25 citations · 109 across the 31 of their papers we have counts for
31 papers
Chevalley-Herbrand formulas and Zp -extensions of a p-principal imaginary quadratic field
Georges Gras
Let be an imaginary quadratic field and let be a prime number, split in into . We assume that the -class group of i…
Exceptional case of the non semi-simple Real Abelian Main Conjecture
Georges Gras
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…
On the Real Abelian Main Conjecture in the non semi-simple case
Georges Gras
Let be a real cyclic extension of degree divisible by . We analyze the {\it statement} of the "Real Abelian Main Conjecture", for the -class group $\mathcal{H}…
New characterization of the norm of the fundamental unit of
Georges Gras
We give an elementary criterion for the norm of the fundamental unit of , square-free. More precisely, if , $…
Algebraic norm and capitulation of p-class groups in ramified cyclic p-extensions
Georges Gras
We examine the phenomenon of capitulation of the -class group of a real number field in totally ramified cyclic p-extensions of degree . Using an elementary…
The Chevalley-Herbrand formula and the real abelian Main Conjecture
Georges Gras
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", a…
Unlimited lists of fundamental units of quadratic fields -- Applications to some arithmetic properties
Georges Gras
We use the polynomials , , in an elementary process giving arbitrary large lists of {\it fundamental units} of quadratic fields of discriminant…
Invariant generalized ideal classes -- structure theorems for p-class groups in p-extensions
Georges Gras
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 de…
On the -stability of p-class groups along cyclic p-towers of a number field
Georges Gras
Let k be a number field, p2 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 to…
Weber's class number problem and -rationality in the cyclotomic -extension of
Georges Gras
Let be the th layer in the cyclotomic -extension . Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisaw…
Algorithmic complexity of Greenberg's conjecture
Georges Gras
Let be a totally real number field and a prime. We show that the ``complexity'' of Greenberg's conjecture () is of -adic nature governed (under Leopoldt's conje…
The p-rank -conjecture on class groups is true for towers of p-extensions
Georges Gras
Let p2 be a given prime number. We prove, for any number field kappa and any integer e1, the p-rank -conjecture, on the p-class groups Cl\_F, for the family F\_kappa^p…
Successive maxima of the non-genus part of class numbers
Georges Gras
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 , where …
Note on some p-invariants of Q(N^{1/p}) using reflection theorem
Georges Gras
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 and , by comparison with…
Practice of incomplete p-ramification over a number field -- History of abelian p-ramification
Georges Gras
The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and , is well known including numerical ex…
Genus theory and -conjectures on p-class groups
Georges Gras
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…
Test of Vandiver's conjecture with Gauss sums -- Heuristics
Georges Gras
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 i…
Annihilation of tor\_p(G\_S^ab) for real abelian extensions
Georges Gras
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…
Heuristics in direction of a p-adic Brauer--Siegel theorem
Georges Gras
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…
A program to test the p-rationality of any number field
Georges Gras
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 Progr…
The p-adic Kummer-Leopoldt constant - Normalized p-adic regulator
Georges Gras
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…
p-adic approach of Greenberg's conjecture (p-split totally real case)
Georges Gras
Let k be a totally real number field ant let k be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the I…
No Riemann-hurwitz formula for the p-ranks of relative class groups
Georges Gras
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…
On the order modulo p of an algebraic number (for p large enough)
Georges Gras
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…