Massey products and ideal class groups

The class group of a number field K measures how far its ring of integers is from having unique factorization into irreducible elements. It is the quotient of the group of all fractional ideals of K by the subgroups of principal fractional ideals. It is well known from class field theory that the ideal class group is also the Galois group of the maximal unramified abelian extension of K.

We shall consider the maximal cyclotomic extension of a finite algebraic number field and its two abelian extensions, the maximal abelian extension and the maximal abelian extension with certain restricted ramification. We shall investigate the structure of these Galois groups with the action of the cyclotomic Galois group.

Let k be an algebraic number field of finite degree or an algebraic function field of dimension 1 with finite constant field and let K/k be a finite, separably Galois extension field. We call an algebraic extension field L, not necessarily Galois nor necessarily of finite degree, of K an everywhere locally abelian extension, denoted in the following as EL-abelian extension, for brevity, of K over k, if and only if for every finite extension L' of K in L and every valuation v of L it holds that the completion field Lv' of L' with reference to v is a composition field of KV and an abelian extension Mv of kv in Lv', where we denote by kv and KV the closures of k and K in Lv', respectively. Let Q be an algebraic closure of K, BK/k the maximum separably EL-abelian extension of K/k in Q, CK/k the maximum separably central extension of K/k in Q (i.e. the maximum extension CK/k which is separably normal over k, contains K, and has the Galois group of CK/k/K in the center of the Galois group of CK/k/k), Ik and IK the groups of the ideles of k and K, Ak and AK the maximum separably abelian extensions of k and K in Q, respectively. Let DK/k=BK/k nCKIk. Then, the norm residue symbol 0Jk in Ak/k is, if restricted into NK/kIK, extensible into a homomorphism UK/k of NK/kIK into the Galois group G(DK/k/K). We shall study in the present article arithmetical meanings of the extensibility. In ?1, we shall define oUK/k precisely, in ?2, study the kernel and obtain a principal genus theorem (Theorem 2), from which will follow easily a certain generalized formulation of some fundamental theorems in the class field theory.1 Combining it with a known property2 of the quotient group of the connected component of the unit element of the idele class group by the natural image of the maximal compact subgroup of the connected component of the unit element of the idele group, we shall obtain, in ?3, our main result (Theorem 8), which concerns total norm residues in the principal idele group of k for K/k, and which is, the author thinks, an idele-theoretic reconstruction of a research of Scholz.3

Let p be a prime and K be a number field with non-trivial p-class group Cl p K . A crucial step in identifying the Galois group G p G K = of the maximal unramified pro-p extension of K is to determine its two-stage approximation 2 G p K = M , that is the second derived quotient G G M . The family 1 K of abelian type invariants of the p -class groups Cl p L of all unramified cyclic extensions L K of degree p is called the indexp abelianization data (IPAD) of K . It is able to specify a finite batch of contestants for the second p -class group M of K . In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD ( ) 1 2 , K K includes data on unramified abelian extensions L K of degree 2 p and enables sharper bounds for the order of M in the case ( ) Cl , , p K p p p , where current implementations of the p-group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order ( ) 2 K contains information on non-abelian unramified extensions L/K of degree p 2 , or even p 3 , and admits the identification of the p-class tower group G for various infinite series of quadratic fields ( ) K d = with ( ) Cl , p K p p possessing a p-class field tower of exact length 3 p K = as a striking novelty.

Construction of unipotent Galois extensions and Massey products

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∞).

On the torsion points of Drinfeld modules in abelian extensions

In this paper, various extensions of local fields are considered. For arbitrary finite extension K of the field of p-adic numbers, the maximum Abelian extension KAb/K and the corresponding Galois group can be described using the well-known Lubin–Tate theory. It is represented as a direct product of groups obtained using the maximum unramified extension of K and a fully ramified extension obtained using the roots of some endomorphisms of Lubin–Tate formal groups. We consider the so-called “generalized Lubin–Tate formal groups” and extensions obtained by adding the roots of their endomorphisms to the field under consideration. Using the fact that a correctly chosen generalized formal group coincides with the classical one over unramified finite extension Tm of degree m of field K, it was possible to obtain the Galois group of the extension (Tm)Ab/K. The main result of the work, is an explicit description of the Galois group of the extension (Kur)Ab/K, where Kur is the maximum unramified extension of K. Similar methods are also used to study ramified extensions of the field K.

In this paper the authors provide strong evidence for a conjecture of Chinburg in the context of abelian extensions N/K of quadratic imaginary fields K with group G. For abelian groups G, the conjecture specialises to the statement that Chinburg's third invariant Ω(N/K, 3)=0 ∈ Cl(Z[G]). We prove (Theorem 1) that if K has class number 1 (and some technical conditions on the conductor of N/K and its roots of unity hold) then Ω(N/K, 3) lies in the kernel group D(Z[G]). For a special class of genus extensions of prime power degree we prove (Theorem 2) the full conjecture, that Ω(N/K, 3)=0. The proof technique for Theorems 1 and 2 uses Kolyvagin's Euler systems of units applied to certain explicit groups of elliptic units, which essentially determines the difference in Galois structure between units and ideal class groups over the maximal order MG ∈ Q[G]. This is made precise by using Hecke factorisation and working in the factorisability defect class group of the category of finitely generated Z[G]-modules, which is isomorphic to Cl(MG). For Theorem 2, the fact that the class number of N is coprime to [N:K] in such genus extensions N/K allows one to lift these results to Cl(Z[G]), after using cohomological calculations to check that (essentially) an explicit representative for a Tate sequence defining Ω (N/K, 3) has the correct extension class (which is related to the fundamental class of N/K). 1991 Mathematics Subject Classification: 11R33.

Let K be a number field, let A be an abelian variety defined over K and let $$K_\infty /K$$ be a uniform p-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If $$K_\infty $$ contains sufficiently many p-power torsion points of A, then we can compare the ranks and the Iwasawa $$\mu $$ -invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple $$\mathbb {Z}_p$$ -extensions), we can also prove relations between suitably generalised Iwasawa $$\lambda $$ -invariants of the two types of Iwasawa modules. In the literature, two different kinds of generalised Iwasawa $$\lambda $$ -invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple $$\mathbb {Z}_p$$ -extensions.

Frey and Jarden asked if any abelian variety over a number field K has the infinite Mordell–Weil rank over the maximal abelian extension Kab. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve C over K such that #C(Kab) = ∞ and for any abelian variety of GL2-type with trivial character.

We recall the well known injectivity of the profinite Kummer map in the arithmetically relevant cases. There are at least two approaches towards injectivity. The abelian approach relies on the determination of the Kummer map for abelian varieties and their arithmetic, see Corollary 71, and also on the computation of the maximal abelian quotient extension \({\pi }_{{}_{1}}^{\mathrm{ab}}(X/k)\), see Proposition 69, which for later use in Sect. 13.5 we carefully revise also for smooth projective varieties of arbitrary dimension. The second approach is intrinsically anabelian and due to Mochizuki, see Theorem 76.In general, for a geometrically connected variety, the injectivity of the Kummer map (after arbitrary finite scalar extension) implies that the fundamental group must be large in the sense of Kollar, see Proposition 77, which imposes strong geometric constraints on possible higher dimensional anabelian varieties.

The primary purpose of this chapter is the analysis of the group Wq(R) in the case of an arithmetic Dedekind domain R. The characterization of the elements of Br(R)2 as the Brauer classes of Clifford algebras of quadratic spaces over R of rank 4 and trivial Arf invariant is an important step along the way. The quadratic spaces which it provides make it possible to describe Ker Arf by use of the total signature. In the special case where R is the ring of integers in a number field, the quadratic group Qu(R) is closely related to the ideal class group, and this connection together with the description of Ker Arf implies that Wq(R) ≅ Cl(R)2 ⊕ G, where C1(R) is the ideal class group of R and G is a free Abelian group of rank with r the number of real embeddings of the number field. An additional focus is the comparison of the number theory of Wq(R) with that of W(R) and the structure of the quotient W(R)/Wq(R).

In this paper, we study the algebraic structure of principal units in the tower of the maximal abelian extensions of local fields of characteristic zero and the corresponding Galois groups at each level. As an application, we show the finiteness result for the number of coverings with a given degree of the maximal abelian extension of a local field in characteristic zero. The number of p-coverings for \({\mathbb{Q}_p}\) is computed explicitly.

We extend and refine the theory of “organizing modules” of Mazur and Rubin to construct a canonical class of matrices that encodes a range of information about natural families of complexes in arithmetic. We then describe several concrete applications of this theory including the proof of new results on the explicit structures of Galois groups, ideal class groups, and wild kernels in higher algebraic K-theory and the formulation of a range of explicit conjectures concerning both the ranks and Galois structures of Selmer groups of abelian varieties over finite (nonabelian) Galois extensions of number fields.