Let f:X→S be a projective morphism of Noetherian schemes. We assume f purely of relative dimension d and finite Tor-dimensional. We associate to d+1 invertible sheaves $\cal L$1,$\ldots$,$\cal L$d+1 on X a line bundle IX/S($\cal L$1,$\ldots$,$\Cal L$d+1) on S depending additively on the $\cal L$i, commuting to ‘good’ base changes and which represents the integral along the fibres of f of the product of the first Chern classes of the $\cal L$i. If d=0, IX/S($\cal L$) is the norm $\cal N$X/S($\cal L$).

Let π be a cuspidal automorphic representation of GL3($\Bbb A$$\Bbb Q$), unramified at p and of cohomological type at infinity. We construct p-adic L-functions, which interpolate the critical values of L(π,s) and which satisfy a logarithmic growth condition. We obtain these functions as p-adic Mellin transforms of certain distributions μπ on $\Bbb Z$p* having values in some fixed number field and which are of moderate growth. In the p-ordinary case we obtain the bound |μπ(U)|p[les ]|μHaar(U)|p for open subsets U[les ] $\Bbb Z$p*, where μHaar denotes the invariant distribution on $\Bbb Z$p*.

The class field theory for the fraction field of a two-dimensional complete normal local ring with finite residue field is established by S. Saito. In this paper, we investigate the index of the norm group in the K2-idele class group for a finite Abelian extension of such fields and deduce that the existence theorem does not hold for almost fields in this case.

Given a torus action (T2, M) on a smooth manifold, the orbit map evx(t)=t·x for each x∈M induces a homomorphism ev*: $\mathbb Z$2→H1(M;$\mathbb Z$). The action is said to be Rank-k if the image of ev* has rank k ([les ]2) for each point of M. In particular, if ev* is a monomorphism, then the action is called homologically injective. It is known that a holomorphic complex torus action on a compact Kähler manifold is homologically injective. We study holomorphic complex torus actions on compact non-Kähler Hermitian manifolds. A Hermitian manifold is said to be a locally conformal Kähler if a lift of the metric to the universal covering space is conformal to a Kähler metric. We shall prove that a holomorphic conformal complex torus action on a compact locally conformal Kähler manifold M is Rank-1 provided that M has no Kähler structure.