Ideal Class Group Research Articles

Infinite families of cyclotomic function fields with any prescribed class group rank

We prove the existence of the maximal real subfields of cyclotomic extensions over the rational function field k = F q ( T ) whose class groups can have arbitrarily large ℓ n -rank, where F q is the finite field of prime power order q . We prove this in a constructive way: we explicitly construct infinite families of the maximal real subfields k ( Λ ) + of cyclotomic function fields k ( Λ ) whose ideal class groups have arbitrary ℓ n -rank for n = 1, 2, and 3, where ℓ is a prime divisor of q − 1 . We also obtain a tower of cyclotomic function fields K i whose maximal real subfields have ideal class groups of ℓ n -ranks getting increased as the number of the finite places of k which are ramified in K i get increased for i ≥ 1 . Our main idea is to use the Kummer extensions over k which are subfields of k ( Λ ) + , where the infinite prime ∞ of k splits completely. In fact, we construct the maximal real subfields k ( Λ ) + of cyclotomic function fields whose class groups contain the class groups of our Kummer extensions over k . We demonstrate our results by presenting some examples calculated by MAGMA at the end.

Galois coinvariants of the unramified Iwasawa modules of multiple $$\mathbb {Z}_p$$-extensions

For a CM-field $K$ and an odd prime number $p$, let $\widetilde K'$ be a certain multiple $\mathbb{Z}_p$-extension of $K$. In this paper, we study several basic properties of the unramified Iwasawa module $X_{\widetilde K'}$ of $\widetilde K'$ as a $\mathbb{Z}_p[[{\rm Gal}(\widetilde K'/K)]]$-module. Our first main result is a description of the order of a Galois coinvariant of $X_{\widetilde K'}$ in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic $\mathbb{Z}_p$-extension of $K$ under a certain assumption on the splitting of primes above $p$. Second one is that if $K$ is an imaginary quadratic field and $p$ does not split in $K$, we give a necessary and sufficient condition for which $X_{\widetilde K}$ is $\mathbb{Z}_p[[{\rm Gal}(\widetilde K/K)]]$-cyclic under several assumptions on the Iwasawa $\lambda$-invariant and the ideal class group of $K$, where $\widetilde K$ is the $\mathbb{Z}_p^2$-extension of $K$.

English

Let n > 1 be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $$\mathbb{Q}\left( {\sqrt {{x^2} - 2{y^n}} } \right)$$ whose ideal class group has an element of order n. This family gives a counterexample to a conjecture by H. Wada (1970) on the structure of ideal class groups.

The 2-Sylow subgroup of $$K_2O_F$$ for certain quadratic number fields

In this paper, we apply Qin’s theorem for the 4-rank of $$K_2O_F$$ to establish the relation between the 4-rank of the ideal class group of $$F=\mathbb {Q}(\sqrt{d})$$ and the 4-rank of $$K_2O_F$$ provided that all odd prime factors of d are congruent to 1 mod 8. As an application, we give a concise and unified proof of two conjectures proposed by Conner and Hurrelbrink (Acta Arith 73:59–65, 1995).

On the compositum of orthogonal cyclic fields of the same odd prime degree

AbstractThe aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.

On minus quotients of ideal class groups of cyclotomic fields

Let [Formula: see text] be the minus quotient of the ideal class group of the [Formula: see text]th cyclotomic field. In this paper, first, we show that each finite abelian group appears as a subgroup of [Formula: see text] for some [Formula: see text]. Second, we show that, for all pairs of integers [Formula: see text] and [Formula: see text] with [Formula: see text], the kernel of the lifting map [Formula: see text] is contained in the [Formula: see text]-torsion [Formula: see text] of [Formula: see text]. Such an evaluation of the exponent is an individuality of cyclotomic fields.

Elliptic curves and lower bounds for class numbers

Ideal class pairings map the rational points of rank r≥1 elliptic curves E/Q to the ideal class groups CL(−D) of certain imaginary quadratic fields. These pairings imply thath(−D)≥12(c(E)−ε)(log⁡D)r2 for sufficiently large discriminants −D in certain families, where c(E) is a natural constant. These bounds are effective, and they offer improvements to known lower bounds for many discriminants.

On the ideal class group of the normal closure of [formula omitted

For a prime number p and an integer n we determine the Galois cohomology groups of the class group of the normal closure of Q(np) to a certain extent and use this information to prove a result about the group structure of the class group.

Class number divisibility for imaginary quadratic fields

In this note we revisit classic work of Soundararajan on class groups of imaginary quadratic fields. Let $$A,B,g \ge 3$$ be positive integers such that $$\gcd (A,B)$$ is square-free. We refine Soundararajan’s result to show that if $$4 \not \mid g$$ or if A and B satisfy certain conditions, then the number of negative square-free $$D \equiv A \pmod {B}$$ down to $$-X$$ such that the ideal class group of $${\mathbb {Q}} (\sqrt{D})$$ contains an element of order g is bounded below by $$X^{\frac{1}{2} + \epsilon (g) - \epsilon }$$, where the exponent is the same as in Soundararajan’s theorem. Combining this with a theorem of Frey, we give a lower bound for the number of quadratic twists of certain elliptic curves with p-Selmer group of rank at least 2, where $$p \in \{3,5,7\}$$.

Annihilators of the ideal class group of an imaginary cyclic field

For certain infinite family of imaginary cyclic fields we can obtain annihilators of the ideal class group by factoring nontrivial roots of modified Gauss sums. In this paper we investigate whether and when these annihilators live outside the Stickelberger ideal.

Imaginary quadratic number fields with class groups of small exponent

Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor of $8$. We compute all $D$ with $|D|\leq 3.1\cdot 10^{20}$ such that $E(D)\leq 8$.

A Classical Family of Elliptic Curves having Rank One and the $2$-Primary Part of their Tate-Shafarevich Group Non-Trivial

We study elliptic curves of the form x^3+y^3=2p and x^3+y^3=2p^2 where p is any odd prime satisfying p\equiv 2 \mod 9 or p\equiv 5 \mod 9 . We first show that the 3 -part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their 2 -Selmer group to the 2 -rank of the ideal class group of \mathbb{Q}(\sqrt[3]{p}) to obtain some examples of elliptic curves with rank one and non-trivial 2 -part of the Tate-Shafarevich group.

HEURISTICS FOR -CLASS TOWERS OF REAL QUADRATIC FIELDS

AbstractLet $p$ be an odd prime. For a number field $K$, we let $K_{\infty }$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\text{Gal}(K_{\infty }/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$-group occurs as the $p$-class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field $K$ as base. As before, the action of $\text{Gal}(K/\mathbb{Q})$ on the $p$-class tower group of $K$ plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for the groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In the appendix, we show how the probabilities introduced for finite $p$-groups can be extended in a consistent way to the infinite pro-$p$ groups which can arise in both the real and imaginary quadratic settings.

Higher Chern classes in Iwasawa theory

We begin a study of $m$th Chern classes and $m$th characteristic symbols for Iwasawa modules which are supported in codimension at least $m$. This extends the classical theory of characteristic ideals and their generators for Iwasawa modules which are torsion, i.e., supported in codimension at least $1$. We apply this to an Iwasawa module constructed from an inverse limit of $p$-parts of ideal class groups of abelian extensions of an imaginary quadratic field. When this module is pseudo-null, which is conjecturally always the case, we determine its second Chern class and show that it has a characteristic symbol given by the Steinberg symbol of two Katz $p$-adic $L$-functions.

Phase transitions on C∗-algebras arising from number fields and the generalized Furstenberg conjecture

We describe the low-temperature extremal KMS states of the semigroup C∗-algebra of the ax+b semigroup of algebraic integers in a number field in terms of ergodic invariant measures for certain groups of linear toral automorphisms. We classify different behaviours in terms of the ideal class group, the degree, and the unit rank of the field and we also obtain an explicit description of the primitive ideal space of the associated transformation group C∗-algebra for number fields of unit rank at least 2 that are not complex multiplication fields.

Computing the ideal class monoid of an order

There are well known algorithms to compute the class group of the maximal order $\mathcal{O}_K$ of a number field $K$ and the group of invertible ideal classes of a non-maximal order $R$. In this paper we explain how to compute also the isomorphism classes of non-invertible ideals of an order $R$ in a finite product of number fields $K$. In particular we also extend the above-mentioned algorithms to this more general setting. Moreover, we generalize a theorem of Latimer and MacDuffee providing a bijection between the conjugacy classes of integral matrices with given minimal and characteristic polynomials and the isomorphism classes of lattices in certain $\mathbb{Q}$-algebras, which under certain assumptions can be explicitly described in terms of ideal classes.

Elliptic surfaces over ℙ1 and large class groups of number fields

Given a non-isotrivial elliptic curve over [Formula: see text] with large Mordell–Weil rank, we explain how one can build, for suitable small primes [Formula: see text], infinitely many fields of degree [Formula: see text] whose ideal class group has a large [Formula: see text]-torsion subgroup. As an example, we show the existence of infinitely many cubic fields whose ideal class group contains a subgroup isomorphic to [Formula: see text].

Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.

On iterated extensions of number fields arising from quadratic polynomial maps

A post-critically finite rational map ϕ of prime degree p and a base point β yield a tower of finitely ramified iterated extensions of number fields, and sometimes provide an arboreal Galois representation with a p-adic Lie image. In this paper, we take ϕ to be the monic Chebyshev polynomial x2−2, and we examine the size of the 2-part of the ideal class group of extensions in the resulting tower. In some cases, we prove an analogue of Greenberg's conjecture from Iwasawa theory. A key tool is a general theorem on p-indivisibility of class numbers of relative cyclic extensions of degree p2.

Iwasawa theory for class groups of CM fields with p = 2

In this paper, we study Iwasawa theory for p=2. First of all, we show that the classical Iwasawa main conjecture holds true even for p=2 over a totally real field k assuming μ=0 and Leopoldt's conjecture. Using the Iwasawa main conjecture, we study the 2-component of the ideal class group of a CM-field K of finite degree as a Galois module. More precisely, for a CM-field K which is cyclic over the base field k, we determine the Fitting ideal of the minus quotient of the 2-component of the ideal class group. In particular, when k=Q and K/Q is imaginary and cyclic, we prove that the Fitting ideal coincides with the Stickelberger ideal.