This article determines the asymptotic behavior of a class of two variable ‘simultaneous’ lattice point problems in ${\bb R}^n$ for any $n \geqslant 2$. An earlier article [Li-3] has studied a similar problem for a large class of such problems in ${\bb R}^2$.

We study automorphisms of a generic Jacobian Kummer surface. First we analyse the action of classically known automorphisms on the Picard lattice of the surface, then proceed to construct new automorphisms not generated by classical ones. We find 192 such automorphisms, all conjugate by the symmetry group of the (16,6)-configuration.

We express Néron functions and Schneider's local $p$-adic height pairing on an abelian variety $A$ with split multiplicative reduction with theta functions and their automorphy factors on the rigid analytic torus uniformizing $A$. Moreover, we show formulas for the $\rho$-splittings of the Poincaré biextension corresponding to Néron's and Schneider's local height pairings.

Let $R$ be a discrete complete valuation ring, with algebraically closed residue field. Let $X$ be a semi-stable $R$-curve, with smooth generic fibre. In this paper we study tame coverings of $X$.

The jacobian variety of the Catalan curve $y^q = x^p - 1$ is shown to be nondegenerate. As an application, a 0-1-matrix whose determinant computes the relative class number of the $pq$-th cyclotomic field is constructed.