On the Invariant Factors of Class Groups in Towers of Number Fields

Hajir, Farshid; Maire, Christian

doi:10.4153/cjm-2017-032-9

Cited by 7 publications

(7 citation statements)

References 32 publications

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“…Using Class Field Theory it is also possible to produce some free-pro-p extensions with some splitting phenomena, but we do not know if it is possible to construct free pro-p extensions in which infinitely many primes of the base field split completely. In fact this question is related to the work of Ihara [20] and the recent work of the authors [16].…”

Section: 3mentioning

confidence: 98%

Prime decomposition and the Iwasawa MU-invariant

Hajir¹,

Maire²

2018

Math. Proc. Camb. Phil. Soc.

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For Γ = Z p , Iwasawa was the first to construct Γ-extensions over number fields with arbitrarily large µ-invariants. In this work, we investigate other uniform pro-p groups which are realizable as Galois groups of towers of number fields with arbitrarily large µ-invariant. For instance, we prove that this is the case if p is a regular prime and Γ is a uniform pro-p group admitting a fixed-point-free automorphism of odd order dividing p − 1. Both in Iwasawa's work, and in the present one, the size of the µ-invariant appears to be intimately related to the existence of primes that split completely in the tower. A(F),where the limit is taken over all number fields F in L/K with respect the norm map. Then X is a Z p [[Γ]]-module and, thanks to a structure theorem (see section 1.1.2), one attaches a µ-invariant to X , generalizing the well-known µ-invariant introduced by Iwasawa in the classical case Γ ≃ Z p . Iwasawa showed that the size of the µ-invariant is related to the rate of growth of p-ranks of p-class groups in the tower. For the simplest Z p -extensions, i.e. the cyclotomic ones, he conjectured that µ = 0; this was verified for base fields which are abelian over Q by Ferrero and Washington [9] but remains an outstanding problem for more general base fields. Iwasawa initially suspected that his µ-invariant vanishes for all Z p -extensions, but later was the first to construct Z p -extensions with non-zero (indeed arbitrarily large) µ-invariants. It is natural to ask what other p-adic groups enjoy this property. Our present work leads us to the following conjecture:Conjecture 0.1. -Let Γ be a uniform pro-p group having a non-trivial fixed-point-free automorphism σ of order m co-prime to p (in particular if m = ℓ is prime, Γ is nilpotent). Then Γ has arithmetic realizations with arbitrarily large µ-invariant, i.e. for all n ≥ 0, there exists a number field K and an extension L/K with Galois group isomorphic to Γ such that µ L/K ≥ n.Our approach for realizing Γ as a Galois group is to make use of the existence of socalled p-rational fields. See below for the definition, but for now let us just say that the critical property of p-rational fields is that in terms of certain maximal p-extensions with restricted ramification, they behave especially well, almost as well as the base field of rational numbers. As we will show, Conjecture 0.1 can be reduced to finding a p-rational field with a fixed-point-free automorphism of order m co-prime to p. These considerations lead us to formulate the following conjecture about p-rational fields.Conjecture 0.2. -Given a prime p and an integer m ≥ 1 co-prime to p, there exist a totally imaginary field K 0 and a degree m cyclic extension K/K 0 such that K is p-rational.Although we will not need it, we believe K 0 in the conjecture may be taken to be imaginary quadratic; see Conjecture 4.16 below. Our key result is: Theorem 0.3. -Conjecture 0.2 for the pair (p, m) implies Conjecture 0.1 for any uniform pro-p group Γ having a fixed-point-free automorphism of order m.One knows tha...

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Section: 3mentioning

confidence: 98%

Prime decomposition and the Iwasawa MU-invariant

Hajir¹,

Maire²

2018

Math. Proc. Camb. Phil. Soc.

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“…For other approaches about p-ranks in towers as the degree grows, see for instance Hajir [23] and Hajir-Maire [24].…”

Section: Denote By {ℓmentioning

confidence: 99%

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

Gras¹

2020

Preprint

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Let p ≥ 2 be a given prime number. We prove, for any number field κ and any integer e ≥ 1, the p-rank ε-conjecture, on the p-class groups Cℓ F , for the family F p e κ of towers F/κ built as successive degree p cyclic extensions (without any other Galois conditions) such that F/κ be of degree p e , namely:, where D F is the absolute value of the discriminant (Theorem 3.6), and more generally. This Note generalizes the case of the family F p Q (Genus theory and ε-conjectures on p-class groups, J. Number Theory 207, 423-459 (2020)), whose techniques appear to be "universal" for all relative degree p cyclic extensions and use the Montgomery-Vaughan result on prime numbers. Then we prove, for F p e κ , the p-rank ε-conjecture on the cohomology groups H 2 (G F , Z p ) of Galois p-ramification theory over F (Theorem 4.3) and for some other classical finite p-invariants of F , as the Hilbert kernels and the logarithmic class groups.

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“…Remark 7.5. In [21], Hajir and Maire define, in the spirit of an algebraic p-adic Brauer-Siegel theorem, the logarithmic mean exponent of a finite p-group A ≃ r i=1 Z/p ai Z, by the formula M p (A) :=…”

Section: Examples Of Non-galois Totally Real Number Fieldsmentioning

confidence: 99%

Heuristics in direction of a p-adic Brauer--Siegel theorem

Gras

2018

Preprint

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Let p be a fixed prime number. Let K be a totally real number field of discriminant D K and let T K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We conjecture the existence of a constant Cp > 0 such that log( # T K ) ≤ Cp • log( √ D K ) when K varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a p-adic analogue, of the classical Brauer-Siegel Theorem, wearing here on the valuation of the residue at s = 1 (essentially equal to # T K ) of the p-adic ζ-function ζp(s) of K. We shall use a different definition that of Washington, given in the 1980's, and approach this question via the arithmetical study of T K since p-adic analysis seems to fail because of possible abundant "Siegel zeros" of ζp(s), contrary to the classical framework. We give extensive numerical verifications for quadratic and cubic fields (cyclic or not) and publish the PARI/GP programs directly usable by the reader for numerical improvements. Such a conjecture (if exact) reinforces our conjecture that any fixed number field K is p-rational (i.e., T K = 1) for all p ≫ 0. Abelian p-ramification -Main definitions and notationsLet K be a totally real number field of degree d, and let p ≥ 2 be a prime number fulfilling the Leopoldt conjecture in K. We denote by Cℓ K the p-class group of K (ordinary sense) and by E K the group of p-principal global units ε ≡ 1 (mod p|p p) of K.Let's recall from [9,12] the diagram of the so called abelian p-ramification theory, in which K c = KQ c is the cyclotomic Z p -extension of K (as compositum with that of Q), H K the p-Hilbert class field and H pr K the maximal abelian p-ramified (i.e., unramified outside p) pro-p-extension of K.is the group of p-principal units of the completion K p of K at p | p, where p is the maximal ideal of the ring of integers of K p .For any field k, let µ k be the group of roots of unity of k of p-power order. Then put W K := tor Zp U K = p|p µ Kp and W K := W K /µ K , where µ K = {1} or {±1}.

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On the Invariant Factors of Class Groups in Towers of Number Fields

Cited by 7 publications

References 32 publications

Prime decomposition and the Iwasawa MU-invariant

Prime decomposition and the Iwasawa MU-invariant

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

Heuristics in direction of a p-adic Brauer--Siegel theorem

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