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$$\mathcal L$$-invariants and Darmon cycles attached to modular forms. (English) Zbl 1292.11069
Summary: Let $$f$$ be a modular eigenform of even weight $$k\geq 2$$ 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 $$\mathbf{D}^{FM}_f$$ and an $$\mathcal{L}$$-invariant $$\mathcal{L}^{FM}_f$$. 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.

MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties
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References:
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