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Journal of the Ramanujan Mathematical Society

Volume 37, Issue 2, June 2022  pp. 129–138.

The p-rank ε-conjecture on class groups is true for towers of p-extensions of a number field

Authors:  Georges Gras
Author institution:Villa la Gardette, 4 Chemin Château Gagnière, F-38520 Le Bourg d'Oisans

Summary:  Let p ≥ 2 be a given prime number. We prove, for any number field κ and any integer e ≥ 1, the p-rank ε-conjecture, on the class groups Cl{F}, for the family F pe κ of towers F/κ built as successive degree p cyclic extensions (without any other Galois conditions) such that F/κ be of degree pe, namely: # Cl{F} [p] 〈〈 {κ, p{e}, ε} (√D{F}){ε} for all F ∈ F{{p{e}}{κ}}, where D{F} is the discriminant (Theorem 3.6). This Note generalizes the case of the family F{p}{Q} (Genus theory and ε-conjectures on p-class groups, J. Number Theory 207, 423–459 (2020)), whose techniques appear to be “universal” for relative degree p cyclic extensions and use the Montgomery–Vaughan result on prime numbers. Then we prove, for F{p{e}}{κ}, similar p-rank ε-inequalities for 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 kernel and the Jaulent logarithmic class group.

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