The p-rank epsilon-conjecture on class groups is true for p-extensions
Résumé
We prove the p-rank epsilon-conjecture on the class groups Cl_F for the p-extensions F/Kappa of degree p^e and, more generally, for a tower of degree p cyclic fields: #(Cl_F[p]) <<_{Kappa,p^e,epsilon} (√D_F)^epsilon, where D_F is the discriminant and Kappa any base field (Theorem 3.5). This Note generalizes the case e=1 and Kappa=Q (Genus theory and epsilon-conjectures on p-class groups, J. Number Theory 207 (2020), 423--459), whose techniques appear to be ``universal'' for all relative p-cyclic-extensions. Then we prove the p-rank epsilon-conjecture for
the F/Kappa and the cohomology groups H^2(G_F,Z_p) of p-ramification theory (Theorem 4.3).
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