Geometric Detection and Finiteness of Log Crystalline Local Systems

Abstract

We prove a geometric detection theorem for log crystalline local systems in the smooth proper setting with a normal crossings boundary. The associated special-fiber log $F$-isocrystal detects geometric absolute irreducibility and, modulo constant rank-one twists, determines the pullback to the geometric generic fiber. The main ingredient is a determinant argument combining a finite-order rank-one reduction, logarithmic period comparison, and the Plücker embedding. As an application, using Abe--Esnault's finiteness theorem and finiteness of stable lattices, we prove finiteness, up to geometrically constant rank-one twists, of geometrically absolutely irreducible log crystalline local systems of fixed rank. Coefficients may lie in any fixed finite extension of ${\mathbb{Q}}_p$, and no bound on the Hodge--Tate weights is imposed.

arXiv comment13pages; v2 generalizes the results of v1 to the logarithmic setting