Mathematical Sciences: Presidential Young Investigator

本文主要研究具有复乘运算的椭圆曲线上的算术问题，特别是与Birch和Swinnerton-Dyer猜想、岩泽理论和p-进L函数有关的问题。文中提到了三个具体问题。首先证明了不变量SHA是有限的，并证明了它的猜想阶。第二个是研究Birch猜想和Swinnerton-Dyer猜想的p-进类似。第三个是确定复乘法域的阿贝尔扩张中的某些特殊单位，因为这很可能使人们能够将先前的结果推广到具有复乘的高维阿贝尔变种。

This research is primarily concerned with arithmetic on elliptic curves with complex multiplication, especially questions related to the Birch and Swinnerton-Dyer conjecture, Iwasawa theory and p-adic L-functions. Three specific problems are mentioned. The first is to show that the invariant Sha is finite and to give evidence for its conjectured order. The second is to study the p-adic analogues of the Birch and Swinnerton-Dyer conjecture. The third is to determine certain special units in abelian extensions of fields of complex multiplication since these would very likely enable one to extend the earlier results to higher dimensional abelian varieties with complex multiplication.