Asymptotic growth patterns for class field towers

<!-- *** Custom HTML *** --> For an odd prime number $p$ and a number field $k$ which is an elementary abelian $p$-extension of the rationals, we prove the equivalence between the vanishing of all Iwasawa invariants of the cyclotomic $\mathbb{Z}_p$-extension of $k$ and an arithmetical condition described by the linking numbers of primes from a viewpoint of analogies between pro-$p$ Galois groups and link groups. A criterion of Greenberg's conjecture for $k$ of degree $p$ is also described in terms of linking matrices.

On Greenberg's generalized conjecture

Let X be a subgroup of a group Y. The \\emphinterval (X, Y) is the set of subgroups of Y that contain X including X and Y. By local class field theory the interval (N K/k K*, k*) contains a finite number of norm groups for any finite extension K of a 𝔭-adic number field k. In the present work we investigate the number of norm groups in the interval (N K/k K*, k*) for a given finite extension K/k of algebraic number fields. We prove that if K/k is an extension of prime degree, or of degree n such that the normal closure of K over k has the Galois group isomorphic to A n or S n , then the interval (N K/k K*, k*) contains only the obvious two norm groups. Also, the interval (N K/k K*, k*) contains a finite number of norm groups for any Galois extension of degree 4, and there are extensions with Galois groups order 8 for which the corresponding intervals contain a finite number of norm groups. The main theorem in our earlier work states that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois extension of even degree that is not a 2-extension, the so-called 2𝔫-\\emphextensions. In the present work we generalize the main theorem to non-Galois 2𝔫-extensions K/k, and determine some subintervals of (N K/k K*, k*) that contain infinitely many norm groups. We then use this theorem to prove that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois 2-extension K/k with the Galois group that either contains an element of order 8 or contains the quaternion group Q8 of order 8, or Q8 is a homomorphic image of the Galois group.

Let N ( d , G , X ) N(d, G, X) be the number of degree d d number fields K K with Galois group G G and whose discriminant D K D_K satisfies | D K | ≤ X |D_K| \le X . Under standard conjectures in diophantine geometry, we show that N ( 4 , A 4 , X ) ≪ ϵ X 2 / 3 + ϵ N(4, A_4, X) \ll _\epsilon X^{2/3+\epsilon } , and that there are ≪ ϵ N 3 + ϵ \ll _\epsilon N^{3+\epsilon } monic, quartic polynomials with integral coefficients of height ≤ N \le N whose Galois groups are smaller than S 4 S_4 , confirming a question of Gallagher. Unconditionally we have N ( 4 , A 4 , X ) ≪ ϵ X 5 / 6 + ϵ N(4, A_4, X) \ll _\epsilon X^{5/6 + \epsilon } , and that the 2 2 -class groups of almost all Abelian cubic fields k k have size ≪ ϵ D k 1 / 3 + ϵ \ll _\epsilon D_k^{1/3+\epsilon } . The proofs depend on counting integral points on elliptic fibrations.

We prove that the number of unimodular integral n × n matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group Sn is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field F . Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors. The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.

Let p be an odd prime number, k an imaginary abelian field containing a primitive p-th root of unity, and k∞/k the cyclotomic Zp-extension. Denote by L/k∞ the maximal unramified pro-p abelian extension, and by L' the maximal intermediate field of L/k∞ in which all prime divisors of k∞ over p split completely. Let N/k∞ (resp. N'/k∞) be the pro-p abelian extension generated by all p-power roots of all units (resp. p-units) of k∞. In the previous paper, we proved that the Zp-torsion subgroup of the odd part of the Galois group Gal(N ∩ L/k∞) is isomorphic, over the group ring Z p [Gal(k/Q)], to a certain standard subquotient of the even part of the ideal class group of k∞. In this paper, we prove that the same holds also for the Galois group Gal(N'∩L'/k∞).

Let p be an odd prime number and K a number field having a primitive p th root of unity \zeta_p . We prove that Nikshych's non group-theoretical Hopf algebra H_p , which is defined over \mathbb Q(\zeta_p) , admits a Hopf order over the ring of integers \mathcal O_K if and only if there is an ideal I of \mathcal O_K such that I^{2(p-1)} = (p) . This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over \mathcal O_K exists, it is unique and we describe it explicitly.

Let p be an odd prime number, n≥3 be a rational integer, and let f(X)=Xpn+aX+a∈ℤ[X] be an Eisenstein trinomial with respect to p. We prove that the Galois group G of f(X) over the field ℚ of rational numbers, is either the full symmetric group Spn, or AGL(1,pn)≤G≤AGL(n,p). We also show that G≃Spn, except possibly when |pnpn−1+ap(pn−1)pn−1| is a square, and for each prime divisor ℓ of a∕p, p divides the ℓ-adic valuation vℓ(a) of the integer a.

Let p be an odd prime number, and a be an integer divisible by p exactly once. We prove that the Galois group G of the trinomial X^{p^{2}}+aX+a over the field Q of rational number is either the full symmetric group S_{p^{2}} or G lies between AGL(1,p^{2}) and AGL(2,p)$. Furthermore, we establish conditions when G is S_{p^{2}}.

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL2(/3), an extension of SL2(/3) by /2, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.

Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] generates an extension of F that contains all ℓ-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes p in the ring of integers of F such that the order of (α mod p) is prime to ℓ. We compute this density in the general case for several classes of A, including elliptic curves and one-dimensional tori. For example, if F is a number field, A/F is an elliptic curve with surjective 2-adic representation and α ∈ A(F) with α ∉ 2A(F(A[4])), then the density of p with (α mod p) having odd order is 11/21.

A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G other than 1 leaves some letter fixed. It is proved in [20] (and sketched below) that if k is a number field such that SL(2, 5) and one other nonsolvable group Ŝ5 of order 240 are realizable as Galois groups over k, then every Frobenius group is realizable over k. It was also proved in [20] that there exists a quadratic (imaginary) field over which these two groups are realizable. In this paper we prove that they are realizable over the rationals Q, hence we ObtainTHEOREM 1. Every Frobenius group is realizable as the Galois group of an extension of the rational numbersQ.

Soient k un corps de nombres et 𝒞l(k) son groupe des classes. Une extension de k à groupe de Galois isomorphe au groupe alterné A 4 est dite alternée. Soit E/k une extension cyclique de degré 3. On calcule la classe de Steinitz, dans 𝒞l(k), de toute extension alternée contenant E. Sous l’hypothèse que le nombre des classes de k est impair, on détermine l’ensemble de telles classes et on montre que c’est un sous-groupe de 𝒞l(k) lorsque l’anneau des entiers de E est libre sur celui de k ou 3 ne divise pas l’ordre de 𝒞l(k). Ensuite, on montre que l’ensemble des éléments de 𝒞l(k) qui sont réalisables par des classes de Steinitz d’extensions alternées (resp. alternées et modérées) est le groupe 𝒞l(k) tout entier.

Let K be a number field of degree n over ℚ and let d, h, and R be the absolute values of the discriminant, class number, and regulator of K, respectively. It is known that if K contains no quadratic subfield, then $$ h\;R\gg \frac{d^{1/2}}{ \log d}, $$ where the implied constant depends only on n. In Theorem 1, this lower estimate is improved for pure cubic fields. Consider the family $$ {\mathcal{K}}_n $$ , where K ∈ $$ {\mathcal{K}}_n $$ if K is a totally real number field of degree n whose normal closure has the symmetric group S n as its Galois group. In Theorem 2, it is proved that for a fixed n ≥ 2, there are infinitely many K ∈ $$ {\mathcal{K}}_n $$ with $$ h\gg {d}^{1/2}{\left( \log \log d\right)}^{n-1}/{\left( \log d\right)}^n, $$ where the implied constant depends only on n. This somewhat improves the analogous result h ≫ d 1/2/(log d) n of W. Duke [MR 1966783 (2004g:11103)].

Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field which has good reduction at every prime above $p$. Under suitable assumptions, we prove that the $\eta$-eigenspace and the $\bar{\eta}$-eigenspace of mixed signed Selmer group of the elliptic curve are pseudo-isomorphic. As a corollary, we show that the $\eta$-eigenspace is trivial if and only if the $\bar{\eta}$-eigenspace is trivial. Our results can be thought as a reflection principle which relate an Iwasawa module in a given eigenspace with another Iwasawa module in a "reflected" eigenspace.