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

Rotger, Victor, and Seveso, Marco Adamo. "$\mathcal {L}$-invariants and Darmon cycles attached to modular forms." Journal of the European Mathematical Society 014.6 (2012): 1955-1999. <http://eudml.org/doc/277375>.

@article{Rotger2012,
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 $D^\{FM\}_f$ 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 $D_f$ 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.},
author = {Rotger, Victor, Seveso, Marco Adamo},
journal = {Journal of the European Mathematical Society},
keywords = {Darmon point; $\mathcal \{L\}$-invariant; Shimura curves; quaternion algebra; $p$-adic integration; Darmon point; -invariant; Shimura curves; quaternion algebra; -adic integration},
language = {eng},
number = {6},
pages = {1955-1999},
publisher = {European Mathematical Society Publishing House},
title = {$\mathcal \{L\}$-invariants and Darmon cycles attached to modular forms},
url = {http://eudml.org/doc/277375},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Rotger, Victor
AU - Seveso, Marco Adamo
TI - $\mathcal {L}$-invariants and Darmon cycles attached to modular forms
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 6
SP - 1955
EP - 1999
AB - 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 $D^{FM}_f$ 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 $D_f$ 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.
LA - eng
KW - Darmon point; $\mathcal {L}$-invariant; Shimura curves; quaternion algebra; $p$-adic integration; Darmon point; -invariant; Shimura curves; quaternion algebra; -adic integration
UR - http://eudml.org/doc/277375
ER -