Tim Dokchitser, Vladimir Dokchitser
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.
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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。
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 …
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。
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。
Let $E$ be an elliptic curve over a number field $K$. If for almost all primes of $K$, the reduction of $E$ modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces $E$ to have a cyclic isogeny of that degree over $K$. Building upon the work of Sutherland, Anni, and Banwait-Cremona in the case of prime degree, we consider this question for cyclic isogenies of arbitrary degree.
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…
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…
Cycles and Cuts in Supersingular L-Isogeny Graphs
本文研究了超奇异L同余图中的循环与割问题，提出基于Brandt矩阵迹的循环计数方法和贪心邻居算法优化图割，为同余密码学提供了理论支持和算法改进。
Let $E$ be an elliptic curve over $\mathbb{Q}$ such that $\mathrm{End}_{\bar{\mathbb{Q}}}(E)=\mathbb{Z}$ and which admits a non-trivial cyclic $\mathbb{Q}$-isogeny. We prove that, for $p>37$, the residual mod $p$ Galois representation $\barρ_{E,p}:G_{\mathbb{Q}}\rightarrow\mathrm{GL}_2(\mathbb{F}_p)$ is surjective.
Some arithmetic properties of Weil polynomials of the form $t^{2g}+at^g+q^g$
本文研究了Weil多项式形式为$t^{2g}+at^g+q^g$的同源类的局部循环性及有理点群的局部增长。
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。
Let $p$ and $q$ be distinct primes, and let $X_{p,q}$ be the $(q+1)$-regular graph whose nodes are supersingular elliptic curves over $\overline{\mathbb{F}}_p$ and whose edges are $q$-isogenies. For fixed $p$, we compute the distribution of the $\ell$-Sylow subgroup of the sandpile group (i.e.\ Jacobian) of $X_{p,q}$ as $q \to \infty$. We find that the distribution disagrees with the Cohen-Lenstra heuristic in this context. Our proof is via Galois representations attached to modular curves. As a corollary of our result, we give an upper bound on the probability that the Jacobian is cyclic, whi…
We study the isogenies of certain abelian varieties over finite fields with non-commutative endomorphism algebras with a view to potential use in isogeny-based cryptography. In particular, we show that any two such abelian varieties with endomorphism rings maximal orders in the endomorphism algebra are linked by a cyclic isogeny of prime degree.
On supersingular isogeny graphs of Drinfeld modules
本文研究A=𝔽_q[T]上秩2的Drinfeld模的超奇异同构图，证明其连通性，引入完备性数并推导其上、下界，证实相关猜想并反驳旧界，还给出计算该数的算法准则。
CM Drinfeld Modules, Self-isogenous Modular Polynomials, and Volcano Structure
本文研究了CM德拉芬模块的自同源模形式和火山结构，提出了一种计算方法并证明了火山结构在同源图中的出现。
The aim of this paper is to study certain family of elliptic curves $\{\mathscr{X}_H\}_H$ defined over a number field $F$ arising from hyperplane sections of some cubic surface $\mathscr{X}/F$ associated to a cyclic cubic extension $K/F$. We show that each $\mathscr{X}_H$ admits a 3-isogeny $ϕ$ over $F$ and the dual Selmer group $S^{(\hatϕ)}(\hat{\mathscr{X}_H}/F)$ is bounded by a kind of unit/class groups attached to $K/F$. This is proven via certain rational function on the elliptic curve $\mathscr{X}_H$ with nice property. We also prove that the Shafarevich-Tate group $\text{\cyr X} (\hat{\…
New isogenies of elliptic curves over number fields
本文利用Galois表示分析椭圆曲线上循环同源的定义域，证明了在GRH下数域上同源的一致性结果及非CM椭圆曲线的mod-$\ell$ Galois表示性质。
We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows us to connect the resulting Selmer group with the logarithmic class group of the base. These constructions are explicit and suitable for computer experimentation. From a conceptual point of view, the questions that arise here are analogues of "visibility" questions in the sense of Cremona and Mazur.
Elliptic $K$-curves are elliptic curves defined over some field extension $L/K$ that are isogenous to all of their Galois conjugates. We present a new result on $K$-curves $E$ that are given by a $K$-rational orbit $\{τ, -1/Nτ\}$ of the Fricke involution on $Y_0(N)$, giving a simple Diophantine condition on the extension $L/K$ that determines which twists of $E$ allow the isogeny between Galois conjugates to be defined over $L$. To support and illustrate this result, we also discuss parameterizations of cyclic $N$-isogenies corresponding to points on modular curves $X_0(N)$ of genus $0$. These…
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 …
p-Selmer ranks of CM abelian varieties
本文直接证明数域上复乘椭圆曲线的 p∞-Selmer 秩为偶数，并将该结果推广至阿贝尔簇。
We present an account of the construction by S. Sekiguchi and N. Suwa of a cyclic isogeny of affine smooth group schemes unifying the Kummer and Artin-Schreier-Witt isogenies. We complete the construction over an arbitrary base ring. We extend the statements of some results in a form adapted to a further investigation of the models of the group schemes of roots of unity.
We consider the finite set of isogeny classes of $g$-dimensional abelian varieties defined over the finite field $\mathbb{F}_q$ with endomorphism algebra being a field. We prove that the class within this set whose varieties have maximal number of rational points is unique, for any prime even power $q$ big enough and verifying mild conditions. We describe its Weil polynomial and we prove that the class is ordinary and cyclic outside the primes dividing an integer that only depends on $g$. In dimension $3$, we prove that the class is ordinary and cyclic and give explicitly its Weil polynomial, …
Classification of the rank of a certain family of elliptic curves
本文对特定类椭圆曲线的秩进行了完整分类，根据素数参数 $p$ 的余数类，确定其秩为 1 或 3，或 0。
Abelian varieties over finite fields and their groups of rational points
本文研究有限域上自同态环交换的阿贝尔簇的有理点群，证明其结构由自同态环决定，并刻画无平方同源类及对偶性质。
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…
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…
Let X0(N) denote the modular curve classifying elliptic curves with a cyclic N-isogeny, A0(N) its group of algebraic autmorphisms and B0(N) the subgroup of automorphisms coming from matrices acting on the upper halfplane. In a well-known paper, Kenku and Momose showed that A0(N) and B0(N) are equal (all automorphisms come from matrix action) when X0(N) has genus at least 2, except for N = 37 and 63. However, there is a mistake in their analysis of the N = 108 case. In the style of Kenku and Momose, we show that B0(108) is of index 2 in A0(108) and construct an explicit new automorphism of orde…
In this paper, we construct an infinite family of elliptic curves whose rank is exactly two and the torsion subgroup is a cyclic group of order two or three, under the parity conjecture.
By reformulating and extending results of Elkies, we prove some results on $\mathbb Q$-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees~$\ell$ which an elliptic curve without CM may have are those degrees which are already possible over~$\mathbb Q$ itself (in particular, $\ell\le37$), and we show the existence of a bound on the degrees of cyclic isogenies between $\mathbb Q$-curves depending only on the degree of the field. We also prove that the only possible torsion groups of $\mathbb Q$-curves over number fields of degree not divisible …
Cyclic Projective Orbits on Rational Normal Curves and MDS Codes
研究循环算子\(A\)、循环向量\(z\)的Krylov码的MDS轨道段与有理正规曲线的关系，证明特定共轭条件下二者的联系，给出有限域和代数闭域上相关分类及性质，还涉及多项式数量公式、轨迹定义及比例结果。
Conjectures About Cyclic Numbers: Resolutions and Counterexamples
本文系统解决了Cohen关于循环数的22个猜想（证16否6），涵盖渐近、间隙、比值及结构等结果，并完全解决了OEIS中滑动平均为斐波那契数的序列问题，证明Fried猜想2。
As the size of data storing arrays of disks grows, it becomes vital to protect data against double disk failures. A popular method of protection is via the Reed-Solomon (RS) code with two parity words. In the present paper we construct alternative examples of linear block codes protecting against two erasures. Our construction is based on an abstract notion of cone. Concrete cones are constructed via matrix representations of cyclic groups of prime order. In particular, this construction produces EVENODD code. Interesting conditions on the prime number arise in our analysis of these codes. At …
The stacky Batyrev-Manin conjecture and modular curves
本文研究了在有理数域上，模曲线X₀(N)上的叠层Batyrev-Manin猜想成立的条件，并给出了X₀(N)作为平方根栈覆盖叠层曲线的明确描述。
Rank distribution in cubic twist families of elliptic curves
本文研究椭圆曲线三次扭族的秩分布，通过伽罗瓦上同调技术证明存在无限多无立方因子扭参数使对应扭曲线的3-Selmer群为零、秩为0。
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