Semistable Reduction Theorem for Overconvergent $F$-isocrystals over Laurent Series Fields

Abstract

We prove the semistable reduction theorem for ${\mathcal{E}}_K^{†}$-valued and $K$-valued overconvergent $F$-isocrystals over $k((t))$-varieties which were introduced by Lazda and Pál. As an application, we prove the finite dimensionality of ${\mathcal{E}}_K^{†}$-valued rigid cohomology with compact support.