Massey products for elliptic curves of rank 1
Minhyong Kim
对于解析秩为1的Q上的椭圆曲线，利用二级Selmer簇和二次上同调积，得到了p-adic点集内整体积分点的显式解析定义方程.
For an elliptic curve over Q of analytic rank 1, we use the level-two Selmer variety and secondary cohomology products to find explicit analytic defining equations for global integral points inside the set of p-adic points.
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Universal triple Massey products on elliptic curves and Hecke's indefinite theta series
推广数学AG/0003076，我们用Hecke不定θ级数表示椭圆曲线上的线丛之间的泛三重Massey积。我们表明，所有Hecke的不定θ系列出现在这种方式。
Massey products and elliptic curves
本文研究完美域上绝对不可约光滑射影曲线中系数为 $\mathbb{Z}/\ell$ 的高阶 Massey 积消失问题，并完整刻画椭圆曲线上三重 Massey 积非消失的条件。
Massey products in the étale cohomology of number fields
本文研究了数字段整环的étale上同调中3重Massey乘积的公式，发现虚二次域的类群具有无限p-类域塔的例子，并推翻了McLeman的(3,3)猜想。
The Kernel Unipotent Conjecture and the vanishing of Massey products for odd rigid fields
伽罗瓦论中的一个主要难题是可实现为域的绝对伽罗瓦群的投射有限群的特征。最近提出了$n\geq 3$的核$n$ -幂幺猜想和消失$n$ -Massey猜想.这些结构是在过去的四十年里，作为拓扑方法应用于伽罗瓦上同调的副产品而发展起来的。我们证明了这两个定理对奇刚性场都成立。这是第一种情况下，一个显着的家庭的领域，其中两个认证，除了字段的伽罗瓦群的$p$ -最大的扩展是自由亲$p$ -组。我们还证明了秩为2的Demushkin群的核幂幺猜想，并建立了一些进一步的相关结果。
Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks …
Elliptic K3 surfaces with geometric Mordell-Weil rank 15
证明了椭圆曲面y^2=x^3+2（t^8+14t^4+1）x+4t^2（t^8+6t^4+1）的几何Mordell-Weil秩为15.这完成了Kuwata的列表，他给出了具有几何Mordell-Weil秩0，1，.的椭圆K3-曲面的明确例子，十四十六十七十八
Given a rational elliptic surface over a number field, we study the collection of fibers whose Mordell--Weil rank is greater than the generic rank. We give conditions on the singular fibers to assure that the collection of fibers for which the rank jumps of at least 3 is not thin.
On the rank of the fibers of elliptic K3 surfaces
设$X$是定义在数域$k$上的具有两个不同Jacobian椭圆纤维化$π_i$，$i=1,2$的椭圆K3曲面.证明了存在一条椭圆曲线$C\subset X$，使得在$C$基扩张后，$X$在$k$上的类属秩严格大于$X$在$k$上的类属秩.此外，如果$π_j$的类属秩为正，则存在无穷多个秩至少为$π_i$的类属秩加1的$π_i$（$j\neq i$）纤维。
Rank statistics for a family of elliptic curves over a function field
我们证明了由D。当参数$d \to\infty$时，Ulmer趋于无穷大。这也许是出乎意料的，因为A. Brumer，正特征函数域上所有椭圆曲线的平均秩渐近上界为2.3。
Elliptic curves with large rank over function fields
本文给出了\Bbb F_p（t）上Mordell-Weil群具有任意大秩的显式椭圆曲线.我们的方法是证明这些曲线的Birch和Swinnerton-Dyer猜想（或者更确切地说是相关椭圆曲面的Tate猜想），然后使用zeta函数来确定秩。与Shafarevitch和Tate的早期例子相反，我们的曲线不是等平凡的。渐近地，这些曲线对于它们的导体具有最大秩。基于这一事实，我们对数域上的椭圆曲线的秩的增长作了一个猜想。
On the rank of the fibres of rational elliptic surfaces
本文研究定义在数域$k$上的椭圆曲面$π: \mathcal{E}\rightarrow \mathbb{P}^1$，研究了$k$上特殊纤维的秩与$k(\mathbb{P}^1)$上一般纤维的秩的比较问题.我们证明，对于一个大类的理性椭圆曲面，存在无穷多个纤维的秩至少等于一般秩加2。
Given an elliptic surface over a number field, we study the collection of fibres whose Mordell-Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree $1$.
Moderate rank jumps on rational elliptic surfaces via construction of conics
本文通过构造圆锥方法，在有理椭圆面中实现秩的适度跳跃，研究具有高秩纤维的椭圆面家族。
Ordinary elliptic curves of high rank over $\bar F_p(x)$ with constant j-invariant II
证明了对所有奇素数$p$，存在$\bar{\mathbb{F}}_p(x)$上的正常椭圆曲线，它具有任意高的秩和常数$j$ -不变量.这表明，特别是有椭圆曲线与任意高的秩在这些领域，相应的椭圆表面是不是超奇异的。这个结果是由一个定理得出的，该定理指出，对于所有奇素数$p$和$\ell$，在亏格为$(\ell-1)/2$的$\bar{\mathbb{F}}_p$上存在一条超椭圆曲线，其雅可比矩阵与一条普通椭圆曲线的幂同构。
On the Rank of the Elliptic Curve y^2=x(x-p)(x-2)
定义在\Q上的椭圆曲线E是一个代数簇，它形成一个生成的阿贝尔群，并且结构定理意味着E = \Z^r + \Z_{tors}，对于某个r \geq 0;这个值r被称为E的秩。曲线的秩分类是椭圆曲线研究中的经典问题。本文利用2-下降法计算了两类椭圆曲线的秩，其中E由E：y^2 = x（x-p）（x-2）给出，p，p-2为孪生素数.
The rank and generators of Kihara's elliptic curve with torsion $\mathbb{Z}/4\mathbb{Z}$ over $\mathbb{Q}(t)$
对于Kihara发现的$\mathbb{Q}(t)$上的椭圆曲线$E$，在挠群$\mathbb{Z}/4\mathbb{Z}$和秩$\geq 5$上，利用适当的内射特化，我们精确地确定了$E(\mathbb{Q}(t))$的秩和生成元.
Ranks of elliptic curves with prescribed torsion over number fields
研究了2、3、4阶数域上椭圆曲线的Mordell-Weil群的结构。证明了若$T$是群，则二次域上所有具有扭子群$T$的椭圆曲线的类要么为空，要么包含秩为0的曲线和正秩的曲线.我们证明了一个类似的，但稍弱的三次和四次领域的结果。另一方面，我们找到一个群$T$和一个四次域$K$，使得在具有挠子群$T$的$K$上的椭圆曲线中，存在正秩曲线，但不存在秩~0的曲线。我们发现的例子椭圆曲线的正秩和给定的挠在许多以前未知的情况下。我们还证明了所有的椭圆曲线在二次域上的一个点的阶数为13或18和所有的椭圆曲线在四次域上的一个点的阶数为22是同构的伽罗瓦共轭之一，并通过一个现象，我们称之为\emph{false complex multiplication}，有偶数秩。最后，我们讨论了与有限域上的椭圆曲线的关系以及在整数分解中的应用。
Constructing Elliptic Curves over $\mathbb{Q}(T)$ with Moderate Rank
给出了$\mathbb{Q}(T)$上中秩椭圆曲线的几种新构造.特别是，我们构造无穷多个有理椭圆曲面（不属于维尔斯特拉斯形式）秩6以上$\mathbb{Q}$使用多项式的次数为2 $T$。虽然我们的方法生成线性无关点，但我们能够证明秩正好是6 \emph{without}，必须验证点是独立的。该方法具有推广性;然而，高阶曲面不是有理的，我们需要检查构造的点是线性无关的。
We prove results on the Mordell--Weil rank of elliptic curves $y^2=x(x-αa^2)(x-βb^2)$ parametrized by binary quadratic forms $αa^2+βb^2=γc^2$. We express our explicit lower bounds over number fields and offer a detailed description of the corresponding Mordell-Weil group structure in the function field case.
We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa's inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers.
Ranks of elliptic curves over function fields
我们提出的实验证据来支持广泛持有的信念，一半的所有椭圆曲线有无穷多个有理点。用于收集这一证据的方法是一个完善的算法，由于作者是基于刚性和结晶上同调。
The average Mordell-Weil rank of elliptic surfaces over number fields
该研究证明了椭圆面在数域上平均Mordell-Weil秩的猜想，并展示了稀疏集性质及类似结果。
Hodge theory and the Mordell-Weil rank of elliptic curves over extensions of function fields
利用Hodge理论证明了函数域上椭圆曲线的Mordell-Weil群的秩的一个新上界，该上界是对基域进行正则几何伽罗瓦扩张后得到的，改进了Silverman和Ellenberg以前的结果，当基域具有特征零点并且椭圆曲线导体的支撑和扩张的分歧因子的支撑不相交时。
Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples
研究了由勾股三元组$(a,b,c)$参数化的椭圆曲线族$y^2=x(x-a^2)(x-b^2)$.证明了对于一般三元组，$\mathbb{Q}$上的Mordell-Weil群的秩的下界为1，对于某些显式给定的无限族，秩为2.对每个家庭，我们附加一个椭圆表面纤维的投影线。我们表明，下界的秩是最佳的，在这个意义上说，这样的椭圆形表面的每一个通用的纤维，其相应的Mordell-Weil组的功能领域$\mathbb{Q}(t)$有秩1或2，分别。为了证明这一点，我们计算的特征多项式的弗罗贝尼乌斯自同构作用于第二个$\ell$ -进上同调群附加到椭圆形表面的Kodaira尺寸0和1。
Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava-Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$ ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$ ordered by height in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon observed in these databases is that …
Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves
本研究将有理数域上椭圆曲线的Sato-Tate猜想与murmuration现象从阿贝尔Hasse-Weil Zeta函数推广至其p约化的非阿贝尔高秩Zeta函数，证明二者在该场景下同样成立。
Three lectures on elliptic surfaces and curves of high rank
在过去的两年中，我们已经改进了几个（Mordell-Weil）秩记录的椭圆曲线在Q和非常数椭圆曲线在Q（t）。例如，我们发现了曲线E/Q在E（Q）中有28个独立点P_i的第一个例子（之前的记录是24，由R.Martin和W.McMillen在2000年发现），以及Q上的曲线的第一个例子，其Mordell-Weil群同构于（Z/2 Z）x Z ^[18]（之前的曲线的秩记录是15，由Dujella在2002年发现）。在这些讲座中，我们给出了导致这些新记录和相关应用的一些背景、理论和计算工具。一.背景和概述：定理的Mordell（-Weil）和Mazur;秩问题;的方法Neron-Shioda和梅斯特;椭圆曲面和Neron专业化;领域以外的Q。II椭圆曲面和K3曲面：Mordell-Weil和Neron-Severi群;高Neron-Severi秩的K3曲面及其模; Mordell-Weil秩为17的Q上的椭圆K3曲面。高秩K_3曲面及其模的其他应用III计算问题、技术和结果：Niemeier格的切片;寻找和转换高阶K3曲面的模型;寻找良好的专业化。椭圆曲线的新秩记录摘要。
Ordinary elliptic curves of high rank over $\bar F_p(x)$ with constant j-invariant
证明了在Artin原根猜想的假设下，对所有素数p，存在$\bar F_p(x)$上任意高秩且具有常数j不变的正常椭圆曲线.对于奇素数p，这个结果来自一个定理，该定理指出，只要p是（Z/ell Z）^*/<-1>（ell是奇素数）的生成元，就存在$\bar F_p$上的超椭圆曲线，其雅可比矩阵与一个正常椭圆曲线的幂同构。
Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves II: Beyond Riemann Hypothesis
本文提出不依赖黎曼假设的新方法，利用更强渐近界建立椭圆曲线高阶Zeta函数的Sato-Tate定律并表述Murmuration现象。
We discuss the distribution of Mordell--Weil ranks of the family of elliptic curves $y^2=(x+αf^2)(x+βb g^2)(x+γh^2)$ where $f,g,h$ are coprime polynomials that parametrize the projective smooth conic $a^2+b^2=c^2$ and $α,β,γ$ are elements from $\overline{\mathbb{Q}}$. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.
The rank of elliptic surfaces in unramified abelian towers
设E -> C是定义在数域K上的椭圆曲面。对于定义在K上的每个有限覆盖C' --> C，设E' --> C'是拉回.在C' -> C是非分歧阿贝尔覆盖的情况下，在Tate猜想对E'/K成立的假设下，给出了E '（C'/K）的秩的一个强上界.当C是椭圆曲线，映射C '= C --> C是n乘映射时，E'（C '/K）的秩对于每个e > 0都是O（n^e），这可以与O（n^2）的初等界相比较。
The average rank of elliptic $n$-folds
本文研究有限域上高维代数簇上椭圆曲线的秩分布，证明维数至少为3时正秩椭圆曲线密度为零，曲面情形密度至多为 $1-ζ_V(3)^{-1}$。
The $p$-rank stratification of the moduli space of double covers of a fixed elliptic curve
本文研究了固定椭圆曲线双覆盖的模空间中 $p$-阶数分层的结构，证明了闭层的维数与预期一致，并展示了存在光滑双覆盖可取所有可能的 $p$-阶数。
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 …
In the present paper, we introduce some new families of elliptic curves with positive rank arrising from Pythagorean triples.
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。
In this paper, $p$ and $q$ are two different odd primes. First, We construct the congruent elliptic curves corresponding to $p$, $2p$, $pq$, and $2pq,$ then, in the cases of congruent numbers, we determine the rank of the corresponding congruent elliptic curves.
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。
Infinite family of elliptic curves of rank at least 4
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。
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 …
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