Torsion of $\Q$-curves over quadratic fields

On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields

The possible torsion groups of elliptic curves induced by Diophantine triples over quadratic fields, which do not appear over Q, are Z/2Z x Z/10Z, Z/2Z x Z/12Z and Z/4Z x Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these torsion groups.

Torsion points of elliptic curves over multi-quadratic number fields

We compute the Mordell-Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_1(M, MN)$ over every number field which is the composition of quadratic fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.

Torsion of Elliptic Curves over Quadratic Fields

By focusing on the family $E:y^2=x^3+a$, we present strategies for determining the structure of the torsion subgroup of the Mordell-Weil group of an elliptic curve, $E(K)$, over quadratic field $K$. Generalizations of the Nagell-Lutz theorem and Mazur's theorem to curves defined over quadratic fields allows us to determine the full torsion subgroup of $E(K)$ as one of at most three possibilities when $a$ is a square.

The mean number of 3-torsion elements in ray class groups of quadratic fields

We determine the average number of $3$-torsion elements in the ray class groups of fixed (integral) conductor $c$ of quadratic fields ordered by absolute discriminant, generalizing Davenport and Heilbronn's theorem on class groups. A consequence of this result is that a positive proportion of such ray class groups of quadratic fields have trivial 3-torsion subgroup whenever the conductor $c$ is taken to be a squarefree integer having very few prime factors none of which are congruent to $1 \bmod 3$. Additionally, we compute the second main term for the number of $3$-torsion elements in ray cla…

Quadratic torsion subgroups of modular Jacobian varieties

Let $D$ be an odd square-free positive integer and $C$ a divisor of $D$. For any quadratic character $χ$ modulo $C$, we prove that the $χ$-part of the group $J_0(DC)_\text{tor}$ of torsion points of $J_0(DC)$ coincides with the $χ$-part of its cuspidal subgroup, away from those primes of bad reduction or where possible congruences between oldforms and newforms occur.

Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

In a previous paper, the author examined the possible torsions of an elliptic curve over the quadratic fields $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$. Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$, we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves o…

The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having…

On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion o…

On the quadratic twist of elliptic curves with full $2$-torsion

Let $E: y^2=x(x-a^2)(x+b^2)$ be an elliptic curve with full $2$-torsion group, where $a$ and $b$ are coprime integers and $2(a^2+b^2)$ is a square. Assume that the $2$-Selmer group of $E$ has rank two. We characterize all quadratic twists of $E$ with Mordell-Weil rank zero and $2$-primary Shafarevich-Tate groups $(\mathbb Z/2\mathbb Z)^2$, under certain conditions. We also obtain a distribution result of these elliptic curves.

类数1的虚二次域上椭圆曲线的非循环挠群

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。

二次域之 $\mathbb{Z}_p$-扩张上有理椭圆曲线的挠

Torsion of Rational Elliptic Curves over the $\mathbb{Z}_p$-Extensions of Quadratic Fields

本文证明对二次域的 $\mathbb{Z}_p$-扩张（$p>5$），有理椭圆曲线的挠子群不增大，并据此分类了可实现的挠子群。

Torsion groups of Mordell curves over cubic and sextic fields

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.

关于在二次数域中椭圆曲线的 torsion 增长

On the torsion growth in quadratic number fields for elliptic curves defined over the rationals

本文研究了在二次数域中椭圆曲线 torsion 子群的增长问题，通过分析导数与素数之间的关系，揭示了有理数域上椭圆曲线的 torsion 增长特性。

Average analytic rank of elliptic curves with prescribed torsion

We show that average analytic rank of elliptic curves with prescribed torsion $G$ is bounded for every torsion group $G$ under GRH for elliptic curve $L$-functions.

On the discriminant of elliptic curves with non-trivial torsion

For those elliptic curves defined over the rational with non--trivial torsion subgroup, we find a tight relationship between the torsion subgroup itself and a Galois group naturally arising from the curve.

关于存在挠群增长的二次域的分歧

On the Ramification of Quadratic Fields where there is a torsion growth

本文针对在二次域上挠群增长的有理椭圆曲线，结合曲线不变量，计算添加素数阶挠点时二次域的分歧情况，拓展了此前关于椭圆曲线导子素数与二次域判别式素数关联的研究。

二次基变换对椭圆曲线挠点的影响

The Effect of Quadratic Base Change on Torsion of Elliptic Curves

本文研究二次基变换下椭圆曲线挠群的增长，提出快速算法枚举挠群严格增大的二次扩张，并确定任意二次分圆域上的增长情形。

New Rank Records For Elliptic Curves Having Rational Torsion

We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.

The group of rational points on the Holm curve is torsion-free

Using the division polynomials for elliptic curves in Weierstrass form, it shown that the group of rational points on the curve $H: ky(yy - 1) = lx(xx - 1)$ is torsion-free.

二次域上椭圆曲线的挠子群与 Granville 猜想

Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville

本文借助 Bruin-Stoll 的有理点方法，将二次域 Q(√d) 上椭圆曲线挠子群的分类范围扩展至 |d|<800，并据此检验 Granville 关于超椭圆曲线扭形有理点的猜想。

A Simple Method for High-Rank Families of Elliptic Curves with Specified Torsion

We describe a simple, but effective, method for deriving families of elliptic curves, with high rank, all of whose members have the same torsion subgroup structure.

Tamagawa Numbers of elliptic curves with $C_{13}$ torsion over quadratic fields

Let $E$ be an elliptic curve over a number field $K$, $c_v$ the Tamagawa number of $E$ at $v$, and let $c_E=\prod_{v}c_v$. Lorenzini proved that $v_{13}(c_E)$ is postive for all elliptic curves over quadratic fields with a point of order $13$. Krumm conjectured, based on extensive computation, that the $13$-adic valuation of $c_E$ is even for all such elliptic curves. In this note we prove this conjecture and furhtermore prove that there is an unique such curve satisfying $v_{13}(c_E)=2$.

Torsion growth of rational elliptic curves in sextic number fields

We classify the possible torsion structures of rational elliptic curves over sextic number fields.

A complete diophantine characterization of the rational torsion of an elliptic curve

We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non-)existence of integral solutions of a system of diophantine equations.

Average Cyclicity for Elliptic Curves in Torsion Families

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.

On the torsion part in the K-theory of imaginary quadratic fields

We obtain upper bounds for the torsion in the $K$-groups of the ring of integers of imaginary quadratic number fields, in terms of their discriminants.

Elliptic curves with 2-torsion contained in the 3-torsion field

There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X'(6). Our motivation is two-fold: on the one hand, X'(6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well-known), and on the other hand, X'(6)(Q) gives an infinite family of examples of elliptic curves with non-abelian "entanglement fields," which is relevant…

Thue equations and torsion groups of elliptic curves

A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations.

Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization

In this paper, we present several methods for construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve factorization method (ECM).

Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$

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 $X_1(n)$.

Torsion of rational elliptic curves over quadratic fields

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 quadratic number field.

Typically bounding torsion on elliptic curves isogenous to rational $j$-invariant

We prove that the family $\mathcal{I}_{F_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with $F_0$-rational $j$-invariant is typically bounded in torsion. Under an additional uniformity assumption, we also prove that the family $\mathcal{I}_{d_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with degree $d_0$ $j$-invariant is typically bounded in torsion.

A criterion to rule out torsion groups for elliptic curves over number fields

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.

On the cyclic torsion of elliptic curves over cubic number fields (III)

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 $K$.

torsion packet envelope 和代数曲线的有理点

Torsion Packet Envelope and Rational Points of Algebraic Curves

本文提出一种新的方法，利用 torsion packets 确定代数曲线的有理点，并提供了能完全确定有理点的曲线示例。

Torsion points and Galois representations on CM elliptic curves

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic is…

Linearizing torsion classes in the Picard group of algebraic curves over finite fields

We address the problem of computing in the group of $\ell^k$-torsion rational points of the jacobian variety of algebraic curves over finite fields, with a view toward computing modular representations.

Local data of rational elliptic curves with non-trivial torsion

By Mazur's Torsion Theorem, there are fourteen possibilities for the non-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$, such that $E$ may have additive reduction at a prime $p$, we consider a parameterized family $E_T$ of elliptic curves with the property that they parameterize all elliptic curves $E/\mathbb{Q}$ which contain $T$ in their torsion subgroup. Using these parameterized families, we explicitly classify the Kodaira-Néron type, the conductor exponent, and the local Tamagawa number at each prime $p$ where $E/\mathbb{Q}$ has additive reduction. As a consequenc…

有理椭圆曲线在二次域上具有指定 torsion 子群的存在性和不存在性

Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields

本文研究了二次域上椭圆曲线 torsion 子群的存在性问题，证明了某些 torsion 结构在特定条件下不存在，并扩展了此前关于 torsion 分类的结果。

On the small cyclic torsion of elliptic curves over cubic number fields

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 $K$.

The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders

We determine the mean number of 3-torsion elements in the class groups of quadratic orders, where the quadratic orders are ordered by their absolute discriminants. Moreover, for a quadratic order $\mathcal{O}$ we distinguish between the two groups: $\mathrm{Cl}_3(\mathcal{O})$, the group of ideal classes of order $3$; and $\mathcal{I}_3(\mathcal{O})$, the group of ideals of order $3$. We determine the mean values of both $|\mathrm{Cl}_3(\mathcal{O})|$ and $|\mathcal{I}_3(\mathcal{O})|$, as $\mathcal{O}$ ranges over any family of orders defined by finitely many (or in suitable cases, even infin…

Torsion groups of elliptic curves over the $\mathbb Z_p$-extensions of $\mathbb Q$

We determine, for an elliptic curve $E/\mathbb Q$ and for all $p$, all the possible torsion groups $E(\mathbb Q_{\infty, p})_{tors}$, where $\mathbb Q_{\infty, p}$ is the $\mathbb Z_p$-extension of $\mathbb Q$.

亏格3超椭圆曲线的三挠子群与野导子

Three-Torsion Subgroups and Wild Conductors of Genus 3 Hyperelliptic Curves

本文提出用复逼近、同伦延拓和格基约化计算亏格3超椭圆曲线雅可比簇三挠子群的实用方法，并将其应用于计算导子在2处局部指数的野部分。

Torsion groups of elliptic curves over quadratic cyclotomic fields in elementary abelian 2-extensions

Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.

Typically bounding torsion on elliptic curves with rational $j$-invariant

A family $\mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}\in \mathcal{F}$ can be made uniformly bounded after removing from $\mathcal{F}$ those whose number field degrees lie in a subset of $\mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $\mathcal{E}_F$ of elliptic curves defined over number fields and with $F$-rational $j$-invariant is typically bounded in torsion. For any integer $d\in\mathbb{Z…

对于\(p\geq5\)，二次域的\(\Z_p\)-扩张上有理椭圆曲线的挠率，以及\(p = 3\)时的斜率分析

Torsion growth of rational elliptic curves over $\Z_p$-extensions of quadratic fields

研究二次域的\(\Z_p\)-扩张上有理椭圆曲线的挠率，\(p\geq5\)时通过修复证明伽罗瓦点问题及排除障碍证得\(E(L)_{\tors}=E(K)_{\tors}\)，\(p = 3\)时给出斜率分层及相关结论。

$\mathbb Q$-曲线在小奇素数次数数域上的 torsion

Torsion of $\mathbb Q$-curves over number fields of small odd prime degree

本文研究了小奇素数次数数域上$\mathbb Q$-曲线的torsion子群，证明其与有理$j$-invariant椭圆曲线的torsion子群等价，并完成相关分类。

Almost Rational Torsion Points on Semistable Elliptic Curves

If P is an algebraic point on a commutative group scheme A/K, then P is _almost_rational_ if no two non-trivial Galois conjugates sigma(P), tau(P), have sum equal to 2P. In this paper, we classify almost rational torsion points on semistable elliptic curves over Q.

The 12-th roots of the discriminant of an elliptic curve and the torsion points

Given an elliptic curve over a field of characteristic different from 2,3, its discriminant defines a $μ_{12}$-torsor over the field. In this paper, we give an explicit description of this $μ_{12}$-torsor in terms of the 3-torsion points and of the 4-torsion points on the given elliptic curve. %In addition, we show that such a description involves the Weil pairing in a certain way. As an application, we generalize a result of Coates on the 12-th root of the discriminant of an elliptic curve.

椭圆曲线的Tate-Shafarevich群中的可见2-挠元

Visible 2-torsion in the Tate-Shafarevich group of an elliptic curve

本文证明椭圆曲线的Tate-Shafarevich群中任意两个2-挠元可在同一阿贝尔曲面中可见，推广了先前仅知单个2-挠元的结果，并解释了Cremona和Mazur论文中的一些原始观察。

Selmer groups of twists of elliptic curves over K with K-rational torsion points

We generalize a result of Frey [Fre88] on the Selmer group of twists of elliptic curves over Q with Q-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational point. We also provide examples of elliptic curves coming from [Zyw15] that satisfy the conditions of our Theorem B.

Torsion groups of Mordell curves over number fields of higher degree

Mordell curves over a number field $K$ are elliptic curves of the form $ y^2 = x^3 + c$, where $c \in K \setminus \{ 0 \}$. Let $p \geq 5$ be a prime number, $K$ a number field such that $[K:\mathbb{Q}] \in \{ 2p, 3p \}$ and let $E$ be a Mordell curve defined over $K$. We classify all the possible torsion subgroups $E(K)_{\text{tors}}$ for all Mordell curves $E$ defined over $\mathbb{Q}$ when $[K: \mathbb{Q}] \in \{2p, 3p \}$.

Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields

In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.

Bounding the torsion in CM elliptic curves

Merel has shown that the order of torsion subgroup of an elliptic curve over a number field can be bounded in terms of only the degree of the number field. The purpose of this note is to investigate what could be the `right bound'. In this paper we use the result of Deuring on the supersingular primes for a CM elliptic curves to give estimate on the number of torsion points on CM elliptic curves over any number field which depends only linearly with the degree of the number field. This result is also a simple corollary of a result of A. Silverberg bounding the order of torsion elements which w…