Syntomic formalism with coefficients

TL;DR

This paper develops the syntomic formalism with coefficients, providing tools for p-adic étale Abel-Jacobi maps and explicit reciprocity laws for GSp4, by defining and analyzing syntomic polynomial cohomology and its relations to other cohomologies.

Contribution

It introduces syntomic polynomial cohomology with support, Kunneth morphisms, trace maps, cup products, Gysin maps, and establishes relations with Hyodo-Kato and de Rham cohomologies, including comparison morphisms.

Findings

Defined syntomic polynomial cohomology for filtered Frobenius log-isocrystals.

Established relations between syntomic, Hyodo-Kato, and de Rham cohomologies.

Constructed comparison morphisms for p-adic local systems.

Abstract

This paper provides the technical tools needed in ongoing work of the authors to compute p-adic \'etale Abel-Jacobi maps in order to obtain explicit reciprocity laws for GSp4. In particular, we define and study syntomic polynomial cohomology for filtered Frobenius log-isocrystals over proper and semistable schemes over the ring of integers of a local field, with smooth generic fiber, endowed with horizontal divisors. We introduce syntomic polynomial cohomology with support, we define Kunneth morphisms, trace maps and cup products, Gysin maps with respect to divisors and we study some of their properties. We establish the relation with Hyodo-Kato cohomology of the special fiber and de Rham cohomology of the generic fiber. We also introduce overconvergent variants with and without support by restricting to open smooth formal subschemes. Most of all, in case that the filtered log-isocrystal is associated to a p-adic local system on the generic fiber, we establish comparison morphisms between \'etale and syntomic cohomology and compatibilities with Hochschild-Serre morphisms and between Gysin morphisms.