Katz Correspondence for Quasi-Unipotent Overconvergent Isocrystals

Abstract

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.