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

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 M f and an L-invariant L F M 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. 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.

The Teitelbaum conjecture in the indefinite setting

“…We assume to be in the indefinite case and we may also uniformly treat the known N − = 1 or k = 0 cases. We finally remark that, as explained in Section 6.2, we may also deduce from the above theorem the stronger form [RS,Theorem 4.7] (that was indeed stated as a conjecture, in an earlier version of that paper).…”

“…Fix e ∞ ∈ E whose stabilizer in Γ is Γ 0 and write C har (E, V N − k (F )) to denote the space of V N − k (F )-valued harmonic cocyles. As explained in [RS,Lemma 2.8], the evaluation at e ∞ -morphism induces in cohomology:…”

“…We also denote by D k (P 1 (Q p )) is continuous K-dual, which is a left GL 2 (Q p )-module as above. Write D k 0,b (P 1 (Q p )) to denote the subspace of bounded distributions that are zero on P k (K), as defined for example in [RS,T1] or [T2]. As explained in [RS] (and recalled before Theorem 13), there is an identification…”

Darmon cycles and the Kohnen-Shintani lifting

“…Now by Lemma 3.10, [33] we know that H 1 ( Γ, P k−2 ) = 0 and hence we have the following isomorphism…”

“…See Proposition 4.6 and Theorem 4.7 of [33]. By composing with the isomorphism of Theorem 7, we can consider the cohomological Abel Jacobi map as log AJ :…”