Ha Thanh Nguyen Tran
The function $h^0$ for a number field is analogous to the dimension of the Riemann-Roch spaces at divisors on an algebraic curve. We provide a method to compute this function for number fields with unit group of rank at most 2, even with large discriminant. This method is based on using LLL-reduced bases, the "jump algorithm" and Poisson summation formula.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let $C \geq 1$. Inspired by the LLL-algorithm, we define strongly $C$-reduced divisors of a number field $F$ which are generalized from the concept of reduced Arakelov divisors. Moreover, we prove that strongly $C$-reduced Arakelov divisors still retain outstanding properties of the reduced ones: they form a finite, regularly distributed set in the Arakelov class group and the oriented Arakelov class group of $F$.
We generalize the concept of reduced Arakelov divisors and define $C$-reduced divisors for a given number $C \geq 1$. These $C$-reduced divisors have remarkable properties which are similar to the properties of reduced ones. In this paper, we describe an algorithm to test whether an Arakelov divisor of a real quadratic field $F$ is $C$-reduced in time polynomial in $\log|Δ_F|$ with $Δ_F$ the discriminant of $F$. Moreover, we give an example of a cubic field for which our algorithm does not work.
Computing Arakelov class groups
用于计算代数数域的整数环的类群和单位群的Shanks的基础结构算法和Buchmann的算法最自然地被看作是在Arakelov类群内的计算。本文讨论了Arakelov类群和既约Arakelov因子集的基本性质。作为一个应用程序，我们描述Buchmann的算法在这种情况下。
In this article, we consider an analogue of Arakelov theory of arithmetic surfaces over a trivially valued field. In particular, we establish an arithmetic Hilbert-Samuel theorem and studies the effectivity up to R-linear equivalence of pseudoeffective metrised R-divisors.
We use Arakelov theory to define a height on divisors of degree zero on a hyperelliptic curve over a global field, and show that this height has computably bounded difference from the Néron-Tate height of the corresponding point on the Jacobian. We give an algorithm to compute the set of points of bounded height with respect to this new height. This provides an `in principle' solution to the problem of determining the sets of points of bounded Néron-Tate heights on the Jacobian. We give a worked example of how to compute the bound over a global function field for several curves, of genera up t…
Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\overline{D}$ be an adelic Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $ζ_{\mathrm{ess}}(\overline{D})$ of $\overline{D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $ζ_{\mathrm{ess}}(\overline{D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom--Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion …
Existential closedness of $\overline{\mathbb{Q}}$ as a globally valued field via Arakelov geometry
本研究利用算术体积函数可微性与算术Bertini型定理分类闭点高度逼近问题，证明了$\overline{\mathbb{Q}}$作为全局赋值域（GVF）的存在封闭性，引入GVF泛函并给出Zhang不等式等号成立的新判据。
The function $h^0$ for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. In this paper, we prove the conjecture of van der Geer and Schoof about the maximality of $h^0$ at the trivial Arakelov divisor for quadratic extensions of complex quadratic fields.
The size function for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was proved for many number fields with unit groups of rank one. Our research confirms that the conjecture also holds for cyclic cubic fields, which have unit groups of rank two.
Integrals of automorphic Green's functions associated to Heegner divisors
本文给出了正交群$\Orth(2,p)$的算术导数上Heegner因子的次数的显式公式，以及与Heegner因子有关的自守格林函数的积分的显式公式。后者的数量是重要的研究算术程度的Heegner因子的背景下Arakelov几何。特别是，我们得到一个不同的证明和推广的Kudla的结果有关这些数量的傅里叶系数的某些非全纯爱森斯坦系列的重量$1+p/2$的亚群$\Mp_2(\Z)$。
In this paper, we investigate the size function $h^0$ for number fields. This size function is analogous to the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that $h^0$ attains its maximum at the trivial class of Arakelov divisors. This conjecture was proved for all number fields with unit group of rank 0 and 1, and also for cyclic cubic fields which have unit group of rank two. We prove the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two.
Computation of Admissible Arakelov-Green Functions on Metrized Graphs
本文提出一种算法，用于计算度量图上的可积阿拉克-格林函数，扩展了Cinkir的公式，并通过实例展示了算法的实现。
An algorithm is presented to compute isolated values of the divisor summatory function in O(n^(1/3)) time and O (log n) space. The algorithm is elementary and uses a geometric approach of successive approximation combined with coordinate transformation.
Suppose $\mathcal{E} \to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\overline{\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $\mathbb{R}$-divisors $\overline{D}_X$ on the base curve $B$ over a number field, for each $X \in E(k)\otimes \mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $\overline{D}_X\cdot \overline{D}_Y$, as a biquadratic form on $E(k)\otimes \mathbb{R}$. As a consequence, we have the follo…
Computing the divisor summatory function in $N^{9/28+o(1)}$
提出一种解析算法，通过 Voronoï 求和加速指数和计算，在 $N^{9/28+o(1)}$ 期望时间内计算除数求和函数孤立值，优于现有 $O(N^{1/3})$ 方法。
While studying a characteristic number of manifolds we noticed that the calculation was simply computing a multiple of a multinomial coefficient. We were, at the time, interested in computing the greatest common divisor of these characteristic numbers, and from that came this fun and interesting number theory problem. In this short paper, I will give the answer to this problem, which gives the greatest common divisor of certain multinomial coefficients.
We prove a Riemann-Roch theorem of an entirely novel nature for divisors on the Arakelov compactification of the algebraic spectrum of the integers. This result relies on the introduction of three key concepts: the cohomologies (attached to a divisor), their integer dimension, and Serre duality. These notions directly extend their classical counterparts for function fields. The Riemann-Roch formula equates the (integer valued) Euler characteristic of a divisor with a slight modification of the traditional expression in terms of the sum of the degree of the divisor and the logarithm of 2. Both …
Computing the Greatest Common Divisor of Binomial Coefficients $\binom{mn}{mk}$
本文研究了二项式系数 $\binom{mn}{mk}$ 的最大公约数，给出了 $m=2$ 时该 GCD 的完整刻画，并展示了在特定条件下高效计算的方法。
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…
Computation of 2-groups of narrow logarithmic divisor classes of number fields
给出了数域F的0次窄对数除子类2-群的一个计算算法。作为应用，我们在某些情况下计算野生核WK 2（F）的2-秩。
The Riemann zeta function and exact exponential sum identities of divisor functions
本文证明了一个计算两个移位黎曼zeta函数乘积的显式积分公式，由此推导出涉及除数函数的无穷族精确指数和恒等式，并提出了无需算术技巧的矩阵求逆计算除数函数的推测性方法。
We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.
We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations can be done efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps and Kummer maps; and finding a basis for the $l$-torsion of the Pic…
Riemann-Roch for the ring $\mathbb Z$
本文通过改用绝对基S而非S[±1]，改进了Spec Z的Riemann-Roch公式，将Arakelov除子的整值Euler示性数等同于除子度数（以2为底对数）加1，从而支持把整数环Z视为S上的一元多项式环S[X]，且1+1=X+X^2。
Computing bounded solutions to linear Diophantine equations with the sum of divisors
本文提出了一种高效方法，用于计算线性丢番图方程的有界解，并应用于发现超完美数等领域的新型解。
A problem on the largest divisor $d$ of $N$ with $d\leq \sqrt{N}$
针对给定数 N，提出避免因式分解的随机化算法，计算满足条件的除数对 (r,f)，并给出总数公式及比值估计，同时研究 r>N 的情形。
We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight 2-n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of f. Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the cen…
Weighted Erdős-Kac Theorems via Computing Moments
本文推广了计算矩的方法，研究了一大类非负乘性函数加权的加性函数的分布，建立了矩的一致渐近公式，并以此推广了Delange-Halberstam定理，获得了Kubilius-Shapiro定理的加权类似物。
Computation of 2-groups of positive classes of exceptional number fields
本文给出了数域F中具有例外并元位置的正因子类的2-群的一个计算算法。作为应用，我们计算了这类数域的野生核WK ~ 2（F）在K ~ 2（F）中的2-秩。
If \(A \) is a set of natural numbers containing \(0 \), then there is a unique nonempty "reciprocal" set \(B \) of natural numbers (containing \(0 \)) such that every positive integer can be written in the form \(a + b \), where \(a \in A \) and \(b \in B \), in an even number of ways. Furthermore, the generating functions for \(A \) and \(B \) over \(\FF_2 \) are reciprocals in \(\FF_2 [[q]] \). We consider the reciprocal set \(B \) for the set \(A \) containing \(0 \) and all integers such that \(σ(n) \) is odd, where \(σ(n) \) is the sum of all the positive divisors of \(n \). This problem…
Computing sieve integrals using LattE, and the density of integers with a localized divisor
提出基于LattE软件的多项式时间算法，严格计算凸多面体上的筛法积分，并应用于计算具有局部化因子的整数的自然密度。
Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms
本文通过计算 Euler-Kronecker 常数得到除数和函数计数函数渐近展开的二阶项，并确定 Ramanujan 逼近优于 Landau 逼近的条件，进而研究六个尖形式 Fourier 系数的不可整除性。
By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most 2. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić--Zhai and Blomer.
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。
Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb{Z}$ with compact support at infinity gives, up to sign, the leading term of the zeta-function $ζ_X(s)$ at $s=0$. In this paper we consider the category of sheaves $\bar{X}_L$ on this conjectural site for $X=Spec(\mathcal{O}_F)$ the spectrum of a number ring. We show that $\bar{X}_L$ has, under natural topological assumptions, a well defined fundamental group whose abelianization is isomorphic, as a topologic…
Graphs of Hecke operators in mixed ramification
本文研究了在分歧$G$-束模空间上Hecke算子，引入了$\mathcal H$-分歧概念，证明在弱正则条件下，深cusp处的Hecke算子作用在复杂分歧时类似于简单分歧，从而减少研究案例数量。
On a two-variable zeta function for number fields
本文研究了van der Geer和Schoof在代数数域K上引入的两个复变数（w，s）的zeta函数，它是基于用Arakelov因子模拟数域的Riemann-Roch定理.我们主要考虑有理域Q的情况，其中对于w = 1，我们恢复了黎曼ζ函数，并在无穷远处添加了因子。黎曼xi函数的类似物解析地继续为两个复变数的整函数，并且满足保持w固定并将s发送到w-s的函数方程。因此，对于实数w，“临界线”为Re（s）= w/2。对于固定的非负实数w，零点被限制在s中的一条垂直带上，并且具有与zeta零点相同的渐近性。对于负实数w，函数在临界线上是正实数，所以在那里没有零。这一现象是与一个正卷积半群的无限可分概率分布，和Khintchine典型的措施，这个家庭是明确确定的。
We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power sums (e.g., cardinality) or extrema, without direct enumeration of the inverses. We illustrate our algorithm with Euler's totient function $φ(\cdot)$ and the $k$-th power sum of divisors $σ_k(\cdot)$. For example, we can establish that the number of solutions to $σ_1(x) = 10^{1000}$ is 15,512,215,160,488,452,125,793,724,066,873,737,608,071,476, while it is …
Recursively divisible numbers
本文引入递归除数函数$κ_x(n)$并给出几何解释，推导其性质，证明$κ_0(n)$与有序分解数的关系，并通过计算高递归可除性数字恢复已知及提出新数字。
Computing transcendence and linear relations of 1-periods
本文提出一个算法，计算有限个 1-周期间的所有代数系数线性关系，从而判定其超越性与相等性，解决 Kontsevich-Zagier 问题，并推广到自治一阶微分方程的分类。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。