# $L$-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

## Abstract

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 $D_{f}$ and an $L$-invariant $L_{f}$. 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.

## Cite this article

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

DOI 10.4171/JEMS/352