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

  • Victor Rotger

    Universitat Politècnica de Catalunya, Barcelona, Spain
  • Marco Adamo Seveso

    Università degli Studi di Milano, Italy


Let ff be a modular eigenform of even weight k2k\geq 2 and new at a prime pp dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to ff a monodromy module DfFM\mathbf{D}^{FM}_f and an L\mathcal{L}-invariant LfFM\mathcal{L}^{FM}_f. The first goal of this paper is building a suitable pp-adic integration theory that allows us to construct a new monodromy module Df\mathbf{D}_f and L{\mathcal{L}}-invariant Lf{\mathcal{L}}_f, in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two L{\mathcal{L}}-invariants are equal.
Let KK be a real quadratic field and assume the sign of the functional equation of the LL-series of ff over KK is 1-1. The Bloch-Beilinson conjectures suggest that there should be a supply of elements in the Selmer group of the motive attached to ff over the tower of narrow ring class fields of KK. Generalizing work of Darmon for k=2k=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.

Cite this article

Victor Rotger, Marco Adamo Seveso, L\mathcal L-invariants and Darmon cycles attached to modular forms. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1955–1999

DOI 10.4171/JEMS/352