International Congress of Mathematicians

166 Kazuya Katowhere the transition maps of the inverse system are the trace maps. From this, wecan deduce that H c (X,J 7 ®o L s(Aj) is a finitely generated O^-module for any rn.Hence we have Q(N) ® A FF etjC (X, J 7 ®o L s(Aj) = 0 and this gives an identificationcanonical isomorphismNoteQ(A) ® A detX 1 RY et , c (X,T®o L t(A)) = Q(A). (1.3.3)Q(A) = Q(hjnO L [«]/(«" - 1)) D Q(0 L [u]) = L(u). (1.3.4)nBy a formal argument, we can prove the following (1.3.5) (1.3.6) which showrespectively.zeta function = zeta element,zeta value = zeta element,L(X/F q ,T L ,u) = aX,T®o L s(A),A) in Q(A). (1.3.5)If F C (X,Tp) = 0 for any rn, L(X/F q ,J 7 p,u)has no zero or pole at u = 1, andL(X/F q ,T L ,l) = aX,T,0 L ) in F. (1.3.6)2. Tamagawa number conjectureIn 2.1, we describe the generalized version of Tamagawa number conjecture.In 2.2 (resp. 2.3), we consider p-adic zeta elements associated to 1 (resp. 2) dimensionalp-adic representations of Gal(Q/Q), and their relations to (0.2) (resp.(0.3)).2.1. The conjecture. Let X be a scheme of finite type over Z[-]. For acomplex of sheaves J 7 on X for the etale topology, we define the compact supportversion FF etjC (X, J 7 ) of FF et (X, J 7 ) as the mapping fiber ofFF et (Z[-], RfsF) -+ FF et (R, RfiF) ® FF et (Q p ,PRfiF).where / : X —t Spec(Z[-]).It can be shown that for a commutative pro-p ring A and for a ctf A-complexT on X, RY et:C (X, J 7 ) is perfect.The following is a generalized version of the Tamagawa number conjecture[BK] (see [FP], [Pei], [Kai], [Ka 2 ]). In [BK], the idea of Tamagawa number ofmotives was important, but it does not appear explicitly in this version.Conjecture. To any triple (X, A, J 7 ) consisting of a scheme X of finite type overZ[ì], a commutative pro-p ring A, and a ctf A-complex on X, we can associate aA-basis ((X, T, A) ofA(X,T,A) = det^RTet^X, J 7 ),