Geometric Detection and Finiteness of Crystalline Local Systems

Abstract

We prove a geometric detection theorem for crystalline local systems on the generic fiber of a smooth proper scheme over the ring of integers of a $p$-adic field. The special-fiber crystalline realization detects geometric absolute irreducibility and, modulo constant rank-one twists, determines the restriction to the geometric fundamental group. The main ingredient is a determinant argument combining a finite-order rank-one reduction, 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 arithmetic character twists, of geometrically absolutely irreducible 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.