Srinivas Kotyada, Subramani Muthukrishnan
Let $K$ be a cyclic cubic field and $\mathcal{O}_K$ be its ring of integers. In this note we prove that all cyclic cubic number fields with conductors in the interval $ [73, 11971]$ and with class number one are Euclidean.
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In this paper, we investigate the common index divisors of cyclic cubic fields. Let $a,b,c,d$ and $k$ are integers, we then solve the following Thue cubic equations:: \[ax^3+bx^2y+cxy^2+dy^3= k\ \] when $a,bc+d$ are odd and $3$ doesn't divide $v_2(k)$.
We describe a procedure for generating families of cyclic cubic fields with explicit fundamental units. This method generates all known families and gives new ones.
This is the third part of a series of papers discussing the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=39$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.
By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular.
We provide an equivalent condition for the monogenity of the ring of integers of any cyclic cubic field. We show that if a cyclic cubic field is monogenic then it is a simplest cubic field $K_t$ which is the splitting field of a Shanks cubic polynomial $f_t(x):=x^3-tx^2-(t + 3)x-1$ with $t \in \mathbb Z$. Moreover we give an equivalent condition for when $K_t$ is monogenic, which is explicitly written in terms of $t$.
We compute the $p$-adic regulator of cyclic cubic extensions of $\mathbb Q$ with discriminant up to $10^{16}$ for $3<p<100$, and observe the distribution of the $p$-adic valuation of the regulators. We find that for almost all primes, the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions. Based on this random matrix model, a conjecture on the distribution of the valuations of $p$-adic regulators of cyclic cubic fields is stated.
The determination of norm-Euclidean cyclic cubic fields
该研究针对范数欧几里得循环三次域的确定问题，在广义黎曼假设下已知其数量为13，通过建立三次非剩余显式界、改进计算技术，成功完全刻画了所有范数欧几里得循环三次域。
Let $E$ be an elliptic defined over a number field $K$. Then its Mordell-Weil group $E(K)$ is finitely generated: $E(K)\cong E(K)_{tor}\times\mathbb{Z}^r$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=169,143,91,65,77$ or $55$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.
Merel's result on the strong uniform boundedness conjecture made it meaningful to classify the torsion part of the Mordell-Weil groups of all elliptic curves defined over number fields of fixed degree $d$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=49,40,25$ or $22$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.
Monogenic Cyclic Cubic Trinomials
本文研究整数环上的单项循环三次三项式，证明存在两个无限族非等价形式 $x^3+Ax+B$，并确定形如 $x^3+Ax^2+B$ 的仅有四个，均等价于 $x^3-3x+1$。
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.
We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively.
Well-Rounded ideal lattices of cyclic cubic and quartic fields
本文研究了循环三次和四次域具有良圆理想格的判据，证明了循环三次域必含良圆理想，构造了循环四次域的良圆理想族，并给出了特定素理想为良圆的充要条件及奇素数分解。
The Euclidean Algorithm in Cubic Number Fields
在本文中，我们提出了计算立方数域的欧氏极小值的算法;特别地，我们能够找到所有判别式为-999 < d < 10000的范数欧氏立方数域。
Elliptic curves with a point of order 13 defined over cyclic cubic fields
该研究证明了有理数域的三次伽罗瓦扩张上本质上唯一存在具有13阶有理点且不定义在有理数域上的椭圆曲线，为数论中椭圆曲线的定义域与扭点结构研究提供了结论。
Conditionally on the Generalized Riemann Hypothesis (GRH), we prove the following results: (1) a cyclic number field of degree $5$ is norm-Euclidean if and only if $Δ=11^4,31^4,41^4$; (2) a cyclic number field of degree $7$ is norm-Euclidean if and only if $Δ=29^6,43^6$; (3) there are no norm-Euclidean cyclic number fields of degrees $19$, $31$, $37$, $43$, $47$, $59$, $67$, $71$, $73$, $79$, $97$. Our proofs contain a large computational component, including the calculation of the Euclidean minimum in some cases; the correctness of these calculations does not depend upon the GRH. Finally, we …
Group theory of cyclic cubic number fields
本文通过四类三次剩余图研究共享同一导子的循环三次数域四元组和八元组，揭示了其3-类群结构及阶为6561的闭Andozhskii-Tsvetkov群，并利用阿贝尔型不变量进行识别。
Torsion of elliptic curves over cyclic cubic fields
本研究明确了循环三次域、非循环全实三次域与复三次域上椭圆曲线的全部可能挠群，填补了相关数论研究的空白。
Elliptic curves related to cyclic cubic extensions
本文的目的是研究定义在数域$F$上的一类椭圆曲线族$\{\mathscr{X}_H\}_H$，该数域$F$由与循环三次扩张$K/F$相关联的三次曲面$\mathscr{X}/F$的超平面截面产生.我们证明了每个$\mathscr{X}_H$在$F$上都有一个3-等距的$ϕ$，并且对偶Selmer群$S^{(\hatϕ)}(\hat{\mathscr{X}_H}/F)$被一类附着在$K/F$上的单位/类群所包围.通过椭圆曲线$\mathscr{X}_H$上的有理函数证明了这一点，并具有良好的性质。我们还证明了Shafarevich-Tate群$\text{\cyr X} (\hat{\mathscr{X}_H}/\rat)[\hatϕ]$作为特殊情况与$K$的一个类群重合.
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。
In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D varies over cube-free positive integers. We parametrise these points using well known parametrisation of integral points (x,y,z) of the curve X^2+3Y^2=4Z^3 with GCD(y,z)=1. As an accidental byproduct of our result we show that there are infinitely many primes congruent to 1 or 8 modulo 9, can be expressed as sum of two rational cubes.
Ranks of elliptic curves over cyclic cubic, quartic, and sextic extensions
对于给定的群$G$和定义在数域$K$上的椭圆曲线$E$，本文讨论了寻找$K$的$G$ -扩张，使$E$在其上获得秩的问题.我证明了下面的定理，扩展了Fearnley，Kisilevsky和Kuwata的结果：设$n = 3,4,$或$6$。如果$K$包含其$n^{th}$ -单位根，则对于$K$上的任何椭圆曲线$E$，存在无穷多个$\mathbb{Z}/n\mathbb{Z}$ -$K$的扩展，其中$E$获得秩。
Well-Rounded Twists of the Ring of Integers in Cyclic Cubic Fields
针对循环三次域中整数环的良舍入扭曲问题，提出检测理想基良舍入扭曲的新算法，证明特定条件下良舍入扭曲存在性等价于主良舍入理想存在性，并显式计算三类族的扭曲，证明Shanks族无限域有正交良舍入扭曲。
We consider the Galois group $G_2(K)$ of the maximal unramified $2$-extension of $K$ where $K/\mathbb{Q}$ is cyclic of degree $3$. We also consider the group $G^+_2(K)$ where ramification is allowed at infinity. In the spirit of the Cohen-Lenstra heuristics, we identify certain types of pro-$2$ group as the natural spaces where $G_2(K)$ and $G^+_2(K)$ live when the $2$-class group of $K$ is $2$-generated. While we do not have a theoretical scheme for assigning probabilities, we present data and make some observations and conjectures about the distribution of such groups.
In this paper, we study dilation of cyclic polytopes with the vertices defined by a generator of the simplest cubic fields. In particular, for a specific range of values, we give a precise number of the contained lattice points.
Theoretical and Experimental Approach to p-Class Field Towers of Cyclic Cubic Number Fields
本文研究循环三次数域的p-类域塔，通过Magma算法对导子小于100000的域进行实验，结合伽罗瓦上同调与Artin转移等理论技术，证明了三阶段塔的首次出现并确定了第二p-类群及塔长。
Galois module structure of algebraic integers of cyclic cubic fields
研究通过根式通用循环三次多项式确定三次循环域中整数环的伽罗瓦模结构，并显式给出其生成元。
Group theoretic approach to cyclic cubic fields
本文用群论方法研究共享共同导子的循环三次数域四元组，通过三次剩余符号和歧主理想条件确定极大亚交换非分歧3-扩张的自同构群，并借助关系秩判定其是否等同于极大非分歧pro-3-扩张的Galois群。
Cyclic Cubic Points on Higher Genus Curves
本文研究高亏格曲线上伽罗瓦群为 $C_3$ 的三次点，证明其源于结构良好的态射，并给出可计算检验，同时推广至低亏格情形。
Computation of the cover of Shimura curves $X_0(2) \to X(1)$ for the cyclic cubic field of discriminant 13^2
在Elkies的论文“Shimura curves for level 3 subgroups of the（2，3，7）triangle group”的末尾描述了判别式13^2的三次域，我们计算Shimura曲线覆盖的正则模型$X_0(2) \to X(1)$。最后，我们给出了X（1）上某些有理CM点的坐标。
Difference sets and positive exponential sums II: cubic residues in cyclic groups
本文通过构造非负指数和，给出了循环群中差集避免三次剩余的集合基数的上界。
Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials
在本文中，我们突出拉马努金三次多项式（RCP）和一类多项式，Shanks三次多项式（SCP），生成循环三次域之间的连接。在这种方式中，我们提供了一个新的表征RCP和我们表示的任何RCP的零的显式形式，使用三角函数。此外，我们观察到，周期三的循环变换置换这些零。由于这些结果，我们提供了许多新的和美丽的身份。最后，我们将随机CP与高斯周期联系起来，得到了一个新的恒等式，并研究了与随机CP相关的一些整数序列。
Cyclic cubic number fields with harmonically balanced capitulation
本文证明了689347是使循环三次数域的最大非分歧pro-3扩张具有阶为6561且关系秩等于生成元秩的自同构群及调和平衡转移核的最小导子。
We prove that all imaginary biquadratic fields and cyclic quartic fields of class number $1$ are Euclidean.
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…
On cubic polynomials with the cyclic Galois group
本文研究具有循环伽罗瓦群的三次多项式，通过构造二次多项式对根的循环置换引入耦合、相邻类及超类概念，并给出超类的完整描述。
We prove that approximately $96.23\%$ of cubic fields, ordered by discriminant, have genus number one, and we compute the exact proportion of cubic fields with a given genus number. We also compute the average genus number. Finally, we show that a positive proportion of totally real cubic fields with genus number one fail to be norm-Euclidean.
Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) $G$ can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves $E/\Q$ together with cubic fields $K$ so that $G= E(K)_{tors}$.
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。
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.
Let $X$ be a minimal cubic surface over a finite field $\mathbb{F}_q$. The image $Γ$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ is a cyclic subgroup of the Weyl group $W(E_6)$. There are $25$ conjugacy classes of cyclic subgroups in $W(E_6)$, and $5$ of them correspond to minimal cubic surfaces. It is natural to ask which conjugacy classes come from minimal cubic surfaces over a given finite field. In this paper we give a partial answer to this question and present many explicit examples.
We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66.
Cyclic and self-dual codes are important classes of codes in coding theory. Jia, Ling and Xing \cite{Jia} as well as Kai and Zhu \cite{Kai} proved that Euclidean self-dual cyclic codes of length $n$ over $\mathbb{F}_q$ exist if and only if $n$ is even and $q=2^r$, where $r$ is any positive integer. For $n$ and $q$ even, there always exists an $[n, \frac{n}{2}]$ self-dual cyclic code with generator polynomial $x^{\frac{n}{2}}+1$ called the \textit{trivial self-dual cyclic code}. In this paper we prove the existence of nontrivial self-dual cyclic codes of length $n=2^ν\cdot \bar{n}$, where $\bar…
Lenstra introduced the notion of a Euclidean ideal class, which is a generalization of the Euclidean domain. Lenstra also proved that the Euclidean ideal in a number field $K$ implies that the class group of $K$ is cyclic. We construct a family of biquadratic fields with a Euclidean ideal whenever the class number is 2. This extends the families given by Graves, Hsu, Chattopadhyay, and Muthukrishnan.
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.
Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis
在广义黎曼假设（GRH）的假设下，证明了范数欧几里德伽罗瓦三次域正是具有判别式$Δ=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,67^2,103^2,109^2,127^2,157^2$的伽罗瓦三次域.证明的很大一部分是在建立以下更一般的结果：设$K$是奇素数度$\ell$和导体$f$的伽罗瓦数域。假设$ζ_K(s)$的GRH。如果是$38(\ell-1)^2(\log f)^6\log\log f<f$，则$K$不是范数欧几里德。
Large Pólya groups in simplest cubic fields and consecutive bi-qaudratic fields
本文将数域的连续性概念推广到循环三次域和双二次域的参数族，证明存在无穷多个具有大Pólya群的连续域，推广了全实双二次域的已有结果。
In this paper, we highlight that the point group structure of elliptic curves over finite or infinite fields, may be also observed on singular cubics with a quadratic component. Starting from this, we are able to introduce in a very general way a group's structure over any kind of conics. In the case of conics over finite fields, we see that the point group is cyclic and lies on the quadric; the straight line component plays a role which may be not explicitly visible in the algebraic description of point composition, but it is indispensable in the geometric description. Moreover, some applicat…
Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length $n$ over a finite field if and only if $n$ and the field characteristic are even. The enumeration of such codes has been given under both the Euclidean and Hermitian products. However, in each case, the formula for self-dual cyclic codes of length $n$ over a finite field contains a characteristic function which is not easily computed. In this paper, we focus on more efficient ways to enumerate self-dual cyclic codes of lengths $2^νp^r$ and $2^νp^rq^s$, where $ν$…
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