Jian Wang
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 $K$.
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In this paper, we classify torsion groups of rational Mordell curves explicitly over cubic fields as well as over sextic fields. Also, we classify torsion groups of Mordell curves over cubic fields and for Mordell curves over sextic fields, we produce all possible torsion groups.
We prove asymptotic formulas for cyclicity of reductions of elliptic curves over the rationals in a family of curves having specified torsion. These results agree with established conditional results and with average results taken over larger families. As a key tool, we prove an analogue of a result of Vlăduţ that estimates the number of elliptic curves over a finite field with a specified torsion point and cyclic group structure.
We prove that, on average, the monogenicity or $n$-monogenicity of a cubic field has an altering effect on the behavior of the 2-torsion in its class group.
We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, $\Z /21 \Z$, which corresponds to a sporadic point on $X_1(21)$ of degree 3, which is the lowest possible degree of a sporadic point on a modular curve $X_1(n)$.
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 $K$.
Bhargava, Hanke, and Shankar have recently shown that the asymptotic average $2$-torsion subgroup size of the family of class groups of monogenized cubic fields with positive and negative discriminants is $3/2$ and $2$, respectively. In this paper, we provide strong computational evidence for these asymptotes. We then develop a pair of novel conjectures that predicts, for $p$ prime, the asymptotic average $p$-torsion subgroup size in class groups of monogenized cubic fields.
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 $G= E(K)_{tors}$.
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.
We compute all the moments of the p-torsion in the first step of a filtration of the class group defined by Gerth for cyclic fields of degree p, unconditionally for p=3 and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen-Lenstra-Martinet conjectures. In the p=3 case this gives the distribution of the 3-torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Kluners in their proof of the distribution of the 4-torsion in quadratic fields.
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 $K$.
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.
For all positive integers $\ell$, we prove non-trivial bounds for the $\ell$-torsion in the class group of $K$, which hold for almost all number fields $K$ in certain families of cyclic extensions of arbitrarily large degree. In particular, such bounds hold for almost all cyclic degree-$p$-extensions of $F$, where $F$ is an arbitrary number field and $p$ is any prime for which $F$ and the $p$-th cyclotomic field are linearly disjoint. Along the way, we prove precise asymptotic counting results for the fields of bounded discriminant in our families with prescribed splitting behavior at finitely…
On the average $2$-torsion in class groups and narrow class groups of cubic orders with prescribed shape
本文研究了具有指定形状的三次域和三次阶的类群和狭义类群中2- torsion的平均值，通过改进Bhargava和Varma的方法得出了实二次域和复二次域的类群2- torsion平均大小以及实三次域狭义类群2- torsion平均大小的结论。
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 $t$.
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。
Non-cyclic Torsion of Elliptic Curves Over Imaginary Quadratic Fields of Class Number 1
本文研究类数1非分圆虚二次域上具有全2-挠点的椭圆曲线，确定了其在基域二次扩张上的挠群所有可能结构。
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。
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 $x^3-3x+1$。
On Some Diophantine Parameters of the Cyclic Torsion Subgroups of Odd Order of Elliptic Curves over $\mathbb{Q}$
本文研究有理数域上椭圆曲线奇阶循环挠子群，并给出了其若干显式丢番图参数。
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。
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。
Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(3^\infty)$ be the compositum of all cubic extensions of $\mathbb{Q}$. In this article we show that the torsion subgroup of $E(\mathbb{Q}(3^\infty))$ is finite and determine 20 possibilities for its structure, along with a complete description of the $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-inv…
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。
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.
We prove a generalization of a result of Bhargava regarding the average size $\mathrm{Cl}(K)[2]$ as $K$ varies among cubic fields. For a fixed set of rational primes $S$, we obtain a formula for the average size of $\mathrm{Cl}(K)/\langle S \rangle[2]$ as $K$ varies among cubic fields with a fixed signature, where $\langle S \rangle$ is the subgroup of $\mathrm{Cl}(K)$ generated by the classes of primes of $K$ above primes in $S$. As a consequence, we are able to calculate the average sizes of $K_{2n}(\mathcal{O}_K)[2]$ for $n > 0$ and for the relaxed Selmer group $\mathrm{Sel}_2^S(K)$ as $K$ …
Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors.
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.
For a given group $G$ and an elliptic curve $E$ defined over a number field $K$, I discuss the problem of finding $G$-extensions of $K$ over which $E$ gains rank. I prove the following theorem, extending a result of Fearnley, Kisilevsky, and Kuwata: Let $n = 3,4,$ or $6$. If $K$ contains its $n^{th}$-roots of unity then, for any elliptic curve $E$ over $K$, there are infinitely many $\mathbb{Z}/n\mathbb{Z}$-extensions of $K$ over which $E$ gains rank.
Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field
本文研究了定义在给定数域K上的abelian 品种A的内射代数，其关键步骤是提供所有内射映射定义在K(A[2])中的标准。当K=Q且Q(A[2])/Q的伽罗瓦群是素数阶p=2dim(A)+1时，证明了几何内射代数End(A)⊗Q只有有限种可能性。
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.
This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this particular case, we give a description of the possible torsion growth over cubic fields and a completely explicit description of this growth in terms of some invariants attached to a given elliptic curve.
We present a criterion for proving that certain groups of the form $\mathbb Z/m\mathbb Z\oplus\mathbb Z/n\mathbb Z$ do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply this criterion to eliminate certain groups as torsion groups of elliptic curves over cubic and quartic fields. We also use this criterion to give the list of all torsion groups of elliptic curves occurring over a specific cubic field and over a specific quartic field.
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。
Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields when ordered by their absolute discriminants. For an order $\cal O$ in a cubic field, we study the three groups: $\rm Cl_2(\cal O)$, the group of ideal classes of $\cal O$ of order 2; $\rm Cl^+_2(\cal O)$, the group of narrow ideal classes of $\cal O$ of order 2; and ${\cal I}_2(\cal O)$, the group of ideals of $\cal O$ of order 2. We prove that the mean value of the difference $|\rm Cl_2({\cal…
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.
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。
Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as $S$. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves that appear over the field with smallest absolute value of its discriminant and having Galois group $S_3$ and over the field with smallest absolute value of its discriminant and having Galois group $\Z…
The mean number of $2$-torsion elements in the class groups of cubic orders
本文确定了三次序类群中2-挠元的平均个数，并将其推广到任意奇数维情形，给出了可约SL_N(Z)-轨道的渐近计数公式。
We study Tamagawa numbers of elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over cubic fields and of elliptic curves with an $n-$isogeny over $\mathbb{Q}$, for $n\in\{6,8,10,12,14,16,17,18,19,37,43,67,163\}$. Bruin and Najman proved that every elliptic curve with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field is a base change of an elliptic curve defined over $\mathbb{Q}$. We find that Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cu…
Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. $G= \SL_2 (\C) \times \SL_2 (\C)$ or $\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture made in \cite{BV}, lattices in $G$ should have 'little' --- in the very weak sense of 'subexponential in the co-volume' --- torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the \emph{square root} of the volume. This is deduced from a general theorem that compares twisted and untwisted $L^…
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。
The $p$-adic limits of iterated $p$-power cyclic resultants of multivariable polynomials
本文研究多变量多项式迭代 p 次幂循环结式的 p 进收敛性，并将其应用于证明链环 Z_p^d-覆盖挠数的 p 进收敛，还以扭转 Whitehead 链环为例计算了极限，且证明在特定情形下该极限与 Kionke 引入的 p 进挠这一同伦不变量一致。
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。
We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the $L$-function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients.
In this paper we highlight the connection between Ramanujan cubic polynomials (RCPs) and a class of polynomials, the Shanks cubic polynomials (SCPs), which generate cyclic cubic fields. In this way we provide a new characterization for RCPs and we express the zeros of any RCP in explicit form, using trigonometric functions. Moreover, we observe that a cyclic transform of period three permutes these zeros. As a consequence of these results we provide many new and beautiful identities. Finally we connect RCPs to Gaussian periods, finding a new identity, and we study some integer sequences relate…
This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r_2, the number of complex places of K. We also obtain results abo…
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。
Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx C_2 \oplus C_2$ we determine how a given torsion structure can grow in a quadratic extension and the maximum number of quadratic extensions in which it grows. We also classify the torsion structures which occur as the quadratic twist of a given torsion structure.
We show that torsion points of certain orders are not on a theta divisor in the Jacobian variety of a hyperelliptic curve given by the equation $y^2=x^{2g+1}+x$ with $g \geq 2$. The proof employs a method of Anderson who proved an analogous result for a cyclic quotient of a Fermat curve of prime degree.
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 …
For every finite $p$-group $G_p$ that is non-cyclic and non-quaternion and every positive integer $\ell\neq p$ that is greater than $2$, we prove the first non-trivial bound on $\ell$-torsion in class group of every $G_p$-extension. More generally, for every nilpotent group $G$ where every Sylow-$p$ subgroup $G_p\subset G$ is non-cyclic and non-quaternion, we prove a non-trivial bound on $\ell$-torsion in class group of every $G$-extension for every integer $\ell>1$.
What is the true order of growth of torsion in the cohomology of an arithmetic group? Let $D$ be a quaternion over an imaginary quadratic field $F.$ Let $E/F$ be a cyclic Galois extension with $\mathrm{Gal}(E/F) = \langle σ\rangle.$ We prove lower bounds for "the Lefschetz number of $σ$ acting on torsion cohomology" of certain Galois-stable arithmetic subgroups of $D_E^\times.$ For these same subgroups, we unconditionally prove a would-be-numerical consequence of the existence of a hypothetical base change map for torsion cohomology.
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