L-invariants and Darmon cycles attached to modular forms

TL;DR

This paper develops a p-adic integration theory to construct monodromy modules and L-invariants for modular forms, conjectures their equivalence, and explores implications for the Mordell-Weil group over real quadratic fields.

Contribution

It introduces a new p-adic integration framework to define monodromy modules and L-invariants, and extends Darmon's work to higher weights with conjectural equivalences.

Findings

Construction of a p-adic integration theory for modular forms.

Conjecture that two monodromy modules and L-invariants are isomorphic.

A method to produce local cohomology classes related to Mordell-Weil groups.

Abstract

Let f be a modular eigenform of even weight k>0 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_FM(f) and an L-invariant L_FM(f). The first goal of this paper is building a suitable p-adic integration theory that allows us to construct a monodromy module D(f) and an L-invariant L(f) in the spirit of Darmon. We conjecture both monodromy modules are isomorphic, and in particular the two L-invariants are equal. For the second goal of this note we assume the conjecture is true. 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 Mordell-Weil 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.