Ambiguous ideal classes and ambiguous ideals

Multivariate Lucas Polynomials and Ideal Classes in Quadratic Number Fields

In this work, by using Pauli matrices, we introduce four families of polynomials indexed over the positive integers. These polynomials have rational or imaginary rational coefficients. It turns out that two of these families are closely related to classical Lucas and Fibonacci polynomial sequences and hence to Lucas and Fibonacci numbers. We use one of these families to give a geometric interpretation of the 200 years old class number problems of Gauss, which is equivalent to the study of narrow ideal classes in real quadratic number fields.

The prime ideals in every class contain arbitrary large truncated classes

We prove that the prime ideals in every class of a number field contain arbitrary large truncated ideal classes.

Distribution of prime ideals of higher residue degree across ideal classes in the class groups

In this article we investigate the distribution of prime ideals of residue degree bigger than one across the ideal classes in the class group of a number field $L$. A criterion for the class group of $L$ being generated by the classes of prime ideals of residue degree $f>1$ is provided. Further, some consequences of this study on the solvability of norm equations for $L/\mathbb{Q}$ and on the problem of finding annihilators for relative extensions are discussed.

Algorithmic enumeration of ideal classes for quaternion orders

We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of two-sided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2.

二次型和 Genus Theory：与 2-descent 的联系及其在理想类的非平凡特化中的应用

Quadratic forms and Genus Theory : a link with 2-descent and an application to non-trivial specializations of ideal classes

本文将 Genus Theory 扩展至系数在任意 PID 上的扭曲二次型，证明其与超椭圆曲线上 2-descent 映射的关联，并应用于解决除子类特化问题，证明非平凡特化集合密度为 1。

Pulling back torsion line bundles to ideal classes

We prove results concerning the specialisation of torsion line bundles on a variety $V$ defined over $\mathbb{Q}$ to ideal classes of number fields. This gives a new general technique for constructing and counting number fields with large class group.

Two classes of number fields with a non-principal Euclidean ideal

This paper introduces two classes of totally real quartic number fields, one of biquadratic extensions and one of cyclic extensions, each of which has a non-principal Euclidean ideal. It generalizes techniques of Graves used to prove that the number field $\mathbb{Q}(\sqrt{2},\sqrt{35})$ has a non-principal Euclidean ideal.

Parametrization of ideal classes in rings associated to binary forms

We give a parametrization of the ideal classes of rings associated to integral binary forms by classes of tensors in $\mathbb Z^2\tensor \mathbb Z^n\tensor \mathbb Z^n$. This generalizes Bhargava's work on Higher Composition Laws, which gives such parametrizations in the cases $n=2,3$. We also obtain parametrizations of 2-torsion ideal classes by symmetric tensors. Further, we give versions of these theorems when $\mathbb Z$ is replaced by an arbitrary base scheme $S$, and geometric constructions of the modules from the tensors in the parametrization.

Congruences for modular forms mod 2 and quaternionic $S$-ideal classes

We prove many simultaneous congruences mod 2 for elliptic and Hilbert modular forms among forms with different Atkin--Lehner eigenvalues. The proofs involve the notion of quaternionic $S$-ideal classes and the distribution of Atkin--Lehner signs among newforms.

Computing ideal classes representatives in quaternion algebras

Let $K$ be a totally real number field and let $B$ be a totally definite quaternion algebra over $K$. In this article, given a set of representatives for ideal classes for a maximal order in $B$, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in $B$. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of level 30 in an algebra over the real quadratic field $\Q…

Explicit Bound for the Prime Ideal Theorem in Residue Classes

We give explicit numerical estimates for the generalized Chebyshev functions. Explicit results of this kind are useful for estimating of computational complexity of algorithms which generates special primes. Such primes are needed to construct an elliptic curve over prime field using complex multiplication method.

Representing ideal classes of ray class groups by product of prime ideals of small size

We prove that, for every modulus $\mathfrak{q}$, every class of the narrow ray class group $H_{\mathfrak{q}}(\mathbf{K})$ of an arbitrary number field $\mathbf{K}$ contains a product of three unramified prime ideals $\mathfrak{p}$ of degree one with $\mathfrak{N}\mathfrak{p}\le (t(\mathbf{K})\mathfrak{N}\mathfrak{q})^3$, where $t(\mathbf{K})$ is an explicit function of $\mathbf{K}$ described in the paper. To achieve this result, we first obtain a sharp explicit Brun-Titchmarsh Theorem for ray classes and then an equally explicit improved Brun-Titchmarsh Theorem for large subgroups of narrow ra…

Relative Ideal Classes of Arbitrary Order

We adapt a known technique for searching for ideal classes of arbitrary order and then apply it to three families of number fields. We show that a family of cyclic sextic number fields has infinitely many fields in it that contain a relative ideal class of order $r,$ where $r$ is a positive integer relatively prime to the degree of the extension. We then show that the same holds true for a family of cyclic quartic number fields. Though the technique is traditionally applied to Galois extensions, we show how it may be adapted to handle a family of non-Galois cubic number fields and prove that t…

étale 代数中序的分式理想的局部同构类

Local isomorphism classes of fractional ideals of orders in étale algebras

本文研究 étale 代数中序的分式理想局部同构类，借助剩余域上的线性代数对象完成分类，并提出计算类代表元的高效递归算法。

Visibility of ideal classes

Cremona, Mazur, and others have studied what they call visibility of elements of Shafarevich-Tate groups of elliptic curves. The analogue for an abelian number field $K$ is capitulation of ideal classes of $K$ in the minimal cyclotomic field containing $K$. We develop a new method to study capitulation and use it and classical methods to compute data with the hope of gaining insight into the elliptic curve case. For example, the numerical data for number fields suggests that visibility of nontrivial Shafarevich-Tate elements might be much more common for elliptic curves of positive rank than f…

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 …

主理想整环上矩阵相似类对应于理想类的代表元

Representatives of similarity classes of matrices over PIDs corresponding to ideal classes

本文证明当主理想整环上不可约特征多项式对应的阶为极大时，每个矩阵相似类都含有一个接近友矩阵的代表元，其证明结合了一次理想代表元的存在性与 Lenstra 关于极大阶中每个理想类含一次理想的未发表结果。

四元数代数中序的理想类

Ideal classes of orders in quaternion algebras

提出一种算法，计算全局域上四元数代数中任意序的所有右理想类代表元，包括不可逆情形，并推广至更一般代数。

Counting ideals in ray classes

Let $\mathbf{K}$ be a number field and $\mathfrak{q}$ an integral ideal in $\mathcal{O}_{\mathbf{K}}$. A result of Tatuzawa from 1973, computes the asymptotic (with an error term) for the number of ideals with norm at most $x$ in a class of the narrow ray class group of $\mathbf{K}$ modulo $\mathfrak{q}$. This result bounds the error term with a constant whose dependence on $\mathfrak{q}$ is explicit but dependence on $\mathbf{K}$ is not explicit. The aim of this paper is to prove this asymptotic with a fully explicit bound for the error term.

序的Cohen-Macaulay型、生成元与理想类

Cohen-Macaulay type of orders, generators and ideal classes

本文研究Dedekind整环上étale代数中序的Cohen-Macaulay型，给出其界与计算公式，并关联过度序的型与分式理想生成集大小，推广Gorenstein和Bass序的结果，最后分类乘子环为2型的理想类并应用于整矩阵共轭类与有限域上阿贝尔簇同构类的计算。

实数双二次域中的理想类的欧几里得算法：从亏元理论的角度

Euclidean Algorithms for Ideal Classes in Biquadratic fields: A Genus-Theoretic Perspective

本文从亏元理论角度研究实数双二次域中的欧几里得理想类，证明在特定条件下存在欧几里得理想，并展示相关双二次域集合的密度为零。

On equivariant characteristic ideals of real classes

Let $p$ be an odd prime, $F/{\Bbb Q}$ an abelian totally real number field, $F_\infty/F$ its cyclotomic ${\Bbb Z}_p$-extension, $G_\infty = Gal (F_\infty / {\Bbb Q}),$ ${\Bbb A} = {\Bbb Z}_p [[G_\infty]].$ We give an explicit description of the equivariant characteristic ideal of $H^2_{Iw} (F_\infty, {\Bbb Z}_p(m))$ over ${\Bbb A}$ for all odd $m \in {\Bbb Z}$ by applying M. Witte's formulation of an equivariant main conjecture (or "limit theorem") due to Burns and Greither. This could shed some light on Greenberg's conjecture on the vanishing of the $λ$-invariant of $F_\infty/F.$

Ennola三次域中的指定理想类与椭圆点

Prescribed Ideal Classes in Ennola Cubic Fields and Rational Points on Associated Elliptic Curves

本文构造了具有指定理想类阶的Ennola三次域，并通过椭圆曲线和Pell方程获得多个独立有理点及高秩纤维。

Relations in the 2-Class Group of Quadratic Number Fields

We construct a family of ideals representing ideal classes of order 2 in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.

Euclidean ideal classes in Galois number fields of odd prime degree

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least $1$ is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis…

具有循环类群的低次扩张

Low degree extensions with Cyclic class group

本文证明了阿贝尔四次域及某些类数为2的双二次域中存在欧几里得理想类，并探讨了特定三次和二次扩张中的相关性质。

Rings and ideals parametrized by binary n-ic forms

The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied quadratic rings and ideals associated to binary quadratic forms, and found that ideal classes of quadratic rings are exactly parametrized by equivalence classes of integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are parametrized by equivalence classes of integral binary cubic forms. Birch, Merriman, Nakagawa, Corso, Dvornicich, and Simon have all studied rings associated to binary forms of degree n for any n, but it has…

Ideal classes and Cappell-Shaneson homotopy 4-spheres

Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between trace $n$ Cappell-Shaneson matrices and trace $5-n$ Cappell-Shaneson matrices which was suggested by Gompf experimentally. Using this symmetry, we prove that Gompf conjecture for the trace $n$ case is equivalent to the trace $5-n$ case. We confirm Gompf conjecture for the specia…

An Analogue of Gauss Composition for Binary Cubic Forms

Over 200 years ago, Gauss discovered a composition law on the $SL_2({\mathbb Z})$-equivalence classes of primitive binary quadratic forms. Since then, bijections of classes of binary forms have been found with ideal class groups of quadratic rings. This paper uses one such bijection given by Bhargava, relating classes of projective binary cubic forms to the $3$-torsion of an ideal class group, to find an explicit form for a cubic analogue of Gaussian composition.