Finite polynomial cohomology with coefficients

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 this cohomology theory to study the arithmetic of compact Shimura curves over $\mathbb{Q}$ and to simplify the proofs of the works of Darmon–Rotger and Bertolini–Darmon–Prasanna.