$\mathcal L$-invariants and Darmon cycles attached to modular forms

Let f be a modular eigenform of even weight k≥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 D_FM(f) and an L-invariant 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 D(f) and L-invariant L(f), in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two 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.