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Georges Gras

Publications and source records attributed to Georges Gras.

At least 37 records · Page 2Linked to original sources
arXiv2019

Test of Vandiver's conjecture with Gauss sums -- Heuristics

The link between Vandiver's conjecture and Gauss sums is well known since the papers of Iwasawa (1975), Thaine (1995-1999) and Angl{è}s-Nuccio (2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver's conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the pth cyclotomic field (1984). Then we provide a specific use of Gauss sums of characters of order p of F\_ell^x and prove new criteria for Vandiver's conjecture to hold (Theorem 1.2(a) using both the sets of exponents of p-irregularity and of p-primarity of suitable twists of the Gauss sums, and Theorem 1.2(b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in \S5.2 new heuristics showing that any counterexample to the conjecture leads to excessive constraints modulo p on the above twists as ell varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in direction of the very probable truth of Vandiver's conjecture. All the calculations are given with their PARI/GP programs.

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arXiv2019

Genus theory and $ε$-conjectures on p-class groups

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.

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arXiv2019

Practice of incomplete p-ramification over a number field -- History of abelian p-ramification

The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and $\infty$, is well known including numerical experiments with PARI/GP programs. The case of ``incomplete p-ramification'' (i.e., when the set S of ramified places is a strict subset of the set P of the p-places) is, on the contrary, mostly unknown in a theoretical point of view. We give, in a first part, a way to compute, for any S $\le$ P, the structure of the Galois group of the maximal S-ramified abelian pro-p-extension H\_(K,S) of any field K given by means of an irreducible polynomial. We publish PARI/GP programs usable without any special prerequisites. Then, in an Appendix, we recall the ``story'' of abelian S-ramification restricting ourselves to elementary aspects in order to precise much basic contributions and references, often disregarded, which may be used by specialists of other domains of number theory. Indeed, the torsion T\_(K,S) of Gal(H\_(K,S)/K) (even if S=P) is a fundamental obstruction in many problems. All relationships involving S-ramification, as Iwasawa's theory, Galois cohomology, p-adic L-functions, elliptic curves, algebraic geometry, would merit special developments, which is not the purpose of this text.

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arXiv2019

A program to test the p-rationality of any number field

Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8) if p > 2 (resp. p = 2) characterizes its p-rationality. Then we give two short, very fast PARI Programs (§§3.1, 3.2) testing if K (defined by an irreducible monic polynomial) is p-rational or not. For quadratic fields we verify some densities related to Cohen-Lenstra-Martinet ones and analyse Greenberg's conjecture on the existence of p-rational fields with Galois groups (Z/2Z) t needed for the construction of some Galois representations with open image. We give examples for p = 3, t = 5 and t = 6 (§§5.1, 5.2) and illustrate other approaches (Pitoun-Varescon, Barbulescu-Ray). We conclude about the existence of imaginary quadratic fields, p-rational for all p $\ge$ 2 (Angelakis-Stevenhagen on the concept of "minimal absolute abelian Galois group") which may enlighten a conjecture of p-rationality (Hajir-Maire) giving large Iwasawa $μ$-invariants of some uniform prop groups. R{é}sum{é} Soit K un corps de nombres. Nous montrons que son corps de classes de rayon modulo p 2 (resp. 8) si p > 2 (resp. p = 2) caract{é}rise sa p-rationalit{é}. Puis nous donnons deux courts programmes PARI (§§3.1, 3.2) trs rapides testant si K (d{é}fini par un polyn{ô}me irr{é}ductible uni-taire) est p-rationnel ou non. Pour les corps quadratiques nous v{é}rifions certaines densit{é}s en relation avec celles de Cohen-Lenstra-Martinet et nous analysons la conjecture de Greenberg sur l'existence de corps p-rationnels de groupes de Galois (Z/2Z) t n{é}cessaires pour la construction de certaines repr{é}sentations galoisiennes d'image ouverte. Nous donnons des exemples pour p = 3, t = 5 et t = 6 (§§5.1, 5.2) et il-lustrons d'autres approches (Pitoun-Varescon, Barbulescu-Ray). Nous concluons sur l'existence de corps quadratiques imaginaires p-rationnels pour tout p $\ge$ 2 (Angelakis-Stevenhagen sur le concept de "groupe de Galois ab{é}lien absolu minimal") qui peut{é}clairerpeut{é}clairer une conjecture de p-rationalit{é} (Hajir-Maire) donnant de grands invariants $μ$ d'Iwasawa relatifs {à} certains prop -groupes uniformes.

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arXiv2018

Normes d'idéaux dans la tour cyclotomique et conjecture de Greenberg

Pre-print of a publication in "Annales mathématiques du Qu{é}bec". Let $k$ be a totally real number field and let $k_\infty$ be its cyclotomic $\mathbb{Z}_p$-extension for $p$ totally split in $k$. This text completes our article entitled: "Approche $p$-adique de la conjecture de Greenberg pour les corps totalement réels" (Annales Mathématiques Blaise Pascal 2017), by means of heuristics on the $p$-adic behavior of the norms, in $k_n/k$, of the ideals in $k_\infty$ ; indeed, this conjecture (on the nullity of the invariants $λ$ et $μ$ of Iwasawa) depends of images in the torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, thus of Artin symbols in a finite extension $F/k$ obtained by Galois descent of ${\mathcal T}_k$. An assumption of distribution of these norms implies $λ=μ=0$. Several statistics and numerical examples in the quadratic case confirm the probable exactness of such properties which constitute the fundamental obstruction for a proof of Greenberg's conjecture in the sole context of Iwasawa's theory.

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arXiv2018

Annihilation of tor\_p(G\_S^ab) for real abelian extensions

Preprint of a paper to appear in "Communications in Advanced Mathematical Sciences". Let K be a real abelian extension of Q. Let p be a prime number, S the set of p-places of K and G\_K,S the Galois group of the maximal S-ramified pro-p-extension of K (i.e., unramified outside p and infinity). We revisit the problem of annihilation of the p-torsion group T\_K:=tor\_Z\_p(G\_K,S^ab) initiated by us and Oriat then systematized in our paper on the construction of p-adic L-functions in which we obtained a canonical ideal annihilator of T\_K in full generality (1978--1981). Afterwards (1992--2014) some annihilators, using cyclotomic units, were proposed by Solomon, Belliard--Nguyen Quang Do, Nguyen Quang Do--Nicolas, All, Belliard--Martin.In this text, we improve our original papers and show that, in general, the Solomon elements are not optimal and/or partly degenerated. We obtain, whatever K and p, an universal non-degenerated annihilator in terms of p-adic logarithms of cyclotomic numbers related to L\_p-functions at s=1 of its primitive characters of K (Theorem 9.4). Some computations are given with PARI programs; the case p=2 is analyzed and illustrated in degrees 2, 3, 4 to test a conjecture.

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arXiv2018

Heuristics in direction of a p-adic Brauer--Siegel theorem

Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K and let T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We conjecture the existence of a constant C\_p>0 such that log(\#T\_K) $\le$ C\_p log(\sqrt(D\_K)) when K varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a p-adic analogue, of the classical Brauer--Siegel Theorem, wearing here on the valuation of the residue at s=1 (essentially equal to \#T\_K) of the p-adic zeta-function zeta\_p(s) of K.We shall use a different definition that of Washington, given in the 1980's, and approach this question via the arithmetical study of T\_K since p-adic analysis seems to fail because of possible abundant "Siegel zeros" of zeta\_p(s), contrary to the classical framework.We give extensive numerical verifications for quadratic and cubic fields (cyclic or not) and publish the PARI/GP programs directly usable by the reader for numerical improvements. Such a conjecture (if exact) reinforces our conjecture that any fixed number field K is p-rational (i.e., T\_K=1) for all p >> 0 .

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arXiv2017

The p-adic Kummer-Leopoldt constant - Normalized p-adic regulator

The p-adic Kummer--Leopoldt constant kappa\_K of a number field K is (assuming the Leopoldt conjecture) the least integer c such that for all n \textgreater{}\textgreater{} 0, any global unit of K, which is locally a p^(n+c)th power at the p-places, is necessarily the p^nth power of a global unit of K. This constant has been computed by Assim \& Nguyen Quang Do using Iwasawa's techniques,after intricate studies and calculations by many authors. We give an elementary p-adic proof and an improvement of these results, then a class field theory interpretation of kappa\_K. We give some applications (including generalizations of Kummer's lemma on regular pth cyclotomic fields) and a natural definition of the normalized p-adic regulator for any K and any p$\ge$2.This is done without analytical computations, using only class field theoryand especially the properties of the so-called p-torsion group T\_K of Abelian p-ramification theory over K.

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arXiv2017

p-adic approach of Greenberg's conjecture (p-split totally real case)

Let k be a totally real number field ant let k$\infty$ be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k, and we obtain important tables of quadratic fields and p for which we can conclude that lambda = mu=0.We show that the number of ambiguous p-classes of kn (nth stage in k$\infty$) is equal to the order of the torsion group T, of the Galois group of the maximal Abelian p-ramified pro-p-extension of k (Theorem 4.2), for all n \textgreater{}\textgreater{} e, where p^e is the exponent of U*/ adh(E) (in terms of local and global units of k). Then we establish analogs of Chevalley's formula using a family (Lambda\_i^n)\_{0$\le$i$\le$m\_n} of subgroups of k* containing E, in which any x is norm of an ideal of kn. This family is attached to the classical filtration of the p-class group of kn defining the algorithm of computation of its order in m\_n steps. From this, we prove (Theorem 6.1) that m\_n $\ge$ (lambda.n + mu.p^n + nu)/v\_p(T\_k) and that the condition m\_n = O(1) (i.e., lambda = mu=0) essentially depends on the P-adic valuations of the (x^(p-1)-1)/p, x in Lambda\_i^n, for P I p, so that Greenberg's conjecture is strongly related to "Fermat quotients" in k*. Heuristics and statistical analysis of these Fermat quotients (Sections 6, 7, 8) show that they follow natural probabilities, linked to T\_k whatever n, suggesting that lambda = mu=0 (Heuristics 7.1, 7.2, 7.3). This would imply that, for a proof of Greenberg's conjecture, some deep p-adic results (probably out of reach now), having some analogy with Leopoldt's conjecture, are necessary before referring to the sole algebraic Iwasawa theory.

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arXiv2017

On the order modulo p of an algebraic number (for p large enough)

Let K/Q be Galois, and let eta in K* whose conjugates are multiplicatively independent. For a prime p, unramified, prime to eta, let np be the residue degree of p and gp the number of P I p, then let o\_P(eta) and o\_p(eta) be the orders of eta modulo P and p, respectively.Using Frobenius automorphisms, we show that for all p\textgreater{}\textgreater{}0, some explicit divisors of p^(np)-1 cannot realize o\_P(eta) nor o\_p(eta), and we give a lower bound of o\_p(eta).Then we obtain that, for all p\textgreater{}\textgreater{}0 such that np \textgreater{}1, Prob(o\_p(eta)\textless{}p) $\le$ 1/p^(gp.(np-1)-epsilon)), where epsilon = O(1/(log\_2(p))); under the Borel--Cantelli heuristic, this leads to o\_p(eta)\textgreater{}p for all p\textgreater{}\textgreater{}0 such that gp.(np-1) $\ge$ 2, which covers the "limit" cases of cubic fields with np=3 and quartic fields with np=gp=2, but not the case of quadratic fields with np=2. In the quadratic case, the natural conjecture is, on the contrary, that o\_p(eta) \textless{} p for infinitely many inert p. Some computations are given with PARI programs.

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arXiv2015

No Riemann-hurwitz formula for the p-ranks of relative class groups

We disprove, by means of numerical examples, the existence of a Riemann-Hurwitz formula for the p-ranks of relative class groups in a p-ramified p-extension K/k of number fields of CM-type containing ?\_p. In the cyclic case of degree p, under some assumptions on the p-class group of k, we prove some properties of the Galois structure of the p-class group of K; but we have found, through numerical experimentation, that some theoretical group structures do not exist in this particular situation, and we justify this fact. Then we show, in this context, that Kida's formula on lambda invariants is valid for the p-ranks if and only if the p-class group of K is reduced to the group of ambiguous classes, which is of course not always the case.

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arXiv2015

On the Bertrandias-Payan module in a p-extension -- capitulation kernel

For a number field K and a prime number p we denote by BP\_K the compositum of the cyclic p-extensions of K embeddable in a cyclic p-extension of arbitrary large degree. Then BP\_K is p-ramified (= unramified outside p) and is a finite extension of the compositum K~ of the Z\_p-extensions of K.We study the transfer map j\_(L/K) (as a capitulation map of ideal classes) for the Bertrandias-Payan module bp\_K:=Gal(BP\_K/K~) in a p-extension L/K (p\textgreater{}2, assuming the Leopoldt conjecture). In the cyclic case of degree p, j\_(L/K) is injective except if L/K is kummerian, p-ramified, non globally cyclotomic but locally cyclotomic at p (Theorem 3.1). We then intend to characterize the condition \#bp\_K divides \#bp\_L^G (fixed points). So we study bp\_L^G when j\_(L/K) is not injective and show that it depends on the Galois group (over K~) of the maximal Abelian p-ramified pro-p-extension of K.We give complete proofs in an elementary way using ideal approach of global class field theory.

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arXiv2014

Etude probabiliste des p-quotients de Fermat

For a fixed integer a>1, we suggest that the probability of nullity of the p-Fermat quotient q(p,a) is much lower than 1/p for any arbitrary large prime number p. For this we use various heuristics, justified by means of numerical computations and analytical results, which may imply the finiteness of the q(p,a) equal to 0 and the existence of integers a such that q(p,a) is different from 0 for all p. However no proofs are obtained concerning these heuristics.

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arXiv2014

Notion de $θ$-régulateurs d'un nombre algébrique. Conjectures p-adiques

Let K/Q be a Galois extension of degree n, of Galois group G, and let $η\in K^\times$. For all large enough prime p, we define, by use of the Frobenius theorem on group determinants, the family $(Δ_p^θ(η) \in \F_p)_θ$ of local $θ$-regulators of $η$, indexed by the Qp-irreducible characters $θ$ of G. At each $Δ_p^θ(η)$ is associated a linear representation $L^θ\simeq δV_θ$, $0 \leq δ\leq φ(1)$, which characterizes some properties of $Δ_p^θ(η)$, including its nullity equivalent to $δ\geq 1$ (Th. 3.11). When $η\in \Q^\times$ and $θ= 1$, $Δ_p^1 (η)$ is the p-Fermat quotient of $η$. When $η$ is a "Minkowski unit", each $Δ_p^θ(η)$, $θ\ne 1$, gives the residue modulo p of the $θ$-component of $p^{1-n} Reg_p (K)$, where Reg(K) is the classical p-adic regulator of K. We suggest that the "probability" of ($Δ_p^θ(η) = 0$ and $L^θ\simeq δV_θ$) is $\frac{O(1)}{p^{f δ^2}}$, where f is a suitable residue degree of p. We conjecture that $p^{1-n} Reg_p(K)$, which measures the order of the p-torsion group in Abelian p-ramification over K, is for p large enough a p-adic unit except perhaps for a set of prime numbers of zero density. For these cases said "of minimal p-divisibility" (Def. 3.17), it remains possible, $η$ being then a "partial local pth power" at p, to propose, in connection with the ABC conjecture, a stronger conjecture leading to the same conclusion for all large enough p (Section 7). Some other conjectural aspects on the Fermat quotient are discussed. We precise and verify these properties through numerical studies on various fields and publish the corresponding "PARI" programs.

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arXiv2013

On the structure of the Galois group of the Abelian closure of a number field

Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group A, we study this property for arbitrary number fields. We show that such a property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p-adic obstructions coming from the global units. By restriction to the p-Sylow subgroups of A, we show that the corresponding study is related to a generalization of the classical notion of p-rational fields. However, we obtain some non-trivial information about the structure of the profinite group A, for every number field, by application of results published in our book on class field theory. This version corrects a technical error discovered by Peter Stevenhagen in a lemma of their first draft (arXiv:1209.6005) as well as in the previous versions of our paper reproducing this lemma; it will also be corrected in their final paper to appear in the proceedings volume of ANTS-X, San Diego 2012. This does not modify the nature of the results.

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arXiv2011

Some works of Furtwängler and Vandiver revisited and Fermat's last theorem

From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q not congruent to 1 mod p, q^(p-1) not congruent to 1 mod p^2, such that for Q dividing q in Q(exp(2 i pi /q-1)), we have Q^(1-c) = A^p . (alpha), with alpha congruent to 1 mod p^2 (where c is the complex conjugation), then Fermat's last theorem holds for p. More generally, the main purpose of the paper is to show that the existence of nontrivial solutions for SFLT implies some strong constraints on the arithmetic of the fields Q(exp(2 i pi /q-1)). From there, we give sufficient conditions of nonexistence that would require further investigations to lead to a proof of SFLT, and we formulate various conjectures. This text must be considered as a basic tool for future researchs (probably of analytic or geometric nature) - This second version includes some corrections in the English language, an in depth study of the case p=3 (especially Theorem 8), further details on some conjectures, and some minor mathematical improvements.

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arXiv2011

Analysis of the classical cyclotomic approach to fermat's last theorem

We give again the proof of several classical results concerning the cyclotomic approach to Fermat's last theorem using exclusively class field theory (essentially the reflection theorems), without any calculations. The fact that this is possible suggests a part of the logical inefficiency of the historical investigations. We analyze the significance of the numerous computations of the literature, to show how they are probably too local to get any proof of the theorem. However we use the derivation method of Eichler as a prerequisite for our purpose, a method which is also local but more effective. Then we propose some modest ways of study in a more diophantine context using radicals; this point of view would require further nonalgebraic investigations.

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