Let k be a field and X = Spec (k[t,t−1]). Katz proved that a differential equations with coefficients in k((t−1)) is uniquely extended to a special algebraic differential equation on X when k is of characteristic 0. He also proved that a finite extension of k((t−1)) is uniquely extended to a special covering of X when k is of any characteristic. These theorems are called canonical extension or Katz correspondence. We shall prove a p-adic analogue of canonical extension for quasi-unipotent overconvergent isocrystals. As a consequence, we can show that the local index of a quasi-unipotent overconvergent is equal to its Swan conductor.

We recall the main result of L. Caporaso, J. Harris, and B. Mazur's 1997 paper of ‘Uniformity of rational points.’ It says that the Lang conjecture on the distribution of rational points on varieties of general type implies the uniformity for the numbers of rational points on curves of genus at least 2. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.

We find explicit free resolutions for the D-modules Dfs and D[s] fs / D[s] fs+1, where f is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper.

An investigation of morphisms that coincide topologically is used to generalize to all characteristics and partly reprove Tamagawa's theorem on the Grothendieck conjecture in anabelian geometry for affine hyperbolic curves. The theorem now deals with π1tame of curves over a finitely generated field and its effect on the sets of isomorphisms. Universal homeomorphisms are formally inverted.

We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq.