Finite polynomial cohomology with coefficients

TL;DR

This paper develops a new finite polynomial cohomology theory with coefficients, establishing foundational properties, an Abel-Jacobi map, and applying it to arithmetic studies of Shimura curves, simplifying existing proofs.

Contribution

It introduces a novel cohomology theory with coefficients, expanding tools for arithmetic geometry and providing streamlined proofs for known results.

Findings

Established basic properties of the cohomology theory

Constructed an Abel-Jacobi map with coefficients

Applied the theory to Shimura curves, simplifying existing proofs

Abstract

We introduce a theory of finite polynomial cohomology with coefficients in this paper. We prove several basic properties and introduce an Abel-Jacobi map with coefficients. As applications, we use such a cohomology theory to study arithmetics of compact Shimura curves over $\mathbb{Q}$, and simplify proofs of the works of Darmon-Rotger and Bertolini-Darmon-Prasanna.