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 X be a G-connected nilpotent simplicial set, where G is a finite Hamiltonian group. We construct a cofibrant equivariant minimal model of X with the strong homotopy type of the injective minimal model of X defined by Triantafillou.

This paper deals with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring of integers of an algebraic closure of the p-adic field. Given a formal group F as above, a formal flow is a series Φ(t,x) satisfying the conditions Φ(0,x)=x and Φ(F(s,t),x)=Φ(s,Φ(t,x)). With this definition, any formal group will act on the disk by left translation, but this paper constructs flows Φ with any specified divisor of fixed points, where a point ξ of the open unit disk is a fixed point of order [les ]n if (x−ξ)n|(Φ(t,x)−x). Furthermore, if γ is an analytic automorphism of the open unit disk with only finitely many periodic points, then there is a flow Φ, an element α of the maximal ideal of the ring of constants, and an integer m such that the m-fold iteration of γ(x) is equal to Φ(α,x). All the formal flows constructed here are actions of the additive formal group on the unit disk. Indeed, if the divisor of fixed points of a formal flow is of degree at least two, then the formal group involved must become isomorphic to the additive group when the base is extended to the residue field of the constant ring.

Let F be a non-Archimedean local field with ring of integers R and prime ideal $\frak p$. Suppose T is a GLn(F)-invariant distribution on ${\frak g}$=Mn(F), the Lie algebra of GLn(F). If T has support in the set of topologically nilpotent elements, then the restriction of T to the set of functions which are compactly supported and invariant under Mn($\frak p$) may be expressed as a linear combination of nilpotent orbital integrals restricted to the same set of functions.

In this paper we study the group K2n(n+1)(F) where F is the function field of a complete, smooth, geometrically irreducible curve C over a number field, assuming the Beilinson–Soulé conjecture on weights. In particular, we compute the Beilinson regulator on a subgroup of K2n(n+1)(F), using the complexes constructed in Compositio Math. 96 (1995), pp. 197–247. We study the boundary map in the localization sequence for n=2 and n=3. We combine our results with results of Goncharov in order to obtain a complete description of the image of the regulator map on K4(3)(C) and K6(4)(C) (which have the same images as K4(C)$⊗ _{\Bbb Z}\, {\Bbb Q}$, and K6(C)$⊗ _{\Bbb Z}\, {\Bbb Q}$, respectively), independent of any conjectures.

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.

The purpose of this paper is to define functional realizations of the Khizhnik–Zamolochikov–Bernard (KZB) connection on the bundle of conformal blocks over the moduli space of curves.

We prove a certain Riemann–Roch-type formula for symmetric powers of Galois modules on Dedekind schemes which, in the number field or function field case, specializes to a formula of Burns and Chinburg for Cassou–Noguès–Taylor operations.

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.

In Ann. of Math. 121 (1985), 111–168, Coleman defines p-adic Abelian integrals on curves. Given a family of curves X/S, a differential ω and two sections s and t, one can define a function λω on S by λω(P)=$\int$s(P)t(P)ωP. In this paper, we prove that λω is locally analytic on S.

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.

Let M be a smooth manifold, ${\cal S}$ the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on ${\cal S}$. This cohomology space is closely related to the Vect(M)-modules, ${\cal D}$λ(M), of linear differential operators on the space of tensor densities on M of degree λ.

It is well-known that quasi-homogeneity is characterized by equality of the Milnor and Tjurina numbers for isolated complex analytic hypersurface singularities and for certain low-dimensional singularities. In this paper we prove that this characterization extends to isolated purely elliptic complete intersection singularities, with bounds on neither the embedding codimension nor the dimension of the singularity.