  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.
