摘要翻译：
设~$a$是与新形式$F$相关联的$J_0(N)$的商，这样$a$(at$S=1$)的特殊$L$-值不为零。我们给出了一个特殊的$l$-值与实际周期$a$的比值的公式，它将这个比值表示为有理数。我们从这个公式的分子中提取一个整数因子；这个因子一般是非平凡的，并且与$F$与正解析秩特征形的某些同余有关。我们使用可见性技巧来证明，在某些假设（包括Birch和Swinnerton-Dyer关于秩的猜想的第一部分）下，如果奇素数$q$除这个因子，那么$q$除Shafarevich-Tate群的阶或$a$的分量群的阶。假定$P$是一个奇素数，使得$P^2$不除$N$,$P$不除$a$的有理扭转子群的阶，并且$F$与$P$上的素理想同余于一个特征形，该特征形的关联阿贝尔变型具有正的Mordell-Weil秩。然后我们证明$P$除上述因素；特别是，$P$除以特殊的$L$-值与实际期间$A$之比的分子。这两个结果都与Birch和Swinnerton-Dyer猜想的第二部分所暗示的一样，从而为该猜想提供了理论依据。
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英文标题：
《A visible factor of the special L-value》
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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~$A$ be a quotient of $J_0(N)$ associated to a newform $f$ such that the special $L$-value of $A$ (at $s=1$) is non-zero. We give a formula for the ratio of the special $L$-value to the real period of $A$ that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of $f$ with eigenforms of positive analytic rank. We use the techniques of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime $q$ divides this factor, then $q$ divides either the order of the Shafarevich-Tate group or the order of a component group of $A$. Suppose $p$ is an odd prime such that $p^2$ does not divide $N$, $p$ does not divide the order of the rational torsion subgroup of $A$, and $f$ is congruent modulo a prime ideal over $p$ to an eigenform whose associated abelian variety has positive Mordell-Weil rank. Then we show that $p$ divides the factor mentioned above; in particular, $p$ divides the numerator of the ratio of the special $L$-value to the real period of $A$. Both of these results are as implied by the second part of the Birch and Swinnerton-Dyer conjecture, and thus provide theoretical evidence towards the conjecture.
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PDF链接：
https://arxiv.org/pdf/0810.2477