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Georges Gras

Publications and source records attributed to Georges Gras.

At least 19 recordsLinked to original sources
arXiv2026

Chevalley-Herbrand formulas and Zp -extensions of a p-principal imaginary quadratic field

Let $k$ be an imaginary quadratic field and let $p \ne 2$ be a prime number, split in $k$ into ${\mathfrak p}{\overline {\mathfrak p}}$. We assume that the $p$-class group of $k$ is trivial. Let $\delta \geq 0$ be the ${\mathfrak p}$-valuation of the ${\overline {\mathfrak p}}$-Fermat quotient of the fundamental ${\mathfrak p}$-unit $x$ of $k$. Let $K/k$ be any bi-ramified ${\mathbb{Z}}_p$-extension and let $p^e$ be the degree of the inertia field of ${\mathfrak p}$, $\overline {\mathfrak p}$ being totally ramified. We prove that if $e \geq \delta$, then $\lambda(K/k) = 1$, $\mu(K/k) = 0$; if $e<\delta$ a characterization is obtained from Iwasawa invariants of the $S^{\mathfrak p}$-class groups. This approach only uses generalizations of Chevalley-Herbrand formulas and the non-nullity of a $p$-adic regulator ${\mathcal R}^{\mathfrak p}_\delta$ in incomplete $p$-ramification. It provides effective and computable results that complement some aspects of Iwasawa theory. Conjecture states that only the cyclotomic ${\mathbb{Z}}_p$-extension is ''exceptional''; justifications are given. A pari/gp program computes $\delta$ and ${\mathcal R}^{\mathfrak p}_\delta$, for $p=3$, $e=1$.

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arXiv2024

Initial layer of the anti-cyclotomic $\mathbb{Z}_3$-extension of $\mathbb{Q}(\sqrt{-m})$ and capitulation phenomenon

Let $k=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic field. We consider the properties of capitulation of the $p$-class group of $k$ in the anti-cyclotomic $\mathbb{Z}_p$-extension $k^{\rm ac}$ of $k$; for this, using a new approach based on the Log$_p$-function (Theorems 2.3, 3.4), we determine the first layer $k_1^{\rm ac}$ of $k^{\rm ac}$ over $k$, and we show that some partial capitulation may exist in $k_1^{\rm ac}$, even when $k^{\rm ac}/k$ is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the $\mathbb{Z}_p$-extensions of $k$, distinct from the cyclotomic one. For $p=3$, we characterize a sub-family of fields $k$ (Normal Split cases) for which $k^{\rm ac}$ is not linearly disjoint from the Hilbert class field (Theorem 5.1). No assumptions are made on the splitting of 3 in $k$ and in $k^*=\mathbb{Q}(\sqrt{3m})$, nor on the structures of their 3-class groups. Four PARI/GP programs (7.1, 7.2, 7.3, 7.4 depending on the classification of Definition 2.10) are given, computing a defining cubic polynomial of $k_1^{\rm ac}$, and the main invariants attached to the fields $k$, $k^*$, $k_1^{\rm ac}$; some relations with Iwasawa's invariants are discussed (Theorem 9.6).

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arXiv2024

On the $\mathbb{Z}_p$-extensions of a totally $p$-adic imaginary quadratic field -- With an appendix by Jean-Fran\c{c}ois Jaulent

Let $k = \mathbb{Q}(\sqrt {-m})$ and $p \geq 3$ split in $k$. We prove new properties of the $\mathbb{Z}_p$-extensions $K/k$, distinct from the cyclotomic one; we do not assume $K/k$ totally ramified, nor the triviality of the $p$-class group of $k$. These properties are governed by the $p$-valuation $\delta_p(k)$ of a Fermat quotient of the fundamental $p$-unit $x$ of $k$, which also yields the order of the logarithmic class group $\# \mathcal{H}_k$ (Thm. 4.2 extended in App.A to the case of imaginary abelian fields of prime-to-$p$ degree), and allows to generalize the Gold-Sands criterion (Sec. 7). These results are related to the first two elements, $\mathcal{H}_{K_n}^1$ and $\mathcal{H}_{K_n}^2$, of the filtrations of the $p$-class groups in $K = \cup_n K_n$, without any argument of Iwasawa's theory, and provide new perspectives since $\# ( \mathcal{H}_{K_n}^2/ \mathcal{H}_{K_n}^1) = \# \widetilde {\mathcal{H}}_k$ for $n$ large enough (Thm 7.1). We give a short proof generalizing a result of Kundu-Washington (Thm. 7.8) on the $p$-class groups in the anti-cyclotomic $\mathbb{Z}_p$-extension $k^{\rm ac}$. We compute, Sec. 9, for $p = 3$, the first layer $k_1^{\rm ac}$ of $k^{\rm ac}$, using the Log$_p$-function, and show (Thms. 9.2,9.4) that capitulation of suitable ``classes'' is possible in $k^{\rm ac}$, suggesting Conjecture 7.10. Finally, we generalize (Thms.10.1,10.7) a result of Ozaki giving large $\lambda$'s invariants. Calculations and programs are gathered App. C.

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arXiv2023

Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification

We consider, for real abelian fields K, the Birch--Tate formula linking the tame kernel \#K\_2(Z\_K) to $\zeta$\_K(-1); we compare, for quadratic and cyclic cubic fields with p=2,3, \#K\_2(\BZ\_K)[p^$\infty$] to the order of the torsion group T\_{K, p} of abelian p-ramification theory given by the residue of $\zeta$\_{K, p}(s) at s=1. This is done via the ``genus theory'' of p-adic pseudo-measures, inaugurated in the 1970/80's and the fact that T\_{K, p} only depends on the p-class group and on the normalized p-adic regulator of K (Theorem A). We apply this to prove a conjecture of Deng--Li giving the structures of K\_2(Z\_K)[2^$\infty$] for an interesting family of real quadratic fields (Theorem B). Then, for p>3, we give a lower bound of the p-rank of K\_2(\BZ\_K) in cyclic p-extensions (Theorem C). Complements, PARI programs and tables are given in an Appendix.

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arXiv2023

Exceptional case of the non semi-simple Real Abelian Main Conjecture

In the papers: "The Chevalley--Herbrand formula and the real abelian Main Conjecture (New criterion using capitulation of the class group),J. Number Theory 248 (2023)" and "On the real abelian main conjecture in the non semi-simple case, arXiv (2023)", we consider real cyclic fields $K$ and primes $\ell \equiv 1 \pmod {2p^N}$ totally inert in $K$, implying implicitly, $K \cap \mathbb{Q}(μ_{p^\infty}^{})^+ = \mathbb{Q}$. In this complementary work, we examine the non linearly disjoint case.

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arXiv2023

Application of the notion of $Φ$-object to the study of p-class groups and p-ramified torsion groups of abelian extensions

We revisit, in an elementary way, the classical statement of various ``Main Conjectures'' for $p$-class groups $\mathcal{H}_K$ and $p$-ramified torsion groups $\mathcal{T}_K$ of abelian fields $K$, in the non semi-simple case $p \mid [K : \mathbb{Q}]$. The classical ``algebraic'' definition of the $p$-adic isotopic components, $\mathcal{H}^{\rm alg}{K,φ}$, used in the literature, is inappropriate with respect to analytical formulas. For that reason we have introduced, in the 1970's, an ``arithmetic'' definition, $\mathcal{H}^{\rm ar}{K,φ}$, in perfect correspondence with all analytical formulas and giving a natural ``Main Conjecture'', still unproved for real fields in the non semi-simple case. The two notions coincide for relative class groups $\mathcal{H}_K^-$ and groups $\mathcal{T}_K$ since, in $p$-extensions, transfer maps are injective for these groups but not necessarily for real class groups. Numerical evidence of the gap between the two notions is given (Examples A.2.2, A.2.3) and PARI calculations corroborate that the true Real Main Conjecture for $K$ writes on the form $\# \mathcal{H}^{\rm ar}_{K,φ} = \# (\mathcal{E}_K / \hat{\mathcal{E}}_K \, \mathcal{F}_{\!K})_φ$, in terms of units $\mathcal{E}_K$, $\hat{\mathcal{E}}_K$ (units of the strict subfields) and $\mathcal{F}_K$ (Leopoldt's cyclotomic units). A recent approach, conjecturing the capitulation of $\mathcal{H}_K$ in some auxiliary cyclotomic extensions $K(μ_\ell^{})$, proves the difficult real case.

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arXiv2023

On the Real Abelian Main Conjecture in the non semi-simple case

Let $K/\mathbb{Q}$ be a real cyclic extension of degree divisible by $p$. We analyze the {\it statement} of the "Real Abelian Main Conjecture", for the $p$-class group $\mathcal{H}_K$ of $K$, in this non semi-simple case. The classical {\it algebraic} definition of the $p$-adic isotopic components $\mathcal{H}^{\rm alg}_{K,φ}$, for irreducible $p$-adic characters $φ$, is inappropriate with respect to analytical formulas, because of capitulation of $p$-classes in the $p$-sub-extension of $K/\mathbb{Q}$. In the 1970's we have given an {\it arithmetic} definition, $\mathcal{H}^{\rm ar}_{K,φ}$, and formulated the conjecture, still unproven, $\# \mathcal{H}^{\rm ar}_{K,φ} = \# (\mathcal{E}_K / \mathcal{E}^\circ_K \, \mathcal{F}_{\!K})_{φ_0}$, in terms of units $\mathcal{E}_K$ then $\mathcal{E}^\circ_K$ (generated by units of the strict subfields of $K$) and cyclotomic units $\mathcal{F}_K$, where $φ_0$ is the tame part of $φ$. We prove that the conjecture holds as soon as there exists a prime $\ell$, totally inert in $K$, such that $\mathcal{H}_K$ capitulates in $K(μ_\ell^{})$, existence having been checked, in various circumstances, as a promising new tool.

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arXiv2023

Unlimited lists of fundamental units of quadratic fields -- Applications to some arithmetic properties

We use the polynomials $m_s(t) = t^2 - 4 s$, $s \in \{-1, 1\}$, in an elementary process giving arbitrary large lists of {\it fundamental units} of quadratic fields of discriminants listed in ascending order. More precisely, let $\mathbf{B} \gg 0$; then as $t$ grows from $1$ to $\mathbf{B}$, for each {\it first occurrence} of a square-free integer $M \geq 2$, in the factorization $m_s(t) =: M r^2$, the unit $\frac{1}{2} \big(t + r \sqrt{M}\big)$ is the fundamental unit of norm $s$ of $\mathbb{Q}(\sqrt M)$, even if $r >1$ (Theorem 4.1). Using $m_{sν}(t) = t^2 - 4 s ν$, $ν\geq 2$, the algorithm gives arbitrary large lists of {\it fundamental solutions} to $u^2 - M v^2= 4sν$ (Theorem 4.11). We deduce, for $p>2$ prime, arbitrary large lists of {\it non $p$-rational} quadratic fields (Theorems 6.3, 6.4, 6.5) and of degree $p-1$ imaginary fields with non-trivial $p$-class group (Theorems 7.1,7.2). PARI programs are given to be copied and pasted.

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arXiv2023

New characterization of the norm of the fundamental unit of $\mathbb{Q}(\sqrt M)$

We give an elementary criterion for the norm of the fundamental unit $\varepsilon_K$ of $K=\mathbb{Q}(\sqrt M)$, $M$ square-free. More precisely, if $\varepsilon_K = a+b\sqrt M$, $a,b \in \mathbb{Z}$ or $\frac{1}{2}\mathbb{Z}$, its norm ${\bf S}_K$ only depends on $m := {\bf gcd} \big(\frac{a+1}{{\bf gcd}(a+1,b)}, M\big)$ and $m' := {\bf gcd} \big(\frac{a-1}{{\bf gcd}(a-1,b)}, M\big)$ as follows when $-1$ is a global norm: ${\bf S}_K = -1$ if and only if $m=m'=1$ (resp. $m=m'=2$) for $M$ odd (resp. even) (Theorems 1.1 or 2.4).

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arXiv2022

Algebraic norm and capitulation of p-class groups in ramified cyclic p-extensions

We examine the phenomenon of capitulation of the $p$-class group $H_K$ of a real number field $K$ in totally ramified cyclic p-extensions $L/K$ of degree $p^N$. Using an elementary property of the algebraic norm $\nu_{L/K}$, we show that the kernel of capitulation is in relation with the "complexity" of the structure of $H_L$ measured via its exponent $p^e(L)$ and the length $m(L)$ of the usual filtration $\{H_L^i\}_{i \ge 0}$ associated to $H_L$ as $Z_p[Gal(L/K)]$-module. We prove that a sufficient condition of capitulation is given by $e(L) \in [1, N-s(L)]$ if $m(L) \in [p^s(L), p^(s(L)+1)-1]$ for $s(L) \in [0, N-1]$ (Theorem 1.1); this improves the case of "stability" $\#H_L = \#H_K$ (i.e., $m(L) = 1$, $s(L)=0$, $e(L) = e(K)$) (Theorem 1.2). Numerical examples (with PARI programs) showing most often capitulation of $H_K$ in $L$, are given, taking the simplest abelian $p$-extensions $L < K(\mu_\ell$), with primes $\ell=1$ (mod $2p^N$) over cubic fields with $p=2$ and real quadratic fields with $p=3$. Some conjectures on the existence of non-zero densities of such $\ell$'s are proposed (Conjectures 1.4, 2.4). Capitulation property of other arithmetic invariants is examined.

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arXiv2022

The Chevalley-Herbrand formula and the real abelian Main Conjecture

The Main Theorem for abelian fields (often called Main Conjecture despite proofs in most cases) has a long history which has found a solution by means of "elementary arithmetic", as detailed in Washington's book from Thaine's method having led to Kolyvagin's Euler systems. Analytic theory of real abelian fields $K$ says (in the semi-simple case) that the order of the $p$-class group $\mathcal{H}_K$ is equal to the $p$-index of cyclotomic units $(\mathcal{E}_K : \mathcal{F}_K)$. We have conjectured (1977) the relations $\# \mathcal{H}_φ= (\mathcal{E}_φ: \mathcal{F}_φ)$ for the isotypic $p$-adic components using the irreducible $p$-adic characters $φ$ of $K$. We develop, in this article, new promising links between: (i) the Chevalley-Herbrand formula giving the number of ``ambiguous classes'' in $p$-extensions $L/K$, $L \subset K(μ_\ell^{})$ for the auxiliary prime numbers $\ell \equiv 1 \pmod {2p^N}$ inert in $K$; (ii) the phenomenon of capitulation of $\mathcal{H}_K$ in $L$; (iii) the real Main Conjecture $\# \mathcal{H}_φ= (\mathcal{E}_φ: \mathcal{F}_φ)$ for all~$φ$. We prove that the real Main Conjecture is trivially fulfilled as soon as $\mathcal{H}_K$ capitulates in $L$ (Theorem \ref{thmppl}). Computations with PARI programs support this new philosophy of the Main Conjecture. The very frequent phenomenon of capitulation suggests Conjecture 1.2.

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arXiv2021

On the $λ$-stability of p-class groups along cyclic p-towers of a number field

Let k be a number field, p$\ge$2 a prime and S a set of tame or wild finite places of k. We call K/k a totally S-ramified cyclic p-tower if Gal(K/k)=Z/p^NZ and if S non-empty is totally ramified. Using analogues of Chevalley's formula (Gras, Proc. Math. Sci. 127(1) (2017)),we give an elementary proof of a stability theorem (Theorem 3.1 for generalized p-class groups X\_n of the layers k\_n$\le$K:let $λ$=max(0, \#S-1-$ρ$) given in Definition 1.1; then\#X\_n = \#X\_0 x p^{$λ$ n} for all n in [0,N], if and only if \#X\_1=\#X\_0 x p^$λ$. This improves the case $λ$ = 0 of Fukuda (1994), Li--Ouyang--Xu--Zhang (2020), Mizusawa--Yamamoto (2020),whose techniques are based on Iwasawa's theory or Galois theory of pro-p-groups. We deduce capitulation properties of X\_0 in the tower (e.g. Conjecture 4.1). Finally we apply our principles to the torsion groups T\_n of abelian p-ramification theory. Numerical examples are given.

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arXiv2021

Invariant generalized ideal classes -- structure theorems for p-class groups in p-extensions

We give, in Sections 2 and 3, an english translation of: {\it Classes généralisées invariantes}, J. Math. Soc. Japan, 46, 3 (1994), with some improvements and with notations and definitions in accordance with our book: {\it Class Field Theory: from theory to practice}, SMM, Springer-Verlag, $2^{\rm nd}$ corrected printing 2005. We recall, in Section 4, some structure theorems for finite $\mathbb{Z}_p[G]$-modules ($G \simeq \mathbb{Z}/p\,\mathbb{Z}$) obtained in: {\it Sur les $\ell$-classes d'idéaux dans les extensions cycliques relatives de degré premier $\ell$}, Annales de l'Institut Fourier, 23, 3 (1973). Then we recall the algorithm of local normic computations which allows to obtain the order and (potentially) the structure of a $p$-class group in a cyclic extension of degree $p$. In Section 5, we apply this to the study of the structure of relative $p$-class groups of Abelian extensions of prime to $p$ degree, using the Thaine--Ribet--Mazur--Wiles--Kolyvagin "principal theorem", and the notion of "admissible sets of prime numbers" in a cyclic extension of degree $p$, from: {\it Sur la structure des groupes de classes relatives}, Annales de l'Institut Fourier, 43, 1 (1993). In conclusion, we suggest the study, in the same spirit, of some deep invariants attached to the $p$-ramification theory (as dual form of non-ramification theory) and which have become standard in a $p$-adic framework. Since some of these techniques have often been rediscovered, we give a substantial (but certainly incomplete) bibliography which may be used to have a broad view on the subject.

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arXiv2021

Algorithmic complexity of Greenberg's conjecture

Let $k$ be a totally real number field and $p$ a prime. We show that the ``complexity'' of Greenberg's conjecture ($λ= μ= 0$) is of $p$-adic nature governed (under Leopoldt's conjecture) by the finite torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, by means of images in ${\mathcal T}_k$ of ideal norms from the layers $k_n$ of the cyclotomic tower (Theorem (5.2)). These images are obtained via the formal algorithm computing, by ``unscrewing'', the $p$-class group of~$k_n$. Conjecture (5.4) of equidistribution of these images would show that the number of steps $b_n$ of the algorithms is bounded as $n \to \infty$, so that Greenberg's conjecture, hopeless within the sole framework of Iwasawa's theory, would hold true ``with probability $1$''. No assumption is made on $[k : \mathbb{Q}]$, nor on the decomposition of $p$ in $k/\mathbb{Q}$.

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arXiv2020

Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

Let $K$ be the $N$th layer in the cyclotomic $\widehat{\mathbb{Z}}$-extension $\widehat{\mathbb{Q}}$. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington,\,$\ldots$) analyse the behavior of the $p$-class groups ${\mathcal C}_K$. We revisit this problem, in a more conceptual form, since computations show that the $p$-torsion group ${\mathcal T}_K$ of the Galois groups of the maximal abelian $p$-ramified pro-$p$-extension of $K$ (Tate--Shafarevich group of $K$) is often non-trivial; this raises questions since $\# {\mathcal T}_K = \# {\mathcal C}_K\, \# {\mathcal R}_K$ where ${\mathcal R}_K$ is the normalized $p$-adic regulator. We give a new method testing ${\mathcal T}_K \ne 1$ (Theorem 4.6, Table 6.2) and characterize the $p$-extensions $F$ of $K$ in $\widehat{\mathbb{Q}}$ with ${\mathcal C}_F \ne 1$ (Theorem 7.5 and Corollary 7.6). We publish easy to use programs, justifying again the eight known examples, and allowing further extensive computations.

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arXiv2020

The p-rank $ε$-conjecture on class groups is true for towers of p-extensions

Let p$\ge$2 be a given prime number. We prove, for any number field kappa and any integer e$\ge$1, the p-rank $ε$-conjecture, on the p-class groups Cl\_F, for the family F\_kappa^p^e of towers F/kappa built as successive degree p cyclic extensions (without any other Galois conditions) such that F/kappa be of degree p^e, namely: #(Cl\_F[p])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, where D\_F is the absolute value of the discriminant (Theorem 3.6) and, more generally, #(Cl\_F[p^r])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, for any r$\ge$1 fixed. This Note generalizes the case of the family F\_Q^p (Genus theory and $ε$-conjectures on p-class groups, J. Number Theory 207, 423--459 (2020)), whose techniques appear to be "universal" for all relative degree p cyclic extensions and use the Montgomery--Vaughan result on prime numbers. Then we prove, for F\_kappa^p^e, the p-rank $ε$-conjecture on the cohomology groups H^2(G\_F,Z\_p) of Galois p-ramification theory over F (Theorem 4.3) and for some other classical finite p-invariants of F, as the Hilbert kernels and the logarithmic class groups.

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arXiv2019

Successive maxima of the non-genus part of class numbers

Some PARI programs have bringed out a property for the non-genus part of the class number of the imaginary quadratic fields, with respect to $(\sqrt D\,)^{\varepsilon}$, where $D$ is the absolute value of the discriminant and $\varepsilon \in ]0, 1[$, in relation with the $\varepsilon$-conjecture. The general Conjecture 3.1, restricted to quadratic fields, states that, for $\varepsilon \in ]0, 1[$, the successive maxima, as $D$ increases, of $\frac{H}{2^{N-1} \cdot (\sqrt D\,)^{\varepsilon}}$, where $H$ is the class number and $N$ the number of ramified primes, occur only for prime discriminants (i.e., $H$ odd); we perform computations giving some obviousness in the selected intervals. For degree $p>2$ cyclic fields, we define a "mean value" of the non-genus parts of the class numbers of the fields having the same conductor and obtain an analogous property on the successive maxima. In Theorem 2.5 we prove, under an assumption (true for $p=2, 3$), that the sequence of successive maxima of $\frac{H}{p^{N-1} \cdot (\sqrt D\,)^{\varepsilon}}$ is infinite. Finally we consider cyclic or non-cyclic abelian fields of degrees $4, 8, 6, 9, 10, 30$ to test the Conjecture 3.1. The successive maxima of $\frac{H}{(\sqrt D\,)^{\varepsilon}}$ are also analyzed.

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arXiv2019

Note on some p-invariants of Q(N^{1/p}) using reflection theorem

Let p > 2 be a prime number and let N be any rational integer. We consider the p-class groups Cl(L), Cl(M) of the fields $L:=Q(N^{1/p})$ and $M:=Q(N^{1/p},μ_p)$, by comparison with the p-torsion groups T(L) and T(M) of the abelian p-ramification theory, in the framework of the reflection theorem, and obtain relations between the ranks of the isotypic components (Theorem 2.6). For p=3, we characterize the integers N such that L is 3-rational (i.e., T(L)=1), giving the following values: $N=3$; $N=3^d \ell$, $\ell = -1+ 3u$; $N=3^d \ell$, $\ell=(1+3a)^2+27b^2$, with $\ell$ prime and $uab$ prime to 3 (Theorem 2.18). We show that the 3-class group Cl(L) is trivial if and only Cl(M) is trivial (Theorem 2.19). We give various tables with PARI/GP programs computing the structure of Cl(L), Cl(M), T(L), T(M), and of the logarithmic class groups (Appendix A, B, C).

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