108/12/18 蔡國榮 博士(國立台灣大學數學系)
Special Values of L-series and their Arithmetic Incarnations
Dr. Kwok-Wing Tsoi
(Department of Mathematics, National Taiwan University)
One of the most mysterious discoveries in modern number theory is the interplay between algebraic and analytic invariants of geometric objects. For instance, the Birch and Swinnerton-Dyer conjecture, one of the Millennium Prize Problems, relates the leading term of the Hasse-Weil L-series of an elliptic curve with the order of its Tate-Shafarevich group.
A systematic study of the web of problems of this sort is suggested by Iwasawa. In particular, under the philosophy of Iwasawa theory, analytic invariants have natural `incarnations’ as special elements in a relevant first etale cohomology group. In this talk, I will survey these ideas in the most classical setting of cyclotomic fields in which these special elements are given by explicit `cyclotomic units’.
If time permits, I will discuss a recent joint work with D. Burns and T. Sano that addresses a question of L. Washington regarding the structure of the group of units of cyclotomic integers and how our approach also finds applications in K-theory of integers of absolute abelian extensions.
日 期：108年12月18日(星期三) 16:10~17:00