New Rank Records For Elliptic Curves Having Rational Torsion

Counting rational points on elliptic curves with a rational 2-torsion point

Let $E/\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if $E$ has full rational $2$-torsion, the number $N_E(B)$ of rational points with Weil height bounded by $B$ is $\exp\left(O\left(\frac{\log B}{\sqrt{\log\log B}}\right)\right)$. In this paper we exploit the method of descent via $2$-isogeny to extend this result to elliptic curves with just one nontrivial rational $2$-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound $N_{E}(B) = \exp\left(O\left(\frac{\log B}{\log\log B}\right)\…

计算具有一个标记有理3 - 挠点的奇数亏格2曲线

Counting odd genus $2$ curves with a marked rational $3$-torsion point

研究有理数域上特定条件的亏格2曲线，通过朴素高度排序计数，核心方法是基于曲线的首一魏尔斯特拉斯模型及雅可比行列式的有理3 - 挠点条件。

Explicit classification for torsion subgroups of rational points of elliptic curves

The classification of elliptic curves E over the rationals Q is studied according to their torsion subgroups E_{tors}(Q) of rational points. Explicit criteria for the classification are given when E_{tors}(Q) are cyclic groups with even orders. The generator points P of E_{tors}(Q) are also explicitly presented in each case. These results, together with recent results of K. Ono, completely solve the problem of the mentioned explicit classification when E has a rational point of order 2.

Another look at rational torsion of modular Jacobians

We study the rational torsion subgroup of the modular Jacobian $J_0(N)$ for $N$ a square-free integer. We give a new proof of a result of Ohta on a generalization of Ogg's conjecture: for a prime number $p \nmid 6N$, the $p$-primary part of the rational torsion subgroup equals that of the cuspidal subgroup. Whereas previous proofs of this result used explicit computations of the cardinalities of these groups, we instead use their structure as modules for the Hecke algebra.

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.

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.

Constructing Superelliptic Curves with non-trivial rational Torsion on their Jacobians

In this paper, we describe the construction of superelliptic curves with a rational point of prescribed order on their jacobians. The construction is based on Hensel's Lemma and produces for a given integer $N$ a superelliptic curve of genus linear in $N$ with a rational $N$-division point on the jacobian. The method is illustrated with multiple examples.

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…

Torsion subgroups of rational elliptic curves over the compositum of all $D_4$ extensions of the rational numbers

Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(D_4^\infty)$ be the compositum of all extensions of $\mathbb{Q}$ whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of $D_4$. In this article we first show that $\mathbb{Q}(D_4^\infty)$ is in fact the compositum of all $D_4$ extensions of $\mathbb{Q}$ and then we prove that the torsion subgroup of $E(\mathbb{Q}(D_4^\infty))$ is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have…

$\mathbb{Q}_p$ 上具有有理 3 阶挠点或有理 3 次同源的椭圆曲线的密度

The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny

本文通过计算 $p$-adic 体积积分和分析模曲线，确定了随机 Weierstrass 方程在 $p$-adic 数域上定义椭圆曲线具有非平凡 3 阶挠点或 3 次同源的概率，并探讨了 $\ell>3$ 素数的情形。

Torsion of rational elliptic curves over quartic Galois number fields

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be a number field of degree four that is Galois over $\mathbb{Q}$. The goal of this article is to classify the different isomorphism types of $E(K)_{\text{tors}}$.

Rational torsion points on Jacobians of modular curves

Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb{Q}$ and its Jacobian variety $J_0(3p)$ over $\mathbb{Q}$. Let $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ be the group of rational torsion points on $J_0(3p)$ and the cuspidal group of $J_0(3p)$, respectively. We prove that the $3$-primary subgroups of $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ coincide unless $p\equiv 1 \pmod 9$ and $3^{\frac{p-1}{3}} \equiv 1 \!\pmod {p}$.

An Errata for: Torsion subgroups of rational elliptic curves over the compositum of all $D_4$ extensions of the rational numbers

In [2], the author claims that the fields $\mathbb{Q}(D_4^\infty)$ defined in the paper and the compositum of all $D_4$ extensions of $\mathbb{Q}$ coincide. The proof of this claim depends on a misreading of a celebrated result by Shafarevich. The purpose is to salvage the main results of [2]. That is, the classification of torsion structures of $E$ defined over $\mathbb{Q}$ when base changed to the compositum of all $D_4$ extensions of $\mathbb{Q}$ main results of [2]. All the main results in [2] are still correct except that we are no longer able to prove that these two fields are equal.

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…

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。

Higher descents on an elliptic curve with a rational 2-torsion point

Let $E$ be an elliptic curve over a number field $K$. Descent calculations on $E$ can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this group. The general method of 4-descent, developed in the PhD theses of Siksek, Womack and Stamminger, has been implemented in Magma (when $K={\mathbb Q}$) and works well for elliptic curves with sufficiently small discriminant. By extending work of Bremner and Cassels, we describe the improvements that can be made when $E$ has a rational 2-torsion point. In parti…

Rational torsion of generalised modular Jacobians of odd level

We consider the generalised Jacobian $J_{0}(N)_{\mathbf{m}}$ of the modular curve $X_{0}(N)$ of level $N$, with respect to the modulus $\mathbf{m}$ consisting of all cusps on the modular curve. When $N$ is odd, we determine the group structure of the rational torsion $J_{0}(N)_{\mathbf{m}}(\mathbb{Q})_{tor}$ up to $2$-primary and $l$-primary parts for any prime $l$ dividing $N$. Our results extend those of Wei--Yamazaki for squarefree levels and Yamazaki--Yang for prime-power levels.

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…

The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains

Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup of the Jacobian variety $J_0(\frak{n})$ of the Drinfeld modular curve $X_0(\frak{n})$. We prove that for any prime number $\ell$ not dividing $q(q-1)$, the $\ell$-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the $\ell$-primary part of the cuspidal divisor class group for any prime $\ell$ not dividing $q-1$.

Large torsion subgroups of split Jacobians of curves of genus two or three

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P^1 of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X^4 + Y^4 + Z^4) - 96914(X^2 Y^2 + X^2 Z^2 + Y^2 Z^2) = 0, whose Jacobian has 864 rational torsion points. This paper has appeared in Forum Math. 12 (2000) 315-364.

Rational torsion points on Jacobians of Shimura curves

Let $p$ and $q$ be distinct primes. Consider the Shimura curve $\mathcal{X}$ associated to the indefinite quaternion algebra of discriminant $pq$ over $\mathbb{Q}$. Let $J$ be the Jacobian variety of $\mathcal{X}$, which is an abelian variety over $\mathbb{Q}$. For an odd prime $\ell$, we provide sufficient conditions for the non-existence of rational points of order $\ell$ on $J$. As an application, we find some non-trivial subgroups of the kernel of an isogeny from the new quotient of $J_0(pq)$ to $J$.

Torsion of elliptic curves with rational $j$-invariant defined over number fields of prime degree

Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying Galois representations of $E$ and of it's quadratic twists.

具有四元数乘法的阿贝尔曲面上的有理挠点

Rational torsion points on abelian surfaces with quaternionic multiplication

本文研究具有非分裂四元数乘法的阿贝尔曲面的有理挠点，证明其挠子群为12-挠且阶至多18，并在GL₂-型条件下给出完整分类。

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

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…

Torsion of rational elliptic curves over different types of 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 $G= E(K)_{tors}$.

On the torsion of Drinfeld modules of rank two

We study rational points and torsion points on Drinfeld modular curves defined over rational function fields. As a consequence we derive a conjecture of Schweizer describing completely the torsion of Drinfeld modules of rank two over $\Bbb F_2(T)$ implying Poonen's uniform boundedness conjecture in this particular case.

Torsion of rational elliptic curves over the maximal abelian extension of Q

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\mathbb{Q}^{ab}$ be the maximal abelian extension of $\mathbb{Q}$. In this article we classify the groups that can arise as $E(\mathbb{Q}^{ab})_{\text{tors}}$ up to isomorphism. The method illustrates techniques for finding explicit models of modular curves of mixed level structure. Moreover we provide an explicit algorithm to compute $E(\mathbb{Q}^{ab})_{\text{tors}}$ for any elliptic curve $E/\mathbb{Q}$.

Computing the Cassels-Tate Pairing for Genus Two Jacobians with Rational Two Torsion Points

In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption that all points in $J[2]$ are $K$-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating we can indeed improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing.

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。

Groups of generalized $G$-type and applications to torsion subgroups of rational elliptic curves over infinite extensions of $\mathbb{Q}$

Recently there has been much interest in studying the torsion subgroups of elliptic curves base-extended to infinite extensions of $\mathbb{Q}$. In this paper, given a finite group $G$, we study what happens with the torsion of an elliptic curve $E$ over $\mathbb{Q}$ when changing base to the compositum of all number fields with Galois group $G$. We do this by studying a group theoretic condition called generalized $G$-type, which is a necessary condition for a number field with Galois group $H$ to be contained in that compositum. In general, group theory allows one to reduce the original prob…

Structure of the cuspidal rational torsion subgroup of J_1(p^n)

In this article, we determine the structure of the $p$-primary subgroup of the cuspidal rational torsion subgroup of the Jacobian $J_1(p^n)$ of the modular curve $X_1(p^n)$ for a regular prime $p$.

按导子排序且具有有理2-挠点的椭圆曲线及平均秩的有界性

Elliptic curves with a rational 2-torsion point ordered by conductor and the boundedness of average rank

本文改进了按导子计数椭圆曲线的方法，将其应用于具有有理2-挠点的小族椭圆曲线，为无平方因子指标和有界Szpiro比率的曲线证明了计数定理，且假设条件比朴素推广更宽松。

Rational torsion in elliptic curves and the cuspidal subgroup

Let $A$ be an elliptic curve over $\Q$ of square free conductor $N$. We prove that if $A$ has a rational torsion point of prime order $r$ such that $r$ does not divide $6N$, then $r$ divides the order of the cuspidal subgroup of $J_0(N)$.

Watkins猜想对于具有有理扭转的椭圆曲线

Watkins's conjecture for elliptic curves with a rational torsion

本文研究了具有有理扭转的椭圆曲线薄族上的Watkins猜想，量化了满足该猜想的具有有理$\ell$-扭转的椭圆曲线，并将研究推广至包含任意正整数$M$的更广泛秩不等式情形。

具有有理j不变量的椭圆曲线在ℚ的极大初等阿贝尔2扩张上的挠率

Torsion of elliptic curves with rational $j$-invariant over the maximal elementary abelian 2-extension of $\mathbb{Q}$

本文针对定义在有理数域所有二次扩张合成域上、j不变量为非0或1728的有理数的椭圆曲线，完成了对其可能挠子群结构的分类。

Families of elliptic curves with rational torsion points of even order

This paper is a follow up of arXiv:1702.02255 [math.NT]. We construct explicitly versal families of elliptic curves with rational points of order 4, 6, 8, 10, 12 respectively.

Minimal models of rational elliptic curves with non-trivial torsion

In this paper, we explicitly classify the minimal discriminants of all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that they parameterize all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion point. We follow this by giving admissible change of variables, which give a global minimal model for $E$. We also provide necessary and sufficient conditions on the parameters of these families to determine the primes at which $E$ has additive reduction. In addition, we use thes…

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.

Cartan images and $\ell$-torsion points of elliptic curves with rational $j$-invariant

Let $\ell$ be an odd prime and $d$ a positive integer. We determine when there exists a degree-$d$ number field $K$ and an elliptic curve $E/K$ with $j(E)\in\mathbb{Q}\setminus\{0,1728\}$ for which $E(K)_{\mathrm{tors}}$ contains a point of order $\ell$. We also determine when there exists such a pair $(K,E)$ for which the image of the associated mod-$\ell$ Galois representation is contained in a Cartan subgroup or its normalizer, conditionally on a conjecture of Sutherland. We do the same under the stronger assumption that $E$ is defined over $\mathbb{Q}$.

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.

Galois criterion for torsion points of Drinfeld modules

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal $\fl$ of $\F_q[T]$, the question essentially asks whether, up to isogeny, a Drinfeld module $ϕ$ over $\F_q(T)$ contains a rational $\fl$-torsion point if the reduction of $ϕ$ at almost all primes of $\F_q[T]$ contains a rational $\fl$-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is $2$, but negative if the rank…

$\boldsymbol{\text{有理数域}}$分圆扩张上有理椭圆曲线的挠群

Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of $\mathbb{Q}$

本文研究有理椭圆曲线在有理数域分圆扩张上的挠群分类问题，对任意素数p给出同构分类，p-1不被小整数整除时得到更精细结果，还提供了排除阿贝尔数域上未实现挠结构的工具。

Torsion growth over cubic fields of rational elliptic curves with complex multiplication

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.

Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus

We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve $X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it. When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational torsion points $\widetilde{J}_0(N)(\mathbb{Q})_{\mathrm{Tor}}$ tends to be much smaller than the classical Jacobian.

无需使用高度界计算平面四次曲线的 torsion

Computing torsion for plane quartics without using height bounds

本文提出了一种无需高度界即可计算平面四次曲线雅可比簇有理 torsion 子群的算法，通过模素数降解和小数域搜索实现，已在 Magma 中实现并应用于大量数据集。

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。

Torsion of rational elliptic curves over cubic 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 cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors.

On the torsion of rational elliptic curves over quartic fields

Let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G, which possible groups G <= H could appear such that H=E(K)_tors, for [K:Q]=4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields.

Continued fractions for rational torsion

We exhibit a method to use continued fractions in function fields to find new families of hyperelliptic curves over the rationals with given torsion order in their Jacobians. To show the utility of the method, we exhibit a new infinite family of curves over $\mathbb Q$ with genus two whose Jacobians have torsion order eleven. {\bf In this updated version, we correct an error in the initial paper:} The ``new" family claimed in the original Theorem~1 was pointed out by Professor D.~Lorenzini to have elements isomorphic with elements in Flynn's family (as defined in the paper); his guess that the…

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…

On the torsion of rational elliptic curves over sextic fields

Given an elliptic curve $E/\mathbb{Q}$ with torsion subgroup $G = E(\mathbb{Q})_{\rm tors}$ we study what groups (up to isomorphism) can occur as the torsion subgroup of $E$ base-extended to $K$, a degree 6 extension of $\mathbb{Q}$. We also determine which groups $H = E(K)_{\rm tors}$ can occur infinitely often and which ones occur for only finitely many curves. This article is a first step towards a complete classification of torsion growth of over sextic fields.

$\mathbb Q$-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。

无有理定义CM的数域上椭圆曲线挠群的增长

Growth of Torsion Groups of Elliptic Curves Over Number Fields without Rationally Defined CM

本文在不假设广义黎曼假设的条件下，证明对无有理定义CM的二次域K，存在仅依赖K的素数界，使得最小素因子超过该界的任意次数扩张不会增加椭圆曲线的挠群，推广了已有结果。

Torsion points and isogenies on CM elliptic curves

Let $\mathcal{O}$ be an order in the imaginary quadratic field $K$. For positive integers $M \mid N$, we determine the least degree of an $\mathcal{O}$-CM point on the modular curve $X(M,N)_{/K(ζ_M)}$ and also on the modular curve $X(M,N)_{/\mathbb{Q}(ζ_M)}$: that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally defined. To prove these results we establish several new theorems on rational cyclic isogenies of CM elliptic curves. In particular, we extend a result of Kwon that …