International Congress of Mathematicians

168 Kazuya KatoHowever in the middle of C and the p-adic world,(a) there is a 1 dimensional FT-vector space AK (M) constructed by the Bettirealization and the de Rham realization of M, and F'-groups (or motivic cohomologygroups) associated to M.Yet oo be an Archimedean place of K. Then(b) there is an isomorphismA K (M)® K K 00^Kvoconstructed by Hodge theory and F'-theory.Let w be a place of K lying over p, let M w be the representation of Gal(Q/Q)over K w associated to M, and let F be a Gal(Q/Q)-stable OK W -lattice in M w .Then(c) there is an isomorphismA K (M) ®K K„ -+ det^FF etjC (Z[-], j,M w )— p=detö Kw RY et , c (Z[-],;j*T) ® 0liw K WPwhere j : Spec(Q) —¥ Spec(Z[|]), constructed by p-adic Hodge theory and F'-theory.See [FP] how to construct (a)-(c) (constructions require some conjectures).The part (2.1.2) of the conjecture is:(d) there exists a F'-basis ((M) of AK(M) (called the rational zeta elementassociated to M), which is sent to lim s _s. 0 s~ e L(M, s) under the isomorphism(b) where e is the order of L(M,s) at s = 0, and to £(Z[^], j»T,OK W )in det^FF etjC (Z[ì], j*M w ) under the isomorphism (c).The existence of ((M) having the relation with lim s _s. 0 s~ e L(M, s) was conjecturedby Beilinson [Be].How zeta functions and p-adic zeta elements are related is illustrated in thefollowing diagram.zeta functions side (Betti) < — (de Rham)regulatorp-adic Hodge theory(F'-theory) • (etale) p-adic zeta elements side.Chern classWe have the following picture.?automorphic rep