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

## TL;DR

This paper investigates the impact of genus theory on the p-rank epsilon-conjecture for p-class groups, proving the conjecture for degree p cyclic fields and analyzing potential obstructions in the proof.

## Contribution

It proves the p-rank epsilon-conjecture for degree p cyclic fields and analyzes the role of genus theory and exceptional classes as obstructions.

## Key findings

- Proved the p-rank epsilon-conjecture for degree p cyclic extensions.
- Identified the role of exceptional p-classes as potential obstructions.
- Compared epsilon-conjectures with p-adic Brauer-Siegel type conjectures.

## 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.