椭圆曲线二次扭的p部分BSD猜想问题

The Birch and Swinnerton-Dyer conjecture is one of the seven Millennium Prize Problems, which is widely recognized as one of the most challenging mathematical problems. The conjecture relates arithmetic data associated to an elliptic curve E over a number field K to the behaviour of the Hasse--Weil L-function L(E,s) of E at s=1. More specifically, it is conjectured that the rank of the Abelian group E(K) of points of E is the order of the zero of L(E,s) at s=1, and the first non-zero coefficient in the Taylor expansion of L(E,s) at s=1 is given by more refined arithmetic data attached to E over K. It describes a link between the arithmetic property and the analytic property of Abelian varieties. Moreover, it is very meaningful for the research of modern number theory and arithmetic algebraic geometry. Recently, there has been some important progress on this direction, specifically for the p-part of the exact formula of the Birch--Swinnerton-Dyer (p is odd), Rubin, Skinner and Urban has made remarkable progress by Iwasawa theory. However, Iwasawa theory says nothing for the 2-part of the Birch--Swinnerton-Dyer conjecture until now. In this project, we aim to work on the 2-part of the Birch--Swinnerton-Dyer conjecture by modular symbols method, in particular, for the quadratic twists of elliptic curves with analytic rank zero. Then we plan to generalize the modular symbols method to apply on the p-part of the exact Birch--Swinnerton-Dyer formula. Moreover, we shall combine the above results to apply on the distribution of ranks in families of quadratic twists of elliptic curves by classical analytic methods. We remark here that the modular symbols method does not only could apply to the elliptic curves with complex multiplication, but also to the elliptic curves without complex multiplication. We plan that throughout this project, we could prove that the full Birch--Swinnerton-Dyer conjecture holds for large families of quadratic twists of elliptic curves with analytic rank zero, by combining the methods of Iwasawa theory and modular symbols.

BSD猜想，是当今数学界研究的热点问题，也是千禧年七大猜想之一。它描述了阿贝尔簇的算术性质与解析性质之间的联系，对于数论和算术代数几何的研究具有重要意义。近几年，关于BSD猜想的研究已有了诸多重要进展，其中关于p部分BSD猜想的精确表达式(这里p为奇素数)，Rubin, Skinner, Urban 等数学家运用岩泽理论已经给出了非常好的结果，但对于2部分BSD猜想，目前岩泽理论是无效的。本项目旨在引入模符号的方法，首先对解析秩为零时BSD猜想的精确形式的2部分展开深入研究，然后对p部分BSD猜想进行探讨，与此同时，将所得结果结合经典解析方法运用到椭圆曲线二次扭族中秩的分布问题。模符号的方法不局限于复乘椭圆曲线，对于不满足复乘性质的椭圆曲线仍然是有效的。计划通过本项目的实施，将模符号的方法与岩泽理论相结合，完整证明秩零椭圆曲线特定的二次扭族所对应的BSD猜想精确表达式成立。

BSD猜想，是当今数学界研究的热点问题，是千禧年七大猜想之一。它描述了阿贝尔簇的算术性质与解析性质之间的联系，对于数论和算术代数几何的研究具有重要意义。近几年，关于BSD猜想的研究已有了诸多重要进展，其中关于p部分BSD猜想的精确表达式(这里p为奇素数)，Rubin, Skinner, Urban, Wan 等数学家运用岩泽理论已经给出了非常好的结果，但对于2部分BSD猜想，应用岩泽理论比较复杂。本项目引入模符号主要对解析秩为零时BSD猜想的精确形式的2部分展开深入研究，并应用到BSD全猜想。模符号的方法不局限于复乘椭圆曲线，对于不满足复乘性质的椭圆曲线仍然是有效的。本项目最终给出了无穷多族非复乘椭圆曲线满足秩零BSD全猜想。