摘要翻译：
设$E$是导体$N$的$\q$上的解析秩为一的最优椭圆曲线，即$E$的$L$-函数$L_E(s)$在$s=1$处消失为一阶。设$k$是一个二次虚域，其中除$n$的所有素数都分裂，使得$e$上的$l$-函数在$s=1$处消失。假定在同一导体$n$上有另一个最佳椭圆曲线，其Mordell-Weil秩大于1，且其相关的新形式与$e$模整数$r$相关的新形式全等。可见性理论随后表明，在某些附加假设下，$R$将Shafarevich-Tate组的$E$的顺序划分为$K$。我们证明，在一些类似的假设下，$R$将$E$的Shafarevich-Tate群的序划分为$K$。我们证明了在近似的假设下，$R$也将$E$上的Shafarevich-Tate群的Birch和Swinnerton-Dyer{em猜想}阶划分为$K$,这为分析秩一情形下Birch和Swinnerton-Dyer猜想的第二部分提供了新的理论证据。
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英文标题：
《Visibility and the Birch and Swinnerton-Dyer conjecture for analytic
  rank one》
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作者：
Amod Agashe
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Number Theory        数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ having analytic rank one, i.e., such that the $L$-function $L_E(s)$ of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ vanishes to order one at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ of the same conductor $N$ whose Mordell-Weil rank is greater than one and whose associated newform is congruent to the newform associated to $E$ modulo an integer $r$. The theory of visibility then shows that under certain additional hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ also divides the Birch and Swinnerton-Dyer {\em conjectural} order of the Shafarevich-Tate group of $E$ over $K$, which provides new theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank one case.
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PDF链接：
https://arxiv.org/pdf/0810.2487