Finite polynomial cohomology for general varieties

TL;DR

This paper introduces a modified finite polynomial cohomology for general varieties over p-adic fields, extending previous theories to include cases without good reduction and providing formulas for p-adic regulator maps.

Contribution

It develops a new cohomology theory replacing 1 - Phi with other polynomials, generalizing finite polynomial cohomology to all varieties over p-adic fields.

Findings

Defines a modified cohomology theory for arbitrary varieties.

Provides formulas for p-adic regulator maps on curves and their products.

Extends finite polynomial cohomology beyond good reduction cases.

Abstract

Nekovar and Niziol have introduced in [arxiv:1309.7620] a version of syntomic cohomology valid for arbitrary varieties over p-adic fields. This uses a mapping cone construction similar to the rigid syntomic cohomology of the first author in the good-reduction case, but with Hyodo--Kato (log-crystalline) cohomology in place of rigid cohomology. In this short note, we describe a cohomology theory which is a modification of the theory of Nekovar and Niziol, modified by replacing 1 - Phi (where Phi is the Frobenius map) with other polynomials in Phi. This is the analogue for general varieties of the finite-polynomial cohomology defined by the first author for varieties with good reduction. We use this cohomology theory to give formulae for p-adic regulator maps on curves or products of curves, without imposing any good reduction hypotheses.