FINITE CRYSTALLINE HEIGHT REPRESENTATIONS AND SYNTOMIC COMPLEXES (Algebraic Number Theory and Related Topics)

Abstract

Using finite crystalline height representations and their naturally associated invariants, we study local and global syntomic complexes with coefficients. The text is organized as follows. After briefly recalling the 𝑝-adic crystalline comparison theorem and importance of syntomic methods in its proof we pose a question on syntomic complex with coefficients. To answer our question, we quickly recount the theory of finite crystalline height representations developed in [Abh21] and show that Galois cohomology of such representations (upto a twist), is essentially computed by (Fontaine-Messing) syntomic complex with coefficients in the associated 𝑭-isocrystal. In global applications, for smooth (𝑝-adic formal) schemes, we show a comparison between syntomic complex with coefficient in a locally free Fontaine-Laffaille module and complex of 𝑝-adic nearby cycles of the associated étale local system on the (rigid) generic fiber. Proofs of aforementioned results can be found in [Abh22].