Florian Breuer
Let K be a number field and E/K a modular elliptic curve, with modular parametrization $X_0(N) \to E$ defined over K. The purpose of this note is to study the images in E of classes of isogenous points in X_0(N).
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Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in the $\mathbb{Q}$-isogeny class $\mathcal{E}$, and an edge for each cyclic $\mathbb{Q}$-isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny-torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of th…
We count by height the number of elliptic curves over the rationals that possess an isogeny of degree three.
We describe the structure of Tate-Shafarevich groups of a constant elliptic curves over function fields by exploiting the volcano structure of isogeny graphs of elliptic curves over finite fields.
Counting elliptic curves over the rationals with a 7-isogeny
本文按高度计数了有理数域上具有7次循环同源的椭圆曲线数量，分别考虑了有理数域及其代数闭包上的同构类。
We show that the distribution of elliptic curves in isogeny classes of curves with a given value of the Frobenius trace $t$ becomes close to uniform even when $t$ is averaged over very short intervals inside the Hasse-Weil interval.
We work out the complete descent via 4-isogeny for a family of rational elliptic curves with a rational point of order 4; such a family is of the form $y^2 + x y + a y = x^3 + a x^2$ where $\sqrt{-a} \in \mathbb Q^\times$. In the process we exhibit the 4-isogeny and the isogenous curve, explicitly present the principal homogeneous spaces, and discuss examples by computing the rank.
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$ without CM. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in $\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree recorded as a label of the edge. An isogeny-torsion graph is an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the correspond…
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each element of $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in $\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree of the isogeny recorded as a label of the edge. The isogeny-torsion graph associated to $\mathcal{E}$ is the isogeny graph associated to $\mathcal{E}$ where, in addition, we label each vertex with the abstract group structure of t…
Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs
面向基于同源的密码学，本文将层结构引入超奇异椭圆曲线，建立其与四元数代数中 Eichler 序的对应，并研究相关同源图。
We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of $\mathcal{D}$-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.
We describe the neighborhood of the vertex $[E_0]$ (resp. $[E_{1728}]$) in the $\ell$-isogeny graph $\mathcal{G}_\ell(\mathbb{F}_{p^2}, -2p)$ of supersingular elliptic curves over the finite field $\mathbb{F}_{p^2}$ when $p>3\ell^2$ (resp. $p>4\ell^2$) with $E_0: y^2=x^3+1$ (resp. $E_{1728}: y^2=x^3+x$) supersingular.
Selmer groups of families of elliptic curves with an $\ell$-isogeny
本文研究具有素数次有理同源的椭圆曲线族，通过 Tamagawa 比的中心极限定理及其平均值界，证明若干 $\ell$ 值下存在任意大的 $\ell$-Selmer 群。
Maximal curves over finite fields and a modular isogeny
本文证明了在含$11^5$个元素的域上存在达到Hasse-Weil-Serre上界的亏格$7$和$12$曲线，通过推广Chen同源结果计算商模曲线点数并发现多项已知最大点数新纪录。
In this note, we consider an l-isogeny descent on a pair of elliptic curves over Q. We assume that l > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finite- dimensional F_l-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the l-part of the Birch and Swinnerton-Dyer conjectural formula for certain curves of small conductor.
Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves. We prove the analogous results for p=2 under the additional assumption that E is not supersingular at primes above 2.
The possible adelic indices for elliptic curves admitting a rational cyclic isogeny
本文证明了对于允许非平凡有理循环同源映射的非CM椭圆曲线，Zywina关于adelic指数的猜想成立，加强了Lemos对Serre统一问题的结果。
An elliptic curve $E$ defined over a $p$-adic field $K$ with a $p$-isogeny $ϕ:E\rightarrow E^\prime$ comes equipped with an invariant $α_{ϕ/K}$ that measures the valuation of the leading term of the formal group homomorphism $Φ:\hat E \rightarrow \hat E^\prime$. We prove that if $K/\mathbb{Q}_p$ is unramified and $E$ has additive, potentially supersingular reduction, then $α_{ϕ/K}$ is determined by the number of distinct geometric components on the special fibers of the minimal proper regular models of $E$ and $E^\prime$.
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each element of $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one element of $\mathcal{E}$ to another element of $\mathcal{E}$, with the degree recorded as a label of the edge. The isogeny-torsion graph associated to $\mathcal{E}$ is the isogeny graph associated to $\mathcal{E}$ where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\ma…
Models of Elliptic Curves with Maximal 2-adic Image Subject to a Rational 2-Isogeny
本文给出在有理2-同源约束下具有最大2-adic像的非CM椭圆曲线的显式模型，利用模曲线与同源图分类使通用性显式化。
For a fixed $j$-invariant $j_0$ of an elliptic curve without complex multiplication we bound the number of $j$-invariants $j$ that are algebraic units and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. Our bounds are effective. We also modify the problem slightly by fixing a singular modulus $α$ and looking at all $j$ such that $j-α$ is an algebraic unit and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. The number of such $j$ can again be bounded effectively.
Tamagawa Numbers of Elliptic Curves with an $\ell$-isogeny
本文研究具有ℓ-同源映射的椭圆曲线，将其与带ℓ-挠点的椭圆曲线建立联系，并分析其整体Tamagawa数的ℓ-可除性。
Let $n>1$ be an integer such that $X_{0}\!\left( n\right) $ has genus $0$, and let $K$ be a field of characteristic $0$ or relatively prime to $6n$. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a non-trivial isogeny over $\mathbb{Q}$. We achieve this by introducing $56$ parameterized families of elliptic curves $\mathcal{C}_{n,i}(t,d)$ defined over $K(t,d)$, which have the following two properties for a fixed $n$: the elliptic curves $\mathcal{C}_{n,i}(t,d)$ are isogenous over $K(t,d)$, and there are integers $k_{1}$ and $k_{2}$ such tha…
Let h be a p-isogeny of elliptic curves. We describe how to perform h-descents on the nontrivial elements in the Shafarevich-Tate group which are killed by the dual isogeny h'. This makes computation of p-Selmer groups of elliptic curves admitting a p-isogeny over Q feasible for p = 5,7 in cases where an isogeny descent is insufficient and a full p-descent would be infeasible. As an application we complete the verification of the full Birch and Swinnerton-Dyer conjectural formula for all elliptic curves over Q of rank zero or one and conductor less than 5000.
We show that if $E$ is an elliptic curve over $\mathbf{Q}$ with a $\mathbf{Q}$-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves with complex multiplication by $\mathbf{Q}(\sqrt{-7})$. The analogous result with 7 replaced by a prime $p > 7$ was proved by the first author in [7]. The present case $p = 7$ has additional interesting complications. We show that any exceptions correspond to the rational points on a certain curve of genus 12. We then use the method of Chabauty to show that the…
Statistics for 3-isogeny induced Selmer groups of elliptic curves
本文通过改进Davenport-Heilbronn策略，将整二元三次型的统计与3-同源诱导Selmer群联系起来，证明了使椭圆曲线E_a在Q(μ_3)上的3-同源诱导Selmer群维数≤1的六次幂自由整数a具有显式下密度。
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…
Supersingular Isogeny Graphs from Algebraic Modular Curves
本文描述并比较了计算超奇异同源图的算法，推导出一般超奇异同源图邻接矩阵迹的公式，并证明了Nakaya最近提出的猜想。
We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full 3-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations. As an application, we show that the elliptic curves with torsion subgroup of order 3 and rank at least 13, found by Eroshkin, have rank exactly 13.
Let p>3 be a prime and let E, E' be supersingular elliptic curves over F_p. We want to construct an isogeny phi: E --> E'. The currently fastest algorithm for finding isogenies between supersingular elliptic curves solves this problem by performing a "meet-in-the-middle" breadth-first search in the full supersingular 2-isogeny graph over F_{p^2}. In this paper we consider the structure of the isogeny graph of supersingular elliptic curves over F_p. We give an algorithm to construct isogenies between such supersingular elliptic curves that works faster than the usual algorithm. We then discuss …
The elliptic curve $E_k \colon y^2 = x^3 + k$ admits a natural 3-isogeny $ϕ_k \colon E_k \to E_{-27k}$. We compute the average size of the $ϕ_k$-Selmer group as $k$ varies over the integers. Unlike previous results of Bhargava and Shankar on $n$-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on $k$; this sensitivity can be precisely controlled by the Tamagawa numbers of $E_k$ and $E_{-27k}$. As consequences, we prove that the average rank of the curves $E_k$, $k\in\mathbb Z$, is less than 1.21 and over $23\%$ (resp. $41\%$) of the cur…
Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this paper we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve $E$ that runs, under certain heuristics, in time $O((\log p)^2p^{1/2})$. The algorithm works by first finding two cycles of a certain form in the supersingular $\ell$-isogeny graph $G(p,\ell)$, generating an order $Λ\subseteq \operatorname{End}(E)$. Then all maxi…
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。
Weber modular curves and modular isogenies
本文研究韦伯函数定义的模曲线及其关联的模多项式、SL₂(ℤ)作用及椭圆曲线参数化，旨在探讨其确定的同源图，特别是奇异椭圆曲线的性质，并应用于密码学中的高效同源计算及显式伽罗瓦表示。
We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to compute Phi_n with n over 5000, and Phi_n mod m with n over…
Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves
本文利用可嵌入自同态环的虚二次序及素数分裂条件，在超奇异同源图中构造自环和环，分析其长度并给出意外2-环对应的素数上界。
Consider a rational point on an elliptic curve under an isogeny. Suppose that the action of Galois partitions the set of its pre-images into n orbits. It is shown that all such points above a certain height have their denominator divisible by at least n distinct primes. This generalizes Siegel's theorem and more recent results of Everest et al. For multiplication by a prime l, it is shown that if n>1 then either the point is l times a rational point or the elliptic curve emits a rational l-isogeny.
Let $p$ be a prime and $K$ a number field of degree $p$. We count the number of elliptic curves, up to $\bar{K}$-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup, or that it has an isogeny of prescribed degree.
Polynomial bounds on torsion from a fixed geometric isogeny class of elliptic curves
该研究针对固定几何同源类的椭圆曲线，证明了其挠率存在多项式界，给出了挠点阶数与挠群大小关于基域在有理数域上扩张次数的具体多项式上界形式。
We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss Conjecture A, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension is a finitely generated $\mathbb{Z}_p$-module. The relationship between this conjecture and Iwasawa's classical $μ=0$ conjecture is clarified. We also present some partial results towards the question whether Conjecture A is invariant under isogenies.
We count the number of rational elliptic curves of bounded naive height that have a rational $N$-isogeny, for $N \in \{2,3,4,5,6,8,9,12,16,18\}$. For some $N$, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack.
Computing isogenies from modular equations in genus two
提出一种算法，利用模方程和显式Kodaira-Spencer同构，在特征足够大且曲线足够一般的条件下，显式计算亏格2曲线上Jacobian的l-同源或循环核同源。
Minimal degree of an isogeny between a supersingular elliptic curve and its conjugate
研究超奇异椭圆曲线与其共轭曲线间同构最小次数，通过新技术计算次数并给出界，此界渐近最优且在多情况下精确，还基于数据给出多个猜想。
The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$
研究椭圆曲线族E_n的3-同源Selmer群，证明对几乎所有n，Sel_φ(E_n)秩为0，而Sel_φ̂(Ê_n)秩由n中模3余2的素因子个数及n模9的同余类决定。
For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of $\mathrm{Sha}(E/K)[p^\infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p \le 3$ were known). T. and V. Dokchitser have showed …
An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a centr…
On the squareness of the discriminant of elliptic curves with an isogeny
本文研究了具有同源映射的椭圆曲线判别式平方性的分类，并确定了两个平方判别式椭圆曲线之间同源关系的 $ N $ 值参数化方法。
Zeta functions of abstract isogeny graphs and modular curves
本文提出抽象同源图这一 Serre 图的不可定向变体，证明其 Ihara 行列式公式类比，并将超奇异椭圆曲线同源图的 Ihara Zeta 函数与模曲线的 Hasse-Weil Zeta 函数联系起来，进而给出点计数与圈计数的显式关系及非回溯圈数的渐近公式。
Counting elliptic curves with a cyclic $m$-isogeny over $\mathbb{Q}$
本文利用解析数论和Tauberian定理，给出了有理数域上具有循环$m$-同源的椭圆曲线在二次扭转和同构意义下的计数渐近式，并获得了幂次节省的误差项。
A formula is given for the dimension of the Selmer group of the rational three-isogeny of elliptic curves of the form y^2=x^3+a(x-b)^2. The formula is in terms of the three-ranks of the quadratic number fields Q(\sqrt{a}) and Q(\sqrt{-3a}) and various aspects of the arithmetic of these number fields. In addition a duality theorem is used to relate the dimension of the Selmer group of the three-isogeny with the dimension of the Selmer group of its dual isogeny.
Ordinary Isogeny Graphs with Level Structure
本文研究了具有层级结构的普通椭圆曲线ℓ-同源图，探讨了层级结构对图结构的影响及理想类群的作用。
On the Zeta functions of supersingular isogeny graphs and modular curves
本文证明了模曲线的 Hasse--Weil zeta 函数与配备级结构的超奇异椭圆曲线同源图的 Ihara zeta 函数之间的关系公式，推广了 Sugiyama 的结果。
Minimal torsion curves in geometric isogeny classes
本文研究固定几何同源类中的最小 torsion 曲线，探讨有理同源类或复乘同源类中点的最小度数特性。
Mazur's isogeny theorem
本文综述了 Mazur 同源定理及其证明，该定理刻画了允许有理同源的素数次数，是椭圆曲线理论和费马大定理证明的重要基础。
We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an "isogeny estimate", providing an explicit bound on the degree of a minimal isogeny between two isogenous elliptic curves. We also give several corollaries of these two results.
Radical isogenies and modular curves
该研究将Tate正规形下的根式同源原始框架转化为模曲线语言，解决了Onuki和Moriya提出的X₀(N)上根式同源公式的公开问题，深化了根式同源与模曲线的关联研究。
Uniform polynomial bounds on torsion from rational geometric isogeny classes
本文证明了对于几何同源于至少一条有理椭圆曲线的椭圆曲线族，其挠群大小存在一致多项式界，推广了固定有理几何同源类的结果，并为有理$j$-不变量情形获得了最优指数界。
Isogeny graphs of elliptic curves in characteristic zero
该研究对特征零域上End_K E≅ℤ的椭圆曲线的同源图进行分类，分解问题为p-主同源图分类，引入p- blooming不变量，还应用于刻画潜在复乘椭圆曲线的同源图等场景。
An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a product formula, in terms of congruence considerations involving that polynomial, for the size of such an isogeny class. In this paper, we give a new, transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality. This answers a question posed by N. Katz.
The remarkable structure and computationally explicit form of isogeny graphs of elliptic curves over a finite field has made them an important tool for computational number theorists and practitioners of elliptic curve cryptography. This expository paper recounts the theory behind these graphs and examines several recently developed algorithms that realize substantial (often dramatic) performance gains by exploiting this theory.
Fix an elliptic curve $E_0$ without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of $E_0^g$, also defined over the algebraic numbers, under all isogenies between $E_0^g$ and some fiber of the $g$-th fibered power $\mathcal{A}$ of the elliptic scheme, where $g$ is a fixed natural number. As a special case of a slightly more general result, we characterize the subvarieties (of arbitrary dimension) inside $\mathcal{A}$ that have potentially Zari…
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