International Congress of Mathematicians

ICM 2002 • Vol. II • 149-162Equivariant Bloch-Kato Conjecture andNon-abelian Iwasawa Main ConjectureA. Huber* G. Kings 1 'AbstractIn this talk we explain the relation between the (equivariant) Bloch-Katoconjecture for special values of L-functions and the Main Conjecture of (nonabelian)Iwasawa theory. On the way we will discuss briefly the case of Dirichletcharacters in the abelian case. We will also discuss how "twisting" in thenon-abelian case would allow to reduce the general conjecture to the caseof number fields. This is one the main motivations for a non-abelian MainConjecture.2000 Mathematics Subject Classification: 11G40, 11R23, 19B28.Keywords and Phrases: Iwasawa theory, L-function, Motive.1. IntroductionThe class number formula expresses the leading coefficient of a Dedekind-(-function of a number field F in terms of arithmetic invariants of F:CF(0)* =JALWF(h the class number, Rp the regulator, wp the number of roots of unity in F). Byworkof Lichtenbaum, Bloch, Beilinson, and Kato among others, the class numberformula has been generalized to other F-functions of varieties (or even motives)culminating in the Tamagawa number conjecture by Bloch and Kato.Iwasawa, on the other hand, initiated the study of the growth of the classnumbers in towers of number fields. His decisive idea was to consider the classgroup of the tower as a module under the completed group ring of the Galois groupof the tower. From his work evolved the "Main Conjecture" describing this growthin terms of the p-adic F-function.*Math. Institut, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany. E-mail:huber@mathematik.uni-leipzig.detNWF I-Mathematik, Universität Regensburg, 93040 Regensburg, Germany. E-mail:guido.kings@mathematik.uni-regensburg.de