# Genus theory and ε-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 ε-conjecture for p-class groups and, a fortiori, for a proof of the strong ε-conjecture'': #Cl_K <<_(d,p,ε) (√D_K)^ε, 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 form of the ε-conjecture: #(Cl_K/Cl_K^p) <<_(d,p,ε) (√D_K)^ε, for d=p and the family of degree p cyclic extensions of Q (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 ε-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.
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Cited literature [182 references]

https://hal.archives-ouvertes.fr/hal-02059441
Contributor : Georges Gras <>
Submitted on : Sunday, March 17, 2019 - 11:05:44 AM
Last modification on : Tuesday, July 9, 2019 - 1:15:37 AM
Long-term archiving on : Tuesday, June 18, 2019 - 12:35:08 PM

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• HAL Id : hal-02059441, version 2
• ARXIV : 1903.02922

### Citation

Georges Gras. Genus theory and ε-conjectures on p-class groups. 2019. ⟨hal-02059441v2⟩

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