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

Victor Rotger   ; Marco Adamo Seveso

Journal of the European Mathematical Society, Tome 14 (2012) no. 6, pp. 1955-1999

Résumé

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 DfFM​ and an L-invariant LfFM​. The first goal of this paper is building a suitable p-adic integration theory that allows us to construct a new monodromy module Df​ and L-invariant Lf​, 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.

Détail

DOI : 10.4171/jems/352

Classification : 14-XX, 00-XX
Keywords: Darmon point, L-invariant, Shimura curves, quaternion algebra, p-adic integration