Let be a modular eigenform of even weight and new at a prime dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to a monodromy module and an -invariant . The first goal of this paper is building a suitable -adic integration theory that allows us to construct a new monodromy module and -invariant , in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two -invariants are equal.
Let be a real quadratic field and assume the sign of the functional equation of the -series of over is . The Bloch-Beilinson conjectures suggest that there should be a supply of elements in the Selmer group of the motive attached to over the tower of narrow ring class fields of . Generalizing work of Darmon for , 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.
Cite this article
Victor Rotger, Marco Adamo Seveso, -invariants and Darmon cycles attached to modular forms. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1955–1999