椭圆曲线的算术性质与Iwasawa理论

本项目的核心研究论题是BSD猜想与椭圆曲线的Iwasawa理论。.本项目拟推广K. Rubin关于虚二次域上带复乘椭圆曲线的工作，对于更为一般的Gross’s Q-curve研究其算术性质;.同时研究小素数特别是p=2时的椭圆曲线Iwasawa理论，研究2-Selmer群与L-函数及Tamagawa数之间的联系。.本项目研究椭圆曲线对应于分圆扩张时p-adic L-函数的mu-不变量。探索带复乘椭圆曲线的本原二维p-adic L-函数.与分圆p-adic L-函数之间的联系，进而研究非交换Iwasawa理论中Mazur's Torsion 猜想。.本项目研究bad reduction处的p-adic Gross-Zagier公式，以及同余数问题。

The main topics of this project are the BSD conjecture and Iwasawa theory for elliptic curves with complex multiplication..We want to generalize the work of K. Rubin on the arithmetic of CM elliptic curves defined on imaginary quadratic fields,.and study the arithmetic of Gross's Q-curves..We will work on the Iwasawa theory for p=2, and study the relation between 2-Selmer groups, .L-functions and the Tamagawa numbers..Furthermore, we wil work on the mu-invariant of p-adic L-functions of elliptic curves for the cyclotomic extension .and study the relation between the primitive two-variable p-adic L-functions and cyclotomic p-adic L-functions,.so we can get useful information about the Mazur's torsion conjecture in non-commutative Iwasawa theory..We will study the p-adic Gross-Zagier formula at primes of bad reduction, and the congruent number problem.