Mathematical Sciences: Presidential Young Investigator

本文主要研究了椭圆曲线的复乘法运算，特别是与Birch和Swinnerton-Dyer猜想、Iwasawa理论和p进l函数有关的问题。文中提到了三个具体问题。第一个是证明不变量Sha是有限的，并为它的猜想次序提供证据。第二步是研究Birch猜想和Swinnerton-Dyer猜想的p进类比。第三个是确定复乘法域的阿贝尔扩展中的某些特殊单位，因为这些很可能使人们能够将先前的结果扩展到具有复乘法的高维阿贝尔变体。

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.