Genus theory and ε-conjectures on p-class groups

Georges Gras (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

We suspect that the "genus part" of the class number of a number field K is an obstruction for an "easy proof" of a "strong ε-conjecture" for p-class groups: # (Cℓ_K ⊗ Z_p) ≪_(d,p,ε) (√ D_K)^ε for all K totally real of degree d. We analyze the weight of genus theory in this inequality by means of an infinite family of p-cyclic fields with many ramified primes, then prove the usual p-rank form of the ε-conjecture: # (Cℓ_K ⊗ F_p) ≪_(d,p,ε) (√ D_K)^ε , for d = p and the family of p-cyclic extensions of Q (Theorem 2.4), then sketch the case of arbitrary base fields. The possible obstruction, for the strong form, is the order of magnitude of the set of "exceptional" p-classes given by a well-known non-predictible algorithm. 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.

Mots clés: Class groups; epsilon-conjecture; Class field theory; p-ramification; Genus theory

Domaines: Théorie des nombres [math.NT]

Georges Gras. Genus theory and ε-conjectures on p-class groups. 2019. ⟨hal-02059441v1⟩