Summary: Let \(f\) be a modular eigenform of even weight \(k\geq 2\) and new at a prime \(p\) dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to \(f\) a monodromy module \(\mathbf{D}^{FM}_f\) and an \(\mathcal{L}\)-invariant \(\mathcal{L}^{FM}_f\). The first goal of this paper is building a suitable \(p\)-adic integration theory that allows us to construct a new monodromy module \(\mathbf{D}_f\) and \({\mathcal{L}}\)-invariant \(\mathcal{L}_f\), in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two \(\mathcal{L}\)-invariants are equal.
Let \(K\) be a real quadratic field and assume the sign of the functional equation of the \(L\)-series of \(f\) over \(K\) is \(-1\). The Bloch-Beilinson conjectures suggest that there should be a supply of elements in the Selmer group of the motive attached to \(f\) over the tower of narrow ring class fields of \(K\). Generalizing work of Darmon for \(k=2\), we give a construction of local cohomology classes which we expect to arise from global classes and satisfy an explicit reciprocity law, accounting for the above prediction.