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

Abstract

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 p-class groups Cℓ F , for the family F p e κ of towers F/κ built as successive degree p cyclic extensions (without any other Galois conditions) such that F/κ be of degree p e , namely:, where D F is the absolute value of the discriminant (Theorem 3.6), and more generally. 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 all relative degree p cyclic extensions and use the Montgomery-Vaughan result on prime numbers. Then we prove, for F p e κ , 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.