松田 茂樹

Let V be a complete discrete valuation ring of mixed characteristic. We classify arithmetic D-modules on Spf(V[[t]]) up to certain kind of 'analytic isomorphism'. This result is used to construct canonical extensions (in the sense of Katz and Gabber) for objects of this category.

Let k be a field and X = Spec (k[t,t]). Katz proved that a differential equations with coefficients in k((t)) 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)) 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.

Let X be a proper smooth surface over an algebraically closed field of positive characteristic and U be a complement of a simple normal crossing divisor. For a smooth l-adic sheaf F on U, Deligne proved a formula calculating the Euler characteristic of F by local invariants. Kato gave another formula for the Euler characteristic in case where F is of rank 1, using class field theory. In this paper, we show that local terms of these formulas coincide, admitting certain conjectures for vanishing cycles.