Computing Arakelov class groups

On class groups of random number fields

The main aim of the present paper is to disprove the Cohen--Lenstra--Martinet heuristics in two different ways and to offer possible corrections. We also recast the heuristics in terms of Arakelov class groups, giving an explanation for the probability weights appearing in the general form of the heuristics. We conclude by proposing a rigorously formulated Cohen--Lenstra--Martinet conjecture.

Explicit regulator maps on polylogarithmic motivic complexes

We define a regulator map from the weight n polylogarithmic motivic complex to the weight n Deligne complex of an algebraic variety X. The regulator map is constructed explicitly via the classical polylogarithms with some funny combinations of Bernoulli numbers as coefficients. This leads to conjectures on special values of L-functions at s=n which reduce to Zagier's conjecture when X is of dimension zero over Q. The cone of this map, shifted by one, should be called the Arakelov motivic complex of X. Its last cohomology group is identified with a group of codimension n Arakelov cycles on X.

Computing Hilbert Class Polynomials

We present and analyze two algorithms for computing the Hilbert class polynomial $H_D$ . The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for computing $H_D$, and we show that all methods have comparable run times.

计算实二次域的p-类群结构：一种新方法

Computing $p$-Class Group Structure in Real Quadratic Fields: A New Approach

本文提出了一种新方法，用于计算实二次域的p-类群结构，扩展了希尔伯特类域理论并建立了类域理论与理想类之间的联系。

Computing polynomials of the Ramanujan $\mathbf{t_n}$ class invariants

We compute the minimal polynomials of the Ramanujan values $t_n$, where $n\equiv 11 \mod 24$, using Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$, and have much smaller coefficients than the Hilbert polynomials.

在Magma中计算类群和单位群

Computing class groups and unit groups in Magma

本文介绍Magma V2.29中数域类群与单位群的计算实现，基于因子基、关系收集及解析类数评估，并区分不同算法模式的行为。

Computing automorphic forms on Shimura curves over fields with arbitrary class number

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with two examples which illustrate the effectiveness of our algorithms.

计算任意同余子群的经典模形式

Computing Classical Modular Forms for Arbitrary Congruence Subgroups

本文提出了一种高效算法，用于计算任意同余子群的模形式q展开式，并讨论了相关理论和实际应用。

Computing Class Groups of Function Fields Using Stark Units

Let $k$ be a fixed finite geometric extension of the rational function field $\mathbb{F}_q(t)$. Let $F/k$ be a finite abelian extension such that there is an $\Fq$-rational place $\infty$ in $k$ which splits in $F/k$ and let $\mathcal{O}_F$ denote the integral closure in $F$ of the ring of functions in $k$ that are regular outside $\infty$. We describe algorithms for computing the divisor class number and in certain cases for computing the structure of the divisor class group and discrete logarithms between Galois conjugate divisors in the divisor class group of $F$. The algorithms are efficie…

通过广义范数关系的归纳法计算类群

Computing class groups by induction with generalised norm relations

本文提出群代数中的广义范数关系，利用其推导子域间S-单位群关系，并据此设计算法，将类群计算归约至低次域，成功计算若干大数域的类群。

关于$\overline{{\rm Spec}\,\mathbb Z}$的Jacobian

On the Jacobian of $\overline{{{\rm Spec}\,\mathbb Z}}$

本文提出将$\overline{{\rm Spec}\,\mathbb Z}$的Jacobian推广到adele上下文，通过张量积结构和rigidifying数据扩展Picard群。

Hermitian vector bundles and extension groups on arithmetic schemes. II. The arithmetic Atiyah extension

In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element -- the arithmetic Atiyah class -- in a suitable arithmetic extension group. If $\overline{E}$ is a hermitian vector bundle on an arithmetic scheme $X$, its arithmetic Atiyah class is an obstruction to the algebraicity of the unitary connection on the vector bundle $E_\C$ over the complex manifold $X(\C)$ that is compatible with its holomorphic structure. We develop basic properties of the ar…

Computing class polynomials for abelian surfaces

We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and their Galois closures, we pursue an approach due to Dupont for evaluating $θ$- constants in quasi-linear time using Newton iterations on the Borchardt mean. We report on experiments with our implementation and present an example with class number 17608.

计算扭曲L-函数的经典模形式的欧拉乘积与系数

Computing Euler products and coefficients of classical modular forms for twisted L-functions

提出一种快速计算经典模形式系数及欧拉乘积的算法，并在大导子三重积L-函数上验证其效率。

计算有限域上代数曲线的类群和亏格

Computing class groups and gonalities of algebraic curves over finite fields

提出实用算法计算有限域上曲线的除子类群和亏格，通过预计算幂级数展开实现大量Riemann-Roch空间的高效分摊计算，速度提升数个数量级。

Computing Néron-Severi groups and cycle class groups

Assuming the Tate conjecture and the computability of étale cohomology with finite coefficients, we give an algorithm that computes the Néron-Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension p cycles for any p.

New techniques for computing the ideal class group and a system of fundamental units in number fields

We describe a new algorithm for computing the ideal class group, the regulator and a system of fundamental units in number fields under the generalized Riemann hypothesis. We use sieving techniques adapted from the number field sieve algorithm to derive relations between elements of the ideal class group, and $p$-adic approximations to manage the loss of precision during the computation of units. This new algorithm is particularily efficient for number fields of small degree for which a speed-up of an order of magnitude is achieved with respect to the standard methods.

Computing Hilbert modular forms over fields with nontrivial class group

In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over $\Q(\sqrt{10})$ and $\Q(\sqrt{85})$ and their Hilbert class fields, we present some new instances of the conjectural Eichler-Shimura construction for totally real fields, and in particular find new examples of modular abelian varieties with everywhere good reduction.