Infinite family of elliptic curves of rank at least 4
Bartosz Naskrecki
我们研究了椭圆曲线$E_t$：$y^2+txy=x^3+tx^2-x+1$的$\mathbb{Q}$ -秩，其中$t$是有理参数.我们证明了对于无穷多个$t$值，$E_t(\mathbb{Q})$的秩至少为4。
We investigate $\mathbb{Q}$-ranks of the elliptic curve $E_t$: $y^2+txy=x^3+tx^2-x+1$ where $t$ is a rational parameter. We prove that for infinitely many values of $t$ the rank of $E_t(\mathbb{Q})$ is at least 4.
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While there has been considerable interest in the problem of finding elliptic curves of high rank over $\mathbb{Q}$, very few parametrized families of elliptic curves of generic rank $\geq 8$ have been published. In this paper we use solutions of certain symmetric diophantine systems to construct several parametrized families of elliptic curves with their generic ranks ranging from at least 8 to at least 12. Specific numerical values of the parameters yield elliptic curves with quite large coefficients and we could therefore determine the precise rank only in a few cases where the rank of the …
Positive-rank elliptic curves arising pythagorean triples
本文引入了一类新的正秩椭圆曲线，它们都是由勾股三元组组成的.
A positive proportion of elliptic curves over $\mathbb{Q}$ have rank one
证明了当$\mathbb{Q}$上的所有椭圆曲线按朴素高度排序时，正比例的代数秩和解析秩都为1.因此，椭圆曲线的平均秩和平均解析秩都是严格正的。
We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.
We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.
Ranks of elliptic curves in families of quadratic twists
证明了椭圆曲线二次扭的秩的无界性等价于某些无穷级数的发散性。
In this paper, under GRH for elliptic $L$-functions, we give an upper bound for the probability for an elliptic curve with analytic rank $\leq a$ for $a \geq 11$, and also give an upper bound of $n$-th moments of analytic ranks of elliptic curves. These are applications of counting elliptic curves with local conditions, for example, having good reduction at $p$.
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.
Upper bounds for moments of analytic ranks of elliptic curves over number fields
研究数域上椭圆曲线解析秩矩的上界，通过证明条件上界及解析秩大于给定界的椭圆曲线密度的条件界来进行研究。
On the rank of certain parametrized elliptic curves
本文研究了\Q上由方程E_{p}：Y^2=（X-p）^3+X^3+（X+p）^3给出的椭圆曲线族，其中p是素数.证明了椭圆曲线的最大秩至多为3，并给出了秩（E_p（\Q））=0或秩（E_p（\Q））=1或秩（E_p（\Q））>=2的条件.证明了该族中存在秩为0，1，2，3的椭圆曲线.
A Simple Method for High-Rank Families of Elliptic Curves with Specified Torsion
我们描述了一个简单而有效的方法，用于推导具有高秩的所有成员具有相同挠子群结构的椭圆曲线族。
On The Rank Of Congruent Elliptic Curves
本文中，$p$和$q$是两个不同的奇素数.首先构造了对应于$p$、$2p$、$pq$和$2pq,$的全等椭圆曲线，然后在全等数的情况下，确定了相应的全等椭圆曲线的秩.
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.
On positive proportion of rank zero twists of elliptic curves over Q
推广\cite{dab2}的思想，利用2-下降法，给出了Q上的三类椭圆曲线，使得这类椭圆曲线的素数扭转的正比例同时为秩零.
Elliptic curves of rank one over number fields
该研究证明了对任意数域 $K$，均存在无穷多条秩恰好为 1 的椭圆曲线，解决了该领域的一个基本存在性问题。
Elliptic curves of unbounded rank and Chebyshev's bias
建立了$\mathbb Q$上椭圆曲线解析秩的定量无界性与高偏椭圆曲线素数族存在性之间的条件等价性.我们表明，有条件的黎曼假设和假设上的多重性的零$L(E,s)$，大的分析秩转化为一个极端的切比雪夫偏见。相反，我们表明，在一定的线性独立性假设下的零点$L(E,s)$，如果高度偏置的椭圆曲线素数种族确实存在，然后黎曼假设成立无限多个椭圆曲线$L$ -功能，并存在椭圆曲线的任意大的秩。
Integral points on a certain family of elliptic curves
利用Thue-Siegel方法得到了一类四次Thue不等式的本原积分解个数的上界.这将提供一个整数点的数量上界的一个家庭的椭圆曲线与j不变等于1728。
Rank of elliptic curves and class groups of real quadratic fields
本文研究了椭圆曲线的秩与实二次域类群非平凡性之间的联系。
An elliptic K3 surface X/Q(t) with Mordell-Weil rank 17, I: Formulas for X and base changes of rank 18 and 19
本文给出Q(t)上Mordell-Weil秩17的椭圆K3曲面X的公式，证明其二次基变换可得到秩18的椭圆曲面，一对复合后得秩19的椭圆曲面，对应无穷多条Q上高秩椭圆曲线。
Rank one elliptic curves and rank stability
本文证明对任意数域的二次扩张，存在无穷多个秩为一的椭圆曲线，推广了相关定理，并通过特化非平凡秩一椭圆曲线族并计算所有秩来获得结果。
On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve
本文推广了Swinnerton-Dyer关于给定椭圆曲线具有特定秩的2-Selmer群的扭数密度的工作.
On a Family Of Elliptic Curves With Positive Rank arising from Pythagorean Triples
本文的目的是引入一个新的椭圆曲线族的形式$y^2=x(x-a^2)(x-b^2)$，具有正秩。我们首先生成一个列表的勾股三元组$(a,b,c)$，然后构造这个家庭的椭圆曲线。证明了这个新族具有正秩，并求出了它们的秩的上界。
We conjecture that, for a fixed prime $p$, rational elliptic curves with higher rank tend to have more points mod $p$. We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.
Let $E$ be an elliptic curve over $\mathbb{Q}$. Then, we show that the average analytic rank of $E$ over cyclic extensions of degree $l$ over $\mathbb{Q}$ with $l$ a prime not equal to $2$, is at most $2+r_{\mathbb{Q}}(E)$, where $r_{\mathbb{Q}}(E)$ is the analytic rank of the elliptic curve $E$ over $\mathbb{Q}$. This bound is independent of the degree $l$ Also, we also obtain some average analytic rank results over $S_d$-fields.
Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization
本文给出了几种构造小次数域上具有大扭群和正秩的椭圆曲线的方法。我们还讨论了这种曲线在椭圆曲线分解方法（ECM）中的潜在应用。
We give upper bounds for the number of rational elliptic surfaces in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan's conjecture (arXiv:2009.08622v2) in the case $m,n\leq2$.
Based on an equation for the rank of an elliptic surface over $\mathbb{Q}$ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank $0$ when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain $L$-functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental …
Infinite rank of elliptic curves over $\mathbf{Q}^{\ab}$
如果$E$是定义在二次域$K$上的椭圆曲线，并且$E$的$j$不变量不是0或1728，则$E(\mathbf{Q}^{\ab})$具有无限秩。如果$E$是Legendre形式的椭圆曲线$y^2 = x(x-1)(x-λ)$，其中$\mathbf{Q}(λ)$是三次域，则$E(K \mathbf{Q}^{\ab})$具有无限秩。若$λ\in K$有一个4次极小多项式$P(x)$，且$v^2 = P(u)$是$\bbq$上的正秩椭圆曲线，我们证明了$y^2 = x(x-1)(x-λ)$在$K\bbq^{\ab}$上有无穷秩.
100% of elliptic curves with a marked point have positive rank
本文为带标记点的椭圆曲线具有正秩提供了替代证明，并将该结论扩展至特征不为2或3的global fields。
Conditionally bounding analytic ranks of elliptic curves
在Birch猜想和Swinnerton-Dyer猜想以及广义黎曼假设的假设下，给出了一种确定椭圆曲线秩的方法.作为一个例子，我们计算，根据这些结构，确切的上界曲线，已知有秩至少为20，21，22，23，和24一样大。对于秩至少为28的已知曲线，我们得到的界为30。
Diophantine triples and construction of high-rank elliptic curves
利用丢番图m元组理论，即任意两个不同元素加1的乘积是完满平方的集合，我们构造了Q（t）上秩至少为4的具有三个非平凡扭点的椭圆曲线.通过特殊化，得到了Q上挠群Z/2 Z * Z/2 Z秩为7的椭圆曲线的一个例子.
Let $E$ be an elliptic curve with good reduction at a fixed odd prime $p$ and $K$ an imaginary quadratic field where $p$ splits. We give a growth estimate for the Mordell-Weil rank of $E$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$.
Sums of two biquadrates and elliptic curves of rank $\geq 4$
如果整数$n$以两种不同的方式写成两个双二次曲线之和，则椭圆曲线$y^2=x^3-nx$具有秩$\geq 3$。此外，如果$n$是奇数，奇偶猜想是真的，那么它有偶数秩$\geq 4$。最后，给出了秩为4，5，6，7，8和10的一些例子.
The average analytic rank of elliptic curves with prescribed level structure
本文在Hasse--Weil猜想与广义黎曼假设下，对模叠可由射影直线表示的指定级结构椭圆曲线，给出了其平均解析秩的上界。
A new family of elliptic curves with positive ranks arising from the Heron triangles
本文的目的是引入一类新的正秩椭圆曲线。这些椭圆曲线是用一定的有理数，即a、b和c作为具有有理数面积$k$的Heron三角形的边来构造的。证明了这个族的扭群是$\frac{\Bbb{Z}}{2\Bbb{Z}}\times \frac{\Bbb{Z}}{2\Bbb{Z}}$形式的，并且秩是正的.
The Parity of Analytic Ranks among Quadratic Twists of Elliptic Curves over Number Fields
椭圆曲线的解析秩的奇偶性由函数方程L（E，s）中的根数给出。在任意数域上固定一条椭圆曲线，并考虑它的二次扭曲族，很自然地会问这个族的平均解析秩是多少。这个数的下界由平均根数给出。本文研究了这类曲线族的根数，并给出了这类曲线族中偶数秩曲线所占比例的一个渐近公式。然后，我们的结果被用来支持一个猜想的平均解析秩在这个家庭的椭圆曲线。
High rank elliptic curves with prescribed torsion group over quadratic fields
定义在二次数域上的椭圆曲线的挠群有26种可能。我们给出了具有给定扭群的高秩椭圆曲线的例子，这些例子给出了大多数扭群的当前记录。特别地，我们证明了对于每一个可能的挠群，除了可能是Z/15 Z之外，存在一条在某个二次域上具有该挠群且秩>= 2的椭圆曲线。
Elliptic curves of high rank and the Riemann zeta function on the one line
我们描述了一些实验，显示之间的连接高秩椭圆曲线和黎曼ζ函数的一条线。我们还讨论了一对夫妇的统计涉及$L$ -功能的zeta功能的一条线上起着突出的作用。
The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion
本文给出了任意数域K上椭圆曲线的二次扭族的2-Selmer秩分布的一个新结果，该数域K上有一个二阶点，且在其二除域上没有定义循环4-等距线.我们证明了这样一条椭圆曲线的所有二次扭转中至少有一半具有任意大的2-Selmer秩，表明这样一条椭圆曲线的二次扭转族中的2-Selmer秩的分布不同于没有有理二挠或完全有理二挠的椭圆曲线的二次扭转族中的2-Selmer秩的分布.
A conditional determination of the average rank of elliptic curves
在一个比椭圆曲线$L$ -函数的黎曼假设稍强的假设下，我们证明了二次扭族中椭圆曲线的平均解析秩和平均代数秩都恰好是$\frac 12$ .作为推论，我们得到，在这最后一个假设下，伯奇和斯温纳顿-戴尔猜想几乎对我们族中的所有曲线都成立，并且这些曲线的渐近一半具有代数秩$0$，其余一半具有代数秩$1$。在所有椭圆曲线族中，我们也证明了一个类似的结果。解释我们的结果的一种方法是说，椭圆曲线$L$ -函数族中的非实零点对这个族中的平均秩有直接的影响。卡茨-萨尔纳克和杨的结果构成了证明的主要成分。
On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion
本文给出了关于在二除域上没有循环4-等距线的二阶单点椭圆曲线的二次扭族中2-Selmer秩分布的一个新结果。我们证明了这样一条椭圆曲线的所有二次扭转中至少有一半具有任意大的2-Selmer秩，表明这样一条椭圆曲线的二次扭转族中的2-Selmer秩的分布不同于没有有理二挠或完全有理二挠的椭圆曲线的二次扭转族中的2-Selmer秩的分布.
Ranks of elliptic curves over cyclic cubic, quartic, and sextic extensions
对于给定的群$G$和定义在数域$K$上的椭圆曲线$E$，本文讨论了寻找$K$的$G$ -扩张，使$E$在其上获得秩的问题.我证明了下面的定理，扩展了Fearnley，Kisilevsky和Kuwata的结果：设$n = 3,4,$或$6$。如果$K$包含其$n^{th}$ -单位根，则对于$K$上的任何椭圆曲线$E$，存在无穷多个$\mathbb{Z}/n\mathbb{Z}$ -$K$的扩展，其中$E$获得秩。
Elliptic curves with bounded ranks in function field towers
设$k$表示代数闭域。我们重新建设的作者的家庭椭圆曲线的有理函数域$k(t)$。结合组合分析与乌尔默的秩公式，我们证明了，对于所有的，但许多家庭的这些曲线，Mordell-Weil群在$k(t^{1/d})$的秩为零，因为$d$范围在正整数素数的特征$k$。
Quadratic Twists of Elliptic Curves with 3-Selmer Rank 1
1979年戈德菲尔德猜想的一个较弱的形式指出，对于每一条椭圆曲线$E/\mathbb{Q}$，它的二次扭曲$E^{(d)}$的秩为1。利用Galois上同调的工具，我们给出了E和d的准则，该准则使得E的二次扭曲的正比例具有3-Selmer秩1和整体根数-1。然后，我们给出了四个非同构的椭圆曲线的无限族$E_{m, n}$满足这些标准。在Birch和Swinnerton-Dyer猜想的秩部分的条件下，这证明了无限多条椭圆曲线的上述猜想。我们的椭圆曲线很容易明确地给出，我们精确地说明了使用哪些二次扭曲d。此外，我们的方法有可能被推广到其他数域上的椭圆曲线。
Rational Diophantine triples, i.e. rationals a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares, are often used in construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first su…
Let n be a positive integer and t a non-zero integer. We consider the elliptic curve over Q given by E : y 2 = x 3 + tx 2 -- n 2 (t + 3n 2)x + n 6. It is a special case of an elliptic surface studied recently by Bettin, David and Delaunay [2] and it generalizes Washington's family. The point (0, n 3) belongs to E(Q) and we obtain some results about its nondivisibility in E(Q). Our work extends to this two-parameter family of elliptic curves a previous study of Duquesne (mainly stated for n = 1 and t > 0).
We give an infinite family of congruent number elliptic curves, each with rank at least two, which are related to integral solutions of $m^2=n^2+nl+l^2$.
Ranks of elliptic curves in cubic extensions
对于数域K上的椭圆曲线E，证明了E的代数秩在K的无穷多个扩张中上升，这些扩张是通过邻接K的一个元素的一个立方根而得到的.作为一个例子，我们简要地讨论了E=X_1（11）在Q上的性质，以及这个结果与岩泽理论的关系。
A Family of Elliptic Curves With Rank $\geq5$
本文构造了一类秩为$\geq 5$的椭圆曲线.为了做到这一点，我们使用Heron公式来计算三元组$(A^2, B^2, C^2)$，它不一定是三角形的三条边。结果表明，作为椭圆曲线族的参数，这三个正整数$A$、$B$和$C$以及额外的参数$D$满足四次丢番图方程$A^4+D^4=2(B^4+D^4)$。
There are infinitely many elliptic curves over the rationals of rank 2
本文利用显式模型、2-下降法以及 Tao–Ziegler 定理，证明了有理数域上存在无穷多条秩恰好为2的椭圆曲线。
A note on Larsen's conjecture and ranks of elliptic curves
设E是定义在数域K上的椭圆曲线。MichaelLarsen证明了对Gal（\bar K/K）的任意非线性生成子群G，E的Mordell-Weil秩在G固定的数域上是无界的.我们证明了猜想持有K=Q的解析秩和p-无穷Selmer秩的E的每一个奇素数p。对于任意的E/K，我们表明，拉森猜想如下从标准的椭圆曲线的行列，如果K有一个真正的地方或E有非整数j-不变。
Given any positive integer n, it is well known that there always exist triangles with rational sides a, b and c such that the area of the triangle is n. Assuming finiteness of the Shafarevich-Tate group, we first construct a family of infinitely many Heronian elliptic curves of rank exactly 1 from Heron triangles of a certain type. We also explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.
$L$-derivatives of the fixed elliptic curve over rank-one imaginary quadratic fields
研究固定椭圆曲线在虚二次域上的L导数分布，揭示其统计性质及非小导数的存在性。
A family of monogenic $S_4$ quartic fields arising from elliptic curves
我们考虑椭圆曲线的部分挠场（由除多项式的根生成的场）。通过分析椭圆曲线的约简性质，应用Montes算法，得到了整数环的信息。特别地，对于某些单参数非CM椭圆曲线族的部分$3$ -挠场，我们描述了一个幂基。证明了由$T^4 - 6T^2 - αT - 3$给出的四次$S_4$场的单参数族对于$α\in \mathbb{Z}$使得$α\pm 8$是无平方的，是单演的.
We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2. We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending on the factorization and other simple properties of certain polynomials. Finally, we give sharp estimates for the ranks of the considered families and we present several applications, among which there are lists of rational points, generic families with maximal rank and generalizations of former results.
For a non-constant elliptic surface over $\mathbb{P}^1$ defined over $\mathbb{Q}$, it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is non-isotrivial one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann and Setzer. In this note we show that the Legendre elliptic surface has the desired property, conditional on the existence of infin…
A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over Q with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank >= 7, which are both the current records for that k…
On 2-Selmer ranks of quadratic twists of elliptic curves
研究了椭圆曲线的$2$ -Selmer秩.证明了对于任意数域$K$上的椭圆曲线$E$，如果$E$的二次扭转的2-Selmer秩的集合$A_E$包含一个整数$c$，则它包含所有大于$c$且与$c$具有相同奇偶性的整数.我们还找到了$A_E$上的充分条件，使得对于某些数目的$t_E$，$A_E$等于$\Z_{\ge t_E}$。当$E[2]$中的所有点都是有理数时，我们给出了$t_E$的一个上界.
On the infinitude of elliptic curves over a number field with prescribed small rank
该研究证明了在数域上存在无限多个具有指定小秩的椭圆曲线，通过专门化非平凡椭圆曲线并利用Kai的结果实现对坏素数的控制和秩的计算。
On the rank of quadratic twists of elliptic curvers over function fields
我们证明了一个给定的椭圆曲线$E/\Fp_q(C)$在一个函数域上的有限域，有秩$\geq 2$的二次扭曲的数量的定量上界，并为他们的平均秩。主要工具是结构和结果的卡茨和统一版本的Chebotarev密度定理品种在有限域。此外，我们有条件地推导出在某些情况下，导体的程度是无界的一个界。
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