Mathematical Sciences: Presidential Young Investigator

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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.