311 19 16.460180420909375330798097085967676763 -431 21 18.045019802162182082161592477498679286 -479 25 21.425532320359474690178248184779886979 (...) -1118314391 77395 45972.572539103313552220923397893022055 -1130984399 77697 46138.963607764612827282941613333974355 -1139075159 78141 46394.356488699687700451681236043961611 -1184068679 81267 48203.636807716833174323833829210758624 -1229647319 81419 48248.214898940545998582986401841694028 -1237871879 82171 48685.729279564354497202266261569081321 -1250370239 83503 49462.505651262222727040029721940389296

We have no counterexample in the selected interval $D\leq 2\cdot 10^{9}$ and no serious explanation, but if we test the local maxima of \frac{\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}}{(\sqrt{D_{K}}\,)^{\varepsilon}} instead of \frac{2^{-(N-1)}\,\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}}{(\sqrt{D_{K}}\,)^{\varepsilon}}, we have for instance the following normal behaviour:

D=[3, 1; 5, 1] h=2 C=1.8690792417830016333288729232969402612 D=Mat([23, 1]) h=3 C=2.7738186178906946066060851321251971632 D=[3, 1; 13, 1] h=4 C=3.6499202570298105632113852382187054319 D=Mat([47, 1]) h=5 C=4.5411678851245642203257405092290144795 D=Mat([71, 1]) h=7 C=6.2924037512976056357336190628721157852 D=[5, 1; 19, 1] h=8 C=7.1391564091327201771370767959908455517 D=[7, 1; 17, 1] h=10 C=8.8738345254872857598506328581670649858 (...) D=Mat([63839, 1]) h=423 C=320.78438186581184951419633278279394585 D=[23, 1; 2833, 1] h=424 C=321.37826092893978747476207092089453868 D=[113, 1; 607, 1] h=434 C=328.53606501556853121813362584226004750 D=[19, 1; 23, 1; 163, 1] h=440 C=332.76370450428389891215784283045533831 D=[41, 1; 1831, 1] h=454 C=342.90123385517606193005461615124054801 D=[19, 1; 37, 1; 113, 1] h=468 C=352.97586311634473415004028909597003881 D=[43, 1; 1973, 1] h=480 C=361.43179021472306392157190052394995551 D=Mat([88079, 1]) h=487 C=366.35924203761999218970143789845358049

In the same way, if we compute the successive maxima of the $3$-class groups, we obtain a similar result:

{p=3;bD=1;BD=10^9;Cm=0;for(D=bD,BD,e=valuation(D,2);M=D/2^e; if(core(M)!=M,next);if((e==1||e>3)||(e==0&Mod(M,4)!=-1)||(e==2&Mod(M,4)!=1), next);h=qfbclassno(-D);hp=p^valuation(h,p);Cp=hp;if(Cp>Cm,Cm=Cp; C=log(Cp)/log(sqrt(D));print("D=",-D," h=",h," hp=",hp," C=",C)))} D=-23 h=3 hp=3 C=0.70075861284442195481324 D=-199 h=9 hp=9 C=0.83019007976763598642971 D=-983 h=27 hp=27 C=0.95661698654993161545339 D=-3671 h=81 hp=81 C=1.07074359233325762042197 D=-29399 h=243 hp=243 C=1.06778367209896382287404 D=-178559 h=729 hp=729 C=1.09019287826209803280171 D=-2102999 h=2187 hp=2187 C=1.05643959875714455523718 D=-14868719 h=6561 hp=6561 C=1.06436822551851827813563 D=-98311919 h=19683 hh=19683 C=1.07451592116950263349372

Then, for $p=2$:

{p=2;bD=1;BD=10^9;Cm=0;for(D=bD,BD,e=valuation(D,2);M=D/2^e; if(core(M)!=M,next);if((e==1||e>3)||(e==0&Mod(M,4)!=-1)||(e==2&Mod(M,4)!=1), next);h=qfbclassno(-D);hp=p^valuation(h,p);Cp=hp;if(Cp>Cm,Cm=Cp; C=log(Cp)/log(sqrt(D));print("D=",-D," h=",h," hp=",hp," C=",C)))} D=-15 h=2 hp=2 C=0.511916049619630978775355357 D=-39 h=4 hp=4 C=0.756801438067480149325544162 D=-95 h=8 hp=8 C=0.913262080279460212705801846 D=-399 h=16 hp=16 C=0.925899677503555682939700450 D=-791 h=32 hp=32 C=1.038687593312750474942887870 D=-2519 h=64 hp=64 C=1.062075159346033035976072133 D=-10295 h=128 hp=128 C=1.050289653382181398975491576 D=-39431 h=256 hp=256 C=1.048009122470377471769618833 D=-132599 h=512 hp=512 C=1.057783767181715434360601717 D=-328319 h=1024 hp=1024 C=1.091420745999194423260975917 D=-1333631 h=2048 hp=2048 C=1.081244297733198664388474474 D=-4599839 h=4096 hp=4096 C=1.084346236368631648159879902 D=-18855359 h=8192 hp=8192 C=1.075781736259555689965062133 D=-63836951 h=16384 hp=16384 C=1.079918254667737276882538104 D=-266675639 h=32768 hp=32768 C=1.071791801714607295960939150 D=-966467519 h=65536 hp=65536 C=1.072093388756179498237226639

We shall examine elsewhere all these strange phenomena which seems valid for all $p$ and suggest the existence of families for which the $p$-part of the class number has maximal values, so that \displaystyle\frac{{\rm log}(\raise 1.5pt\hbox{{\scriptscriptstyle#}}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p}))}{{\rm log}(\sqrt{D_{K}}\,)}\to 1 as $D_{K}\to\infty$.

4.1.4. Reciprocal study

To try to suggest the existence of analogous families giving huge $p$-class groups, a trick is to consider normic equations, in integers $a,b$, of the form:

[TABLE]

where $q>1$ is any fixed integer and $\rho$ an exponent as large as possible; then when $a\geq 1$ increases, we deduce $b$ and the square free integer $m$. This kind of experiment has shown, in [17, § 5.3], that there exist huge discriminants giving interesting $p$-adic invariants. Moreover the function:

[TABLE]

giving (for any $\varepsilon>0$):

[TABLE]

may constitute an obstruction for the strong $\varepsilon$-conjecture as soon as:

[TABLE]

for an infinite subfamily ${\mathcal{K}}$ of the set of imaginary quadratic fields. But a priori, this does not affect the $p$-rank $\varepsilon$-conjecture.

Of course, the right member of the normic equation being rapidely too large when $\rho$ increases, the experimentation is very limited regarding PARI/GP possibilities. However, even for small values of $\rho$ predicting, a priori, $p$-classes of order aroud $p^{\rho}$ we obtain much large orders, and the following numerical results may be convincing enough about the infiniteness of such utmost examples.

{p=3;rho=4;q=2;Y=4q^(p^rho);ba=1;Ba=sqrt(Y);H=1;for(a=ba,Ba,B=Y-a^2; m=core(B);D=m;if(Mod(m,4)!=-1,D=4m);b=component(core(B,1),2); if(gcd(a,b)>2,next);h=qfbclassno(-D);vh=valuation(h,p);hp=p^vh; if(hp>H,H=hp;Cp=log(hp)/log(sqrt(D));Hp=component(quadclassunit(-D),2); d=component(matsize(Hp),2);L=List;for(k=1,d,c=component(Hp,k); w=valuation(c,p);if(valuation(c,p)!=0,listput(L,p^w))); print("D=",D," a=",a," b=",b," Cp=",Cp," hp=",hp," Hp=",L)))} rho=4 D=9671406556917033397649407 a=1 b=1 Cp=0.152767 hp=81 Hp=[81] D=197375644018714967298967 a=5 b=7 Cp=0.204814 hp=243 Hp=[243] D=9671406556917033397648447 a=31 b=1 Cp=0.229151 hp=729 Hp=[729] D=9671406556917033397648319 a=33 b=1 Cp=0.267343 hp=2187 Hp=[729,3] D=9671406556917033397644647 a=69 b=1 Cp=0.305534 hp=6561 Hp=[243,27] D=9671406556917033397435039 a=463 b=1 Cp=0.343726 hp=19683 Hp=[19683] D=9671406556917033397373783 a=525 b=1 Cp=0.381918 hp=59049 Hp=[59049] D=9671406556917033395993039 a=1287 b=1 Cp=0.420110 hp=177147 Hp=[177147] D=9671406556917033372819119 a=4983 b=1 Cp=0.496494 hp=1594323 Hp=[531441,3] D=9671406556917022018093783 a=106675 b=1 Cp=0.534686 hp=4782969 Hp=[1594323,3] {p=2;rho=6;q=2;Y=4q^(p^rho);ba=1;Ba=sqrt(Y);H=1;for(a=ba,Ba,B=Y-a^2; m=core(B);D=m;if(Mod(m,4)!=-1,D=4m);b=component(core(B,1),2); if(gcd(a,b)>2,next);h=qfbclassno(-D);vh=valuation(h,p);hp=p^vh; if(hp>H,H=hp;Cp=log(hp)/log(sqrt(D));Hp=component(quadclassunit(-D),2); d=component(matsize(Hp),2);L=List;for(k=1,d,c=component(Hp,k); w=valuation(c,p);if(valuation(c,p)!=0,listput(L,p^w))); print("D=",D," a=",a," b=",b," Cp=",Cp," hp=",hp," Hp=",L)))} rho=6 D=8198552921648689607 a=1 b=3 Cp=0.445646 hp=16384 Hp=[512,2,2,2,2,2] D=18446744073709551615 a=2 b=2 Cp=0.468750 hp=32768 Hp=[512,4,2,2,2,2] D=73786976294838206415 a=7 b=1 Cp=0.575757 hp=524288 Hp=[32768,4,2,2] D=73786976294838175135 a=177 b=1 Cp=0.636363 hp=2097152 Hp=[8192,16,2,2,2,2] D=73786976294831146815 a=2657 b=1 Cp=0.666666 hp=4194304 Hp=[16384,8,2,2,2,2,2] D=8198552921599834167 a=20969 b=3 Cp=0.732133 hp=8388608 Hp=[65536,4,2,2,2,2,2] D=73786976293564644495 a=35687 b=1 Cp=0.787878 hp=67108864 Hp=[2048,512,2,2,2,2,2,2] D=73786976290585731943 a=65211 b=1 Cp=0.969696 hp=4294967296 Hp=[33554432,4,2,2,2,2,2]

{p=2;rho=6;q=3;Y=4q^(p^rho);ba=1;Ba=sqrt(Y);H=1;for(a=ba,Ba,B=Y-a^2; m=core(B);D=m;if(Mod(m,4)!=-1,D=4m);b=component(core(B,1),2); if(gcd(a,b)>2,next);h=qfbclassno(-D);vh=valuation(h,p);hp=p^vh; if(hp>H,H=hp;Cp=log(hp)/log(sqrt(D));Hp=component(quadclassunit(-D),2); d=component(matsize(Hp),2);L=List;for(k=1,d,c=component(Hp,k); w=valuation(c,p);if(valuation(c,p)!=0,listput(L,p^w))); print("D=",D," a=",a," b=",b," Cp=",Cp," hp=",hp," Hp=",L)))} rho=6 D=13734735281170049938631396357123 a=1 b=1 Cp=0.154682 hp=256 Hp=[64,2,2] D=53651309692070507572778892020 a=2 b=32 Cp=0.272429 hp=8192 Hp=[64,4,4,2,2,2] D=13734735281170049938631396357108 a=4 b=2 Cp=0.270694 hp=16384 Hp=[64,8,2,2,2,2,2] D=549389411246801997545255854283 a=7 b=5 Cp=0.303662 hp=32768 Hp=[64,4,2,2,2,2,2,2,2] D=38046358119584625868785031460 a=8 b=38 Cp=0.379179 hp=262144 Hp=[512,8,2,2,2,2,2,2] D=13734735281170049938631396327195 a=173 b=1 Cp=0.464048 hp=16777216 Hp=[32768,8,4,2,2,2,2]

We note the exceptional case $D_{K}=73786976290585731943$ with

[TABLE]

giving the large value $C_{K,2}=0.969696$.

One computes that the group ${\mathcal{T}}_{K}$ that we shall study in the next section is isomorphic to $\mathbb{Z}/2^{2}\mathbb{Z}\times(\mathbb{Z}/2\mathbb{Z})^{6}$ which yields $[\widetilde{K}\cap H_{K}:K]=2^{25}$, where $\widetilde{K}$ is the compositum of the $\mathbb{Z}_{2}$-extensions of $K$.

Many families are described by means of parametized radicals as the family of fields $K=\mathbb{Q}(\sqrt{k^{2}-q^{n}})$ with any prime $q\neq 2$, $k^{2}-q^{n}<0$, in which ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$ has, under some conditions on the parameters, an element of order $n$ (see [2, Theorem 3.1] and its bibliography); applied to $n=p^{r}$, we get, for $C_{K,p}$, the upper bound $\frac{O(1)\,r}{p^{r}}\to 0$ as $r\to\infty$, not sufficient to give “bad families”.

It is difficult to say if some of the above huge discriminants may be obtained with explicit parametrized expressions.

4.2. Reminders on $p$-ramification theory

We intend to give now some analogies with the torsion group ${\mathcal{T}}_{K}$ of the Galois group of the maximal abelian $p$-ramified (i.e., unramified outside $p$ and $\infty$) pro-$p$-extension of $K$; this extension contains the $p$-Hilbert class field of $K$ (in the ordinary sense) and the compositum of the $\mathbb{Z}_{p}$-extensions of $K$. This Galois group introduces the “normalized” $p$-adic regulator of $K$ defined in [21, § 5].

Since in this section the non-$p$-part of the class group does not intervene, unless otherwise stated, we shall put, by abuse of notation, ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}:={\mathcal{C}}\hskip-2.0pt{\ell}_{K}\otimes\mathbb{Z}_{p}$.

4.2.1. Structure of the $p$-torsion group ${\mathcal{T}}_{K}$

Let $K$ be any number field and let $p\geq 2$ be a prime number; we denote by ${\mathfrak{p}}\mid p$ the prime ideals of $K$ dividing $p$. Consider the group $E_{K}$ of $p$-principal global units of $K$ (i.e., units $\varepsilon\equiv 1\!\pmod{\prod_{{\mathfrak{p}}\mid p}{\mathfrak{p}}}$). For each ${\mathfrak{p}}\mid p$, let $K_{\mathfrak{p}}$ be the ${\mathfrak{p}}$-completion of $K$ and $\overline{\mathfrak{p}}$ the corresponding prime ideal of the ring of integers of $K_{\mathfrak{p}}$; then let:

U_{K}:=\Big{\{}u\in\displaystyle\mathop{\raise 2.0pt\hbox{\bigoplus}}\limits_{{\mathfrak{p}}\,\mid\,p}K_{\mathfrak{p}}^{\times},\ \,u=1+x,\ x\in\displaystyle\mathop{\raise 2.0pt\hbox{\bigoplus}}\limits_{{\mathfrak{p}}\,\mid\,p}\overline{\mathfrak{p}}\Big{\}}\,\ \ \&\ \ \,W_{K}:={\rm tor}_{\mathbb{Z}_{p}}(U_{K}),

the $\mathbb{Z}_{p}$-module of principal local units at $p$ and its torsion subgroup.

We consider the diagonal embedding $E_{K}\otimes\mathbb{Z}_{p}\relbar\mathrel{\mkern-4.0mu}\rightarrow U_{K}$ whose image is $\overline{E}_{K}$, the topological closure of $E_{K}$ in $U_{K}$.

We assume in this paper that $K$ satisfies the Leopoldt conjecture at $p$. Whence the following $p$-adic result ([21, Lemma 3.1, Corollary 3.2], [16, Lemma III.4.2.4], [26, Définition 2.11, Proposition 2.12]):

Lemma 4.2**.**

Let $\mu_{K}$ be the group of global roots of unity of $p$-power order of $K$. Under the Leopoldt conjecture for $p$ in $K$, we have ${\rm tor}_{\mathbb{Z}_{p}}(\overline{E}_{K})=\mu_{K}$ and the exact sequence (where ${\rm log}$ is the $p$-adic logarithm):

[TABLE]

Put {\mathcal{W}}_{K}:=W_{K}/\mu_{K}\ & \ {\mathcal{R}}_{K}:={\rm tor}_{\mathbb{Z}_{p}}\big{(}{\rm log}(U_{K})/{\rm log}(\overline{E}_{K})\big{)}. Then the above exact sequence becomes 1\to{\mathcal{W}}_{K}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\rightarrow{\rm tor}_{\mathbb{Z}_{p}}\big{(}U_{K}\big{/}\overline{E}_{K}\big{)}\displaystyle\mathop{\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\rightarrow}^{\!\!{\rm log}}{\mathcal{R}}_{K}\to 0.

Let $\widetilde{K}$ be the compositum of the $\mathbb{Z}_{p}$-extensions, $H_{K}$ the $p$-Hilbert class field and $H_{K}^{\rm pr}$ the maximal Abelian $p$-ramified pro-$p$-extension, of $K$. Then let $H_{K}^{\rm bp}$ be the Bertrandias–Payan field (compositum of the $p$-cyclic extensions of $K$ embeddable in $p$-cyclic extensions of arbitrary large degree).

In the following diagram, class field theory yields:

[TABLE]

We denote by $\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{K}$ the subgroup of the $p$-class group ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$ corresponding to ${\rm Gal}(H_{K}/\widetilde{K}\cap H_{K})$ by class field theory.

Then ${\mathcal{R}}_{K}$ is isomorphic to ${\rm Gal}(H_{K}^{\rm bp}/\widetilde{K}H_{K})$:

[TABLE]

We have \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}=\raise 1.5pt\hbox{{\scriptscriptstyle#}}\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{K}\cdot\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K}\cdot\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{W}}_{K} and the following inequalities:

[TABLE]

where $(r_{1},r_{2})$ is the signature of $K$ and $S_{K}$ the set of $p$-places of $K$. So, for a constant degree $d$, the $p$-rank $\varepsilon$-conjecture for the $p$-class groups implies the $p$-rank $\varepsilon$-conjecture for the torsion groups ${\mathcal{T}}_{K}$ and conversely since we have the other inequality:

[TABLE]

For more precise rank formulas for ${\mathcal{T}}_{K}$, see [16, Corollary III.4.2.3] and the reflection theorem that we shall recall in § 4.4.

4.2.2. The $p$-adic Brauer–Siegel conjecture for ${\mathcal{T}}_{K}$

We have proposed in [17], for the totally real case, after extensive numerical computations, the following conjecture:

Conjecture 4.3**.**

Let $p\geq 2$ be prime and let $d$ be a given degree. For any number field $K$ (under Leopoldt’s conjecture), let ${\mathcal{T}}_{K}$ be the torsion group of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $K$. There exists a constant $\widetilde{\mathcal{C}}_{d,p}$ such that:

[TABLE]

We put, for $p$ fixed and for any totally real number field $K$:

[TABLE]

In practice, $\widetilde{\mathcal{C}}_{K,p}$ may be much smaller than $1$ (and it is often [math]), except very sparse cases as that of $K=\mathbb{Q}(\sqrt{19})$ and $p=13599893$, for which ${\mathcal{T}}_{K}={\mathcal{R}}_{K}\simeq\mathbb{Z}/p\mathbb{Z}$, whence \widetilde{\mathcal{C}}_{K,p}=\displaystyle\frac{{\rm log}(\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K})}{{\rm log}(\sqrt{4\times 19})}=\frac{{\rm log}(13599893)}{{\rm log}(\sqrt{4\times 19})}=7.5855. But ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}=1$.

4.2.3. Estimation of $\widetilde{\mathcal{C}}_{F_{N,p},p}$

Put $F:=F_{N,p}$ and $\widetilde{\mathcal{C}}_{F_{N},p}=:\widetilde{\mathcal{C}}_{F}$ for $p$ fixed. Then, from the computations in Subsection 2.3, using for ${\mathcal{T}}_{F}$ the analog of Chevalley’s formula given in [16, Theorem IV.3.3], we can put similarly \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{F}=:p^{N-r+\widetilde{\Delta}(N)}, $\widetilde{\Delta}(N)\geq 0$, where $r\geq 0$ depends on $p$-adic properties of the ramified primes $\ell\neq p$, and we may estimate that, as $N\to\infty$:

[TABLE]

where $c_{p}=\frac{2\,{\rm log}(p)}{p-1}$ and assuming a small order of magnitude of $\widetilde{\Delta}(N)$.

Give a program computing (in ${\sf Cp}$) $\widetilde{\mathcal{C}}_{F}$; in the imaginary quadratic case for $p=2$, the conjectural inequality implies \displaystyle\raise 1.5pt\hbox{{\scriptscriptstyle#}}(\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{F})\leq(\sqrt{D_{F}}\,)^{\widetilde{\mathcal{C}}_{2,2}} (indeed, \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{F}=1 and \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{F}=\raise 1.5pt\hbox{{\scriptscriptstyle#}}\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{F}\cdot{\mathcal{W}}_{F}, where ${\mathcal{W}}_{F}=2$ (resp. $1$) if $m\equiv\pm 1\pmod{8}$ (resp. if not)). So ${\mathcal{C}}\hskip-2.0pt{\ell}_{F}$ (in ${\sf Clres}$) may be larger than \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{F} (in ${\sf Tor}$):

{p=2;n=12;m=1;for(N=2,100,el=prime(N);m=(-1)^((el-1)/2)elm;P=x^2-m; K=bnfinit(P,1);D=abs(m);Kpn=bnrinit(K,p^n);r=1;if(m<0,r=2);L=List; Hpn=component(component(Kpn,5),2);e=component(matsize(Hpn),2);T=1; for(k=1,e-r,c=component(Hpn,e-k+1);if(Mod(c,p)==0,q=p^valuation(c,p); T=T*q;listinsert(L,q,1)));C8=component(K,8);C81=component(C8,1); h=component(C81,1);Clord=component(C81,2);K=bnfnarrow(K); Cl=component(K,2);print("m=",m," Clres=",Clres," Clord =",Clord); print("Structure of T=",L);print("#Tor=",T," Cp=",log(T)/log(sqrt(D))))}

m=-15 Clres=[2] Structure of Tor=[2] #Tor=2 Cp=0.51191604961963097877535535772960454081 m=105 Clres=[2,2]=[2]xClord Structure of Tor=[2,2] #Tor=4 Cp=0.59574824743531323067786608868687642325 m=-1155 Clres=[2,2,2] Structure of Tor=[2,2,2] #Tor=8 Cp=0.58975726471501581115878339498474155345 m=-15015 Clres=[12,2,2,2] Structure of Tor=[2,2,2,2] #Tor=16 Cp=0.57661327808675875001115538902772596330 m=-255255 Clres=[16,2,2,2,2] Structure of Tor=[2,2,2,2,2] #Tor=32 Cp=0.55674390390043840097934284424073618196 m=4849845 Clres=[4,2,2,2,2,2]=[2]xClord Structure of Tor=[4,2,2,2,2,2] #Tor=128 Cp=0.63036067699149527703810495075838580918 m=-111546435 Clres=[42,2,2,2,2,2,2] Structure of Tor=[2,2,2,2,2,2,2] #Tor=128 Cp=0.52369594802182316940823598390366865409 m=-3234846615 Clres=[308,2,2,2,2,2,2,2] Structure of Tor=[4,2,2,2,2,2,2,2] #Tor=512 Cp=0.56978162829823280646049502887398109011 m=100280245065 Clres=[2,2,2,2,2,2,2,2,2]=[2]xClord Structure of Tor=[2,2,2,2,2,2,2,2,2] #Tor=512 Cp=0.49254011775311327297187105815891979776 m=3710369067405 Clres=[34,2,2,2,2,2,2,2,2,2]=[2]xClord Structure of Tor=[2,2,2,2,2,2,2,2,2,2] #Tor=1024 Cp=0.47898799604336105240349464751069989167 m=152125131763605 Clres=[2,2,2,2,2,2,2,2,2,2,2]=[2]xClord Structure of Tor=[4,2,2,2,2,2,2,2,2,2,2] #Tor=4096 Cp=0.50942162732997733185991268818214459006 m=-6541380665835015 Clres=[28284,2,2,2,2,2,2,2,2,2,2,2] Structure of Tor=[2,2,2,2,2,2,2,2,2,2,2,2] #Tor=4096 Cp=0.45680772041407406738242757697825579415 m=307444891294245705 Clres=[14,2,2,2,2,2,2,2,2,2,2,2,2]=[2]xClord Structure of Tor=[2,2,2,2,2,2,2,2,2,2,2,2,2] #Tor=8192 Cp=0.44755741322126195443312989366082458089 m=16294579238595022365 Clres=[2,2,2,2,2,2,2,2,2,2,2,2,2,2]=[2]xClord Structure of Tor=[32,2,2,2,2,2,2,2,2,2,2,2,2,2] #Tor=262144 Cp=0.56407742575833976013164891102271736655 m=-961380175077106319535 Clres=[1210734,2,2,2,2,2,2,2,2,2,2,2,2,2,2] Structure of Tor=[2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] #Tor=32768 Cp=0.43039341330765014032907101774855577316 m=-58644190679703485491635 Clres=[1622526,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] Structure of Tor=[2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] #Tor=65536 Cp=0.42308787191112227068703992014107942867

We do not know an algorithm computing the “exceptional elements” of ${\mathcal{T}}_{F}$ as for $p$-class groups.

The case $p=3$ is similar and gives for instance for the $16$ cyclic cubic fields of conductor $f=7\cdot 13\cdot 19\cdot 31\cdot 37$:

P=x^3+x^2-661054x+49725976 Cl=[3,3,3,3,3] Structure of Tor=[9,3,3,3,3] #Tor=729 Cp=0.45459180523024141673723101157712338880 P=x^3+x^2-661054x+198463201 Cl=[6,6,3,3] Structure of Tor=[9,3,3,3,3] #Tor=729 Cp=0.45459180523024141673723101157712338880 P=x^3+x^2-661054x+97321888 Cl=[3,3,3,3] Structure of Tor=[3,3,3,3,3] #Tor=243 Cp=0.37882650435853451394769250964760282400 P=x^3+x^2-661054x-186270421 Cl=[39,3,3,3,3] Structure of Tor=[9,3,3,3,3] #Tor=729 Cp=0.45459180523024141673723101157712338880 P=x^3+x^2-661054x-79179619 Cl=[3,3,3,3] Structure of Tor=[3,3,3,3] #Tor=81 Cp=0.30306120348682761115815400771808225920 P=x^3+x^2-661054x-188253584 Cl=[6,6,3,3] Structure of Tor=[9,3,3,3,3] #Tor=729 Cp=0.45459180523024141673723101157712338880 P=x^3+x^2-661054x+138968311 Cl=[3,3,3,3,3] Structure of Tor=[3,3,3,3] #Tor=81 Cp=0.30306120348682761115815400771808225920 P=x^3+x^2-661054x-146607161 Cl=[3,3,3,3] Structure of Tor=[3,3,3,3,3] #Tor=243 Cp=0.37882650435853451394769250964760282400 P=x^3+x^2-661054x-158506139 Cl=[3,3,3,3,3] Structure of Tor=[3,3,3,3,3] #Tor=243 Cp=0.37882650435853451394769250964760282400 P=x^3+x^2-661054x-140657672 Cl=[3,3,3,3] Structure of Tor=[3,3,3,3,3] #Tor=243 Cp=0.37882650435853451394769250964760282400 P=x^3+x^2-661054x+186564223 Cl=[6,6,3,3] Structure of Tor=[3,3,3,3] #Tor=81 Cp=0.30306120348682761115815400771808225920 P=x^3+x^2-661054x+81456584 Cl=[3,3,3,3] Structure of Tor=[3,3,3,3,3] #Tor=243 Cp=0.37882650435853451394769250964760282400 P=x^3+x^2-661054x+206395853 Cl=[3,3,3,3] Structure of Tor=[3,3,3,3] #Tor=81 Cp=0.30306120348682761115815400771808225920 P=x^3+x^2-661054x-206102051 Cl=[3,3,3,3] Structure of Tor=[3,3,3,3] #Tor=81 Cp=0.30306120348682761115815400771808225920 P=x^3+x^2-661054x+2130064 Cl=[12,12,3,3] Structure of Tor=[3,3,3,3,3,3] #Tor=729 Cp=0.45459180523024141673723101157712338880 P=x^3+x^2-661054x+27911183 Cl=[3,3,3,3,3] Structure of Tor=[3,3,3,3,3] #Tor=243 Cp=0.37882650435853451394769250964760282400

Now give some examples for $p=2$ and $p=3$ where the constant $\widetilde{\mathcal{C}}_{F}$ is rather large; the degree $p$ cyclic extensions $F$ are not necessary of the form $F_{N}$.

(i) $p=2$, $m$ divides $5\cdot 7\cdots 41\cdot 43$.

m=5005 Clres=[2,2,2] Clord =[2,2] Structure of Tor=[4,2,2] #Tor=16 Cp=0.65098051255057821664489247470558390764 m=1078282205 Clres=[4,2,2,2,2,2,2] Clord =[4,2,2,2,2,2] Structure of Tor=[32,4,2,2,2,2,2] #Tor=4096 Cp=0.79983769534579979852611700457236749058 m=215656441 Clres=[2,2,2,2,2,2] Clord =[2,2,2,2,2] Structure of Tor=[8,2,2,2,2,2] #Tor=256 Cp=0.57794783195778051534152523962133180636 m=436092044389001 Clres=[2,2,2,2,2,2,2,2,2,2] Clord =[2,2,2,2,2,2,2,2,2] Structure of Tor=[32,2,2,2,2,2,2,2,2,2] #Tor=16384 Cp=0.57575701869336876560350400103707496594 m=46189 Clres=[2,2,2] Clord =[2,2] Structure of Tor=[16,2,2] #Tor=64 Cp=0.77443028965455279095387556052161745644 m=221 Clres=[4] Clord =[2] Structure of Tor=[16] #Tor=16 Cp=1.0272342185833848333397010211662592994 m=435656388001 Clres=[2,2,2,2,2,2,2] Clord =[2,2,2,2,2,2] Structure of Tor=[8,2,2,2,2,2,2] #Tor=512 Cp=0.46554453503678235229309396294036417544

We note the case of $m=221$ for which ${\mathcal{C}}\hskip-2.0pt{\ell}_{F}^{\rm res}\simeq\mathbb{Z}/4\mathbb{Z}$, ${\mathcal{C}}\hskip-2.0pt{\ell}_{F}^{\rm ord}\simeq\mathbb{Z}/2\mathbb{Z}$, but ${\mathcal{T}}_{F}\simeq\mathbb{Z}/16\mathbb{Z}$ due to the $2$-adic regulator. This gives the exceptional value $\widetilde{\mathcal{C}}_{F,2}\approx 1.02723422$.

(ii) $p=3$, $f$ divides $13\cdot 19\cdot 31\cdot 37\cdot 43$.

f=10621 P=x^3+x^2-3540x-60579 Cl=[3,3] Structure of Tor=[9,9,3] #Tor=243 Cp=0.33667761382504192691963484073748684274 P=x^3+x^2-4060762x-3150249179 Cl=[3,3,3,3] Structure of Tor=[27,27,3,3,3] #Tor=19683 Cp=0.60601970488507546845534271332747631693 f=12182287 P=x^3+x^2-4060762x+187697459 Cl=[9,9,3,3,3] Structure of Tor=[9,9,3,3,3] #Tor=2187 Cp=0.47134865935505869768748877703248157983 f=12182287 P=x^3+x^2-4060762x+2380509119 Cl=[3,3,3,3] Structure of Tor=[9,3,3,3,3] #Tor=729 Cp=0.40401313659005031230356180888498421129 f=12182287 P=x^3+x^2-4060762x-2309671376 Cl=[3,3,3,3,3] Structure of Tor=[9,9,3,3,3] #Tor=2187 Cp=0.47134865935505869768748877703248157983 f=641173 P=x^3-x^2-213724x-29968901 Cl=[6,6,3] Structure of Tor=[9,3,3,3] #Tor=243 Cp=0.33667761382504192691963484073748684274 f=392977 P=x^3-x^2-130992x+6826156 Cl=[3,3,3] Structure of Tor=[9,3,3,3] #Tor=243 Cp=0.33667761382504192691963484073748684274 f=7657 P=x^3+x^2-2552x+47360 Cl=[9,3,3] Structure of Tor=[9,9,3] #Tor=243 Cp=0.33667761382504192691963484073748684274

4.3. $p$-adic Brauer–Siegel conjecture versus $\varepsilon$-conjectures

The $p$-adic Brauer–Siegel Conjecture 4.3 concerns more essentially the totally real case for the following reasons which are yet given by the rank inequalities (7) and (8).

4.3.1. Analysis by means of CM-fields

We have, with obvious notations and $p\neq 2$:

[TABLE]

but we have the following properties which explain the differences between real fields and non real ones (under the Leopoldt conjecture):

(i) \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K}^{-}=1 since all the units of infinite order of $K$ are real;

(ii) \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{-}=\raise 1.5pt\hbox{{\scriptscriptstyle#}}\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{{}_{-}}\cdot\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\rm Gal}(\widetilde{K}\cap H_{K}/K)^{-} where ${\rm Gal}(\widetilde{K}\cap H_{K}/K)^{-}\simeq{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{-}/\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{{}_{-}}$ may be large (but with bounded rank) since ${\rm Gal}(\widetilde{K}/K)^{-}\simeq\mathbb{Z}_{p}^{\frac{d}{2}}$ contrary to ${\rm Gal}(\widetilde{K}/K)^{+}\simeq\mathbb{Z}_{p}$;

(iii) \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}^{-}=\raise 1.5pt\hbox{{\scriptscriptstyle#}}\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{{}_{-}}\cdot\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{W}}_{K}^{-} is essentially equal to \raise 1.5pt\hbox{{\scriptscriptstyle#}}\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{{}_{-}} since ${\mathcal{W}}_{K}^{-}$ is most often trivial and does not intervene in estimations of class groups for $d$ fixed;

(iv) \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K}^{+} is the main $p$-adic invariant which may be nontrivial for much primes $p$, even if we have conjectured in [24] that it is trivial for all $p$ large enough;

(v) \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{+} is essentially equal to \raise 1.5pt\hbox{{\scriptscriptstyle#}}\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}^{{}_{+}}_{K} since the part of the Hilbert class field, contained in the cyclotomic $\mathbb{Z}_{p}$-extension, is very limited;

(vi) \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}^{+}=\raise 1.5pt\hbox{{\scriptscriptstyle#}}\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{{}_{+}}\cdot\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K}^{+}\cdot\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{W}}_{K}^{+} is thus essentially the product \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{+}\cdot\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K}^{+}.

So if we assume that \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}^{+} is controled, we may consider that \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{+} is much less than \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}^{+} because of the regulator; then, since \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}^{-} is independent of any regulator, it is mesured by a divisor of \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{-} (equal to \raise 1.5pt\hbox{{\scriptscriptstyle#}}\widetilde{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{{}_{-}}) and by \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{W}}_{K}^{-} controled, which explains (from the factor $[\widetilde{K}\cap H_{K}:K]$) a bigger order of magnitude of \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{-} regarding \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}^{-} than \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{+} regarding \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}^{+}.

4.3.2. Computation of $\widetilde{\mathcal{C}}_{K,p}$ for imaginary quadratic fields

We shall illustrate the cases $p=2$, then $p=3$, in various intervals of negative discriminants to observe the local decreasing of the variable $\widetilde{\mathcal{C}}_{K,p}$ (9).

Note that for $p>3$, the group ${\mathcal{W}}_{K}$ is trivial contrary to the cases $p=2$ and $3$ where ${\mathcal{W}}_{K}$ may be $\mathbb{Z}/p\mathbb{Z}$, which must probably be discarded in our considerations.

(a) Case $p=2$. The case $p=2$ is interesting because of the influence of exceptional classes and gives (v_{p}(\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}) in ${\sf vptor}$, $\widetilde{\mathcal{C}}_{K,p}$ in ${\sf Cp}$):

{p=2;bD=10^8;BD=210^8;Lp=log(p);vp=0;n=20; for(D=bD,BD,e=valuation(D,2);M=D/2^e;if(core(M)!=M,next); if((e==1 || e>3)||(e==0 & Mod(M,4)!=-1)||(e==2 & Mod(M,4)==-1),next); m=D;if(e!=0,m=D/4);P=x^2+m;K=bnfinit(P,1);Kpn=bnrinit(K,p^n); C5=component(Kpn,5);Hpn0=component(C5,1);Hpn=component(C5,2); Hpn1=component(Hpn,1);vptor=valuation(Hpn0/Hpn1,p)-(n-1); if(vptor>vp,vp=vptor);if(vptor>=vp,Cp=vptorLp/log(sqrt(D)); print("D=",-D," m=",-m," vptor=",vptor," Cp=",Cp)))}

p=2, Interval [10^6, 2*10^6] D=-1000011 m=-1000011 vptor=3 Cp=0.301029755983435929933445793 D=-1000020 m=-250005 vptor=3 Cp=0.3010295598834164958938994188 D=-1000036 m=-250009 vptor=4 Cp=0.4013722816881976053812061427 D=-1000132 m=-250033 vptor=5 Cp=0.5017118661610285687682449315 (...) D=-1003620 m=-250905 vptor=5 Cp=0.5015854691511746133432777519 D=-1003940 m=-250985 vptor=6 Cp=0.6018886779424325532238667109 D=-1005843 m=-1005843 vptor=7 Cp=0.702107244840955486811297669 D=-1007492 m=-251873 vptor=8 Cp=0.8023131911871206028276870051 (...) D=-1327972 m=-331993 vptor=8 Cp=0.7865966565831969109249007496 D=-1345476 m=-336369 vptor=9 Cp=0.8841001125437591214944446738 D=-1347524 m=-336881 vptor=10 Cp=0.982227596578129040877631145

p=2, Interval [10^7, 2*10^7] D=-10000004 m=-2500001 vptor=3 Cp=0.25802570416574861895099915 (...) D=-10000136 m=-2500034 vptor=3 Cp=0.25802549285588586651933610 D=-10000212 m=-2500053 vptor=4 Cp=0.34403382825873844184039068 D=-10000228 m=-2500057 vptor=5 Cp=0.43004224263528166242354902 (...) D=-10001220 m=-2500305 vptor=5 Cp=0.43003959612044524303603555 D=-10001355 m=-10001355 vptor=7 Cp=0.6020549303754533561270683 (...) D=-10028164 m=-2507041 vptor=10 Cp=0.8599356519990630566433445 D=-11423624 m=-2855906 vptor=10 Cp=0.8530415407428446759627785 D=-11434244 m=-2858561 vptor=11 Cp=0.9382920445879771130663980 D=-19227908 m=-4806977 vptor=11 Cp=0.9092149504336010969244116

p=2, Interval [10^8, 2*10^8] D=-100000011 m=-100000011 vptor=2 Cp=0.15051499693318294862950 D=-100000020 m=-25000005 vptor=3 Cp=0.225772494296692430574969 (...) D=-100000072 m=-25000018 vptor=3 Cp=0.225772487923331962924891 D=-100000120 m=-25000030 vptor=4 Cp=0.301029976053644338278634 (...) D=-100000228 m=-25000057 vptor=4 Cp=0.301029958404363471726365 D=-100000324 m=-25000081 vptor=6 Cp=0.451544914074197327895496 (...) D=-100009811 m=-100009811 vptor=6 Cp=0.45154258866246136591601 D=-100009988 m=-25002497 vptor=7 Cp=0.526799636159210448373252 (...) D=-100042692 m=-25010673 vptor=9 Cp=0.677301796362621931199437 D=-100120215 m=-100120215 vptor=11 Cp=0.8277784989602807429303 D=-100703939 m=-100703939 vptor=11 Cp=0.8275173634473368234393 D=-101091716 m=-25272929 vptor=13 Cp=0.97777114254342282551717 D=-196241540 m=-49060385 vptor=13 Cp=0.94380528108729550144090

One sees some influence of genus theory since, for $D=-101091716$, we have $\widetilde{\mathcal{C}}_{K,2}\approx 0.977771$ because of \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}=2^{13}, but to be put in relation with $\widetilde{\mathcal{C}}_{K,2}\approx 0.982227$ for $D=-1347524$, of the first interval, with \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}=2^{10}.

The structure of the class group given by PARI/GP is ${\sf[1024,2,2,2]}$.

Then consider the program computing the structure of ${\mathcal{T}}_{K}$ [23, Programme I, § 3.2] that we recall for the convenience of the reader (choose ${\sf p}$, ${\sf nt}$ such that ${\sf p^{\rm nt}}$ be a multiple of the exponent of ${\mathcal{T}}_{K}$, then the polynomial ${\sf P}$):

{p=2;nt=32;P=x^2+101091716;K=bnfinit(P,1);Kpn=bnrinit(K,p^nt); S=component(component(Kpn,1),7);r=component(component(S,2),2)+1; Hpn=component(component(Kpn,5),2);L=List;e=component(matsize(Hpn),2); R=0;for(k=1,e-r,c=component(Hpn,e-k+1);if(Mod(c,p)==0,R=R+1; listinsert(L,p^valuation(c,p),1)));print("Structure of T: ",L)}

Then we obtain ${\mathcal{T}}_{K}\simeq\mathbb{Z}/2^{10}\mathbb{Z}\times\mathbb{Z}/2^{2}\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$. The difference comes from ${\mathcal{W}}_{K}\simeq\mathbb{Z}/2\mathbb{Z}$ (since $-101091716\equiv-1\pmod{16}$) and $[\widetilde{K}\cap H_{K}:K]=2$.

(b) Case $p=3$.

{p=3;bD=10^6;BD=210^6;Lp=log(p);vp=0;n=8; for(D=bD,BD,e=valuation(D,2);M=D/2^e;if(core(M)!=M,next); if((e==1 || e>3)||(e==0 & Mod(M,4)!=-1)||(e==2 & Mod(M,4)==-1),next); m=D;if(e!=0,m=D/4);P=x^2+m;K=bnfinit(P,1);Kpn=bnrinit(K,p^n); C5=component(Kpn,5);Hpn0=component(C5,1);Hpn=component(C5,2); Hpn1=component(Hpn,1);vptor=valuation(Hpn0/Hpn1,p)-(n-1); if(vptor>vp,vp=vptor);if(vptor>=vp,Cp=vptorLp/log(sqrt(D)); print("D=",-D," m=",-m," vptor=",vptor," Cp=",Cp)))} p=3, Interval [10^6, 2*10^6] D=-1000011 m=-1000011 vptor=1 Cp=0.1590402916116620131420348520 D=-1000020 m=-250005 vptor=1 Cp=0.1590401880079362819278196125 D=-1000043 m=-1000043 vptor=1 Cp=0.1590399232477087416304663599 (...) D=-1020548 m=-255137 vptor=3 Cp=0.4764198506806243371783107010 D=-1021332 m=-255333 vptor=4 Cp=0.6351912130796763789096457580 D=-1022687 m=-1022687 vptor=5 Cp=0.7939129439088033794338891302 (...) D=-1898859 m=-1898859 vptor=5 Cp=0.7599296140003213455753306574 p=3, Interval [10^7, 10^7+10^6] D=-10002927 m=-10002927 vptor=3 Cp=0.40895365007950900178024254 D=-10003224 m=-2500806 vptor=4 Cp=0.54527052902405852884991426 (...) D=-10065279 m=-10065279 vptor=4 Cp=0.54506139909286008239956811 D=-10066440 m=-2516610 vptor=5 Cp=0.68132187532448910070906763 (...) D=-14316744 m=-3579186 vptor=5 Cp=0.66675746065456780942779488 D=-14547531 m=-14547531 vptor=6 Cp=0.79933316809564910995167969 (...) D=-19767512 m=-4941878 vptor=6 Cp=0.78474406738375920976115602 p=3, Interval [10^8, 10^8+10^7] D=-100075971 m=-100075971 vptor=4 Cp=0.477101585455621088296257 D=-100080003 m=-100080003 vptor=5 Cp=0.596375677517090310811391 (...) D=-100787315 m=-100787315 vptor=5 Cp=0.596147767754605081482216 D=-100867844 m=-25216961 vptor=6 Cp=0.715346318656502973478053 (...) D=-117344127 m=-117344127 vptor=6 Cp=0.709521342553855083402930 D=-119846559 m=-119846559 vptor=7 Cp=0.826835890443221508331985 (...) D=-135140024 m=-33785006 vptor=7 Cp=0.821531794828164970116186 D=-136159455 m=-136159455 vptor=8 Cp=0.938516745792290367614873

For $D_{K}=-136159455$, we get ${\mathcal{W}}_{K}\simeq\mathbb{Z}/3\mathbb{Z}$, ${\mathcal{T}}_{K}\simeq\mathbb{Z}/3^{7}\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ and ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}\simeq\mathbb{Z}/3^{6}\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ (using the instruction ${\sf quadclassunit(-136159455)}$).

4.3.3. Conclusion

Consider a prime $p$ and a fixed degree $d$. Assume, tentatively, a strong $\varepsilon$-conjecture for the groups ${\mathcal{T}}_{K}$ in the case of totaly real number fields $K$ of degree $d$, which implies a strong $\varepsilon$-conjecture for the $p$-class groups. We then have the existence, for all $\varepsilon>0$, of a constant $\widetilde{C}_{d,p,\varepsilon}$ such that:

[TABLE]

then introduce the function $\widetilde{\mathcal{C}}_{K,p}$:

[TABLE]

So, when $D_{K}\to\infty$, we get $\widetilde{\mathcal{C}}_{K,p}=\varepsilon+o(1)$. But in [17, § 5.3], we have proved that there exist explicit infinite families of real quadratic fields for which $\widetilde{\mathcal{C}}_{K,p}\approx 1$ (contradiction).

We obtain, generalizing to arbitrary degrees, the following heuristic:

There is no absolute strong $\varepsilon$-conjecture for the ${\mathcal{T}}_{K}$ groups of totally real number fields $K$ and ${\mathcal{T}}_{K}$ is essentially governed by the normalized $p$-adic regulator ${\mathcal{R}}_{K}$. Nevertheless, as for $p$-class groups, one may conjecture that the exceptions to the strong $\varepsilon$-conjecture are due to sparse subfamilies of density zero.

Recall that, from (7) and (8), the $p$-rank $\varepsilon$-conjecture for the ${\mathcal{T}}_{K}$ does exist if and only if the $p$-rank $\varepsilon$-conjecture does exist for the $p$-class groups ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$.

4.4. Reflection theorem and $p$-rank $\varepsilon$-conjectures

Another justification of the above comments is to recall the reflection theorem [22] which exchanges, roughly speaking, “imaginary components” of $p$-class groups ${\mathcal{C}}\hskip-2.0pt{\ell}$ with “real components” of $p$-torsion groups ${\mathcal{T}}$, and conversely, subject to consider fields $K$ containing the group $\mu_{p}$ of $p$th roots of unity.

In full generality, the following result precises (7) and (8) when $\mu_{p}\subset K$:

Proposition 4.4** ([16, Theorem III.4.2.2]).**

Let $K$ be a number field containing $\mu_{p}$ and fulfilling Leopoldt’s conjecture at $p$. Then we have the rank formula (reflection theorem) 111The mentions “ord”, “res” are related to the case $p=2$; for $p>2$, since $\mu_{p}\subset K$, the two notions coincide. {\rm rk}_{p}({\mathcal{T}}_{K}^{\rm ord})={\rm rk}_{p}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{S\,{\rm res}})+\raise 1.5pt\hbox{{\scriptscriptstyle#}}S-1, where $S$ is the set of $p$-places of $K$ and ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{S\,{\rm res}}$ the $S$-class group ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{\rm res}/c\hskip-1.0pt{\ell}^{\rm res}(S)$.

So, {\rm rk}_{p}({\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{S\,{\rm res}})={\rm rk}_{p}({\mathcal{T}}_{K}^{\rm ord})-(\raise 1.5pt\hbox{{\scriptscriptstyle#}}S-1) yields:

[TABLE]

giving :

[TABLE]

since \raise 1.5pt\hbox{{\scriptscriptstyle#}}S-1 is bounded in the family of number fields, of fixed degree $d$, this relation shows that any $p$-rank $\varepsilon$-conjecture, true for an invariant, is fulfilled by the other. When $\mu_{p}\not\subset K$, one must use the field $K(\mu_{p})$ and the general reflection theorem with $p$-adic characters ([16, Theorem II.5.4.5], [22]).

The reflection theorem has been used in [10], in a different manner, using many split primes in $K(\mu_{p})$, the notion of Arakelov class group (see e.g., [37]), and the following analytic explanation by the authors:

Roughly, the point is that small non-inert primes in a number field represent elements of the class group which tend not to satisfy any relation with small coefficients. Thus the existence of many such primes contributes significantly to the quotient of the class group by its $\ell$-torsion, yielding the desired upper bounds.

It would be interesting to deepen these approaches that have connections through class field theory, complex and $p$-adic analytic methods.

Acknowledgments

We thank Peter Koymans and Carlo Pagano for fruitful information about various approaches (density results v.s. $\varepsilon$-conjectures) concerning $p$-class groups in degree $p$ cyclic fields.