Genus theory and $\epsilon$-conjectures on p-class groups

Abstract

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.