International Congress of Mathematicians

Tamagawa Number Conjecture for zeta Values 165where F c is the etale cohomology with compact supports and tp q is the action ofthe q-th power morphism on X.In the case F = Q p = T, C,(X/F q ,u) = L(X/F q ,J 7 ,u).1.2. p-adic zeta elements in positive characteristic case. Determinantsappear in the theory of zeta functions as above, rather often. The regulator of anumber field, which appears in the class number formula, is a determinant. Suchrelation with determinant is well expressed by the notion of "determinant module".If F is a field, for an F-module V of dimension r, detfl(F) means the 1 dimensionalF-module A r R(V). For a bounded complex C of F-modules whose cohomologiesH m (C) are finite dimensional, detfl(C) means ® TOG z {detfl(F TO (G))}®^1^"*.This definition is generalized to the definition of an invertible F-module detp(C)associated to a perfect complex C of F-modules for a commutative ring F (see[KM]). det^j 1 (G) means the inverse of the invertible module detfl(G).By a pro-p ring, we mean a topological ring which is an inverse limit of finiterings whose orders are powers of p. Let A be a commutative pro-p ring. By a ctfA-complex on X, we mean a complex of A-sheaves on X for the etale topology withconstructible cohomology sheaves and with perfect stalks. For a ctf A-complex Ton X, RY et:C (X, T) ( c means with compact supports) is a perfect complex over A.For a commutative pro-p ring A and for a ctf A-complex J 7 on X, we define thep-adic zeta element ((X, T, A) which is a A-basis of det7 1 v FF etjC (X, T). Considerthe distinguished triangleFF et;C (X,^) -+ RY et , c (X® Fq F q ,F) ^ RY et , c (X ® Fq F q ,F). (1.2.1)Since det is multiplicative for distinguished triangles, (1.2.1) induces an isomorphismdet i - 1 i FF etiC (X,^) ~ det^FFrf^X ® Fq F q ,F) ® A det A RY et , c (X ® Fq F q ,F) s* A.(1.2.2)We define ((X, T, A) to be the image of 1 G A in det 1 A FF etjC (X,T) under (1.2.2).It is a A-basis of the invertible A-module det7 1 v FF etjC (X, T).1.3. Zeta functions and p-adic zeta elements in positive characteristiccase. Let F be a finite extension of Q p , let Op be the valuation ring ofF, and let J 7 be a constructible O^-sheaf on X. We show that the zeta functionL(X/F q ,J 7 p,u) of the F-sheaf Tp = T ®o L L is recovered from a certain p-adiczeta element as in (1.3.5) below. LetA = 0 L [[Gal(F g /F g )]] = hjnO L [Gal(F r /F g )]. (1.3.1)nYet s(A) be the A-module A which is regarded as a sheaf on the etale site of X viathe natural action of Gal(F g /F g ). ThenH C (X,T® 0L s(Aj) - fimF c (X ® Fq F qn ,T) (1.3.2)