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 p-rank ε-conjecture for p-class groups and, a fortiori, for a proof of the "strong ε-conjecture": # (CℓK ⊗Zp) ≪ d,p,ε ( √ DK ) 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 ε-conjecture: # (CℓK ⊗Fp) ≪ d,p,ε ( √ DK ) ε , 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 ε-conjectures with some p-adic conjectures, of Brauer-Siegel type, about the torsion group TK 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.

The p-rank $ε$-conjecture on class groups is true for towers of p-extensions

“…This restriction seems essential because of the existence of very rare fields giving exceptional large invariants M as shown in [7,8,30] for class groups (or [19] for torsion groups T ). This is also justified, in the framework of p-class groups, by the Koymans-Pagano density results [28] as analyzed in [14] for F p Q ; indeed, in any relative degree p cyclic extension, the algorithm defining the filtration (M h ) h≥0 is a priori unbounded, giving possibly large # M contrary to the p-ranks (or the # M[p r ] as seen in Corollary 3.7 which allows to take r ≫ 0, but constant regarding the familly F p e κ ). All the previous results on p-rank ε-inequalities fall within the framework of "genus theory" at the prime p for p-extensions; the case of degree d number fields, when p ∤ d, is highly non-trivial.…”

“…We will perform an induction on the successive degree p cyclic extensions of a tower F ∈ F p e κ , the principles for such p-cyclic steps coming from [14] dealing with the family F p Q . The method involves using fixed points exact sequences, for the invariants considered, and the definition of "minimal relative discriminants" built by means of the Montgomery-Vaughan result on prime numbers.…”

“…; whence, an analogous framework as for the case of degree p cyclic number fields given in [14]. Note that, when κ = Q the primes ℓ, ramified in K/k, are not necessarily such that ℓ ≡ 1 (mod p) (e.g., p = 3,…”

“…To avoid any ambiguity, we shall write Cℓ F ⊗ F p , isomorphic to the "p-torsion group" also denoted Cℓ F [p] in the literature, only giving the p-rank of Cℓ F : rk p (Cℓ F ) := dim Fp (Cℓ F /Cℓ p F ). We refer to our paper [14] for an introduction with some history about the notion of ε-conjecture, initiated by Ellenberg-Venkatesh [11] by means of reflection theorems [20] and Arakelov class groups [37], then developed by many authors as Frei-Pierce-Turnage-Butterbaugh-Widmer-Wood [10,11,12,35,36,41,42], . .…”

Successive maxima of the non-genus part of class numbers

“…The sequence X := For short, one can say that the expression under consideration is locally maximum only when N K (K ∈ F s G,f ) is minimum, which is natural in some sense since, as we know, N K ≫ 0 gives a large genus number dividing H K (and a large discriminant D K ). The study performed in [14], for the degree p cyclic fields, shows that the genus part, and more precisely the p-part of H K , may be an obstruction regarding the ε-conjectures on the behavior of…”