Resumen de L -invariants and Darmon cycles attached to modular forms

Victor Rotger Cerdà , Marco Adamo Seveso

• 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.