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

Abstract

Let $f$ be a modular eigenform of even weight $k\ge 2$ 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 ${\mathbf{D}}_f^{FM}$ and an $\mathcal{L}$-invariant ${\mathcal{L}}_f^{FM}$. The first goal of this paper is building a suitable $p$-adic integration theory that allows us to construct a new monodromy module ${\mathbf{D}}_f$ and $\mathcal{L}$-invariant ${\mathcal{L}}_f$, in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two $\mathcal{L}$-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.