We investigate the Chow groups of zero cycles of products of curves over a p-adic field by means of the Milnor K-groups of their Jacobians as introduced by Somekawa. We prove some finiteness results for CH0(X)/m for X a product of curves over a p-adic field.

For a finite morphism f : X → Y of smooth varieties such that f maps X birationally onto X′=f(X), the local equations of f are obtained at the double points which are not triple. If $\cal C$ is the conductor of X over X′, and $D = Sing(X') ⊂ X'$, $Δ ⊂ X$ are the subschemes defined by $\cal C$, then D and Δ are shown to be complete intersections at these points, provided that $\cal C$ has “the expected” codimension. This leads one to determine the depth of local rings of X′ at these double points. On the other hand, when $\cal C$ is reduced in X, it is proved that X′ is weakly normal at these points, and some global results are given. For the case of affine spaces, the local equations of X′ at these points are computed.

Let Rk(n) denote the number of representations of a natural number n as the sum of three cubes and a kth power. In this paper, we show that R3(n) [Lt ] n5/9+ε, and that R4(n) [Lt ] n47/90+ε, where ε > 0 is arbitrary. This extends work of Hooley concerning sums of four cubes, to the case of sums of mixed powers. To achieve these bounds, we use a variant of the Selberg sieve method introduced by Hooley to study sums of two kth powers, and we also use various exponential sum estimates.

We describe a general method for calculating equivariant Euler characteristics. The method exploits the fact that the γ-filtration on the Grothendieck group of vector bundles on a Noetherian quasi-projective scheme has finite length; it allows us to capture torsion information which is usually ignored by equivariant Riemann–Roch theorems. As applications, we study the G-module structure of the coherent cohomology of schemes with a free action by a finite group G and, under certain assumptions, we give an explicit formula for the equivariant Euler characteristic $\chi ({\cal O}_X)={\rm H}^0(X, {\cal O}_X)-{\rm H}^1(X, {\cal O}_X)$ in the Grothendieck group of finitely generated ${\bf Z}[G]$-modules, when X is a curve over ${\bf Z}$ and G has prime order.