Résume | e describe an explicit rigid analyticuniformization of the maximal toric quotient $J$ of the Jacobian of aShimura curve over $\mathbb Q$ at a prime $\ell$ dividing exactly the level. As acorollary, a proof of a conjecture formulated by Matthew Greenbergis deduced, this allows the construction of local Stark-Heegnerpoints (or, as might also be called, Darmon points) on $J$ and itsquotients.We shall also explain the following arithmetic application of ourresults. Let $E$ be an elliptic curve over $\mathbb Q$ and let $K$ be a realquadratic field such that the root number of $E/K$ is +1. Generalizingprevious work of Bertolini-Darmon-Dasgupta, we establish a formularelating the special value of the $L$-series of $E$ over ring class fieldextensions $H/K$ at the critical point with the specialization ofDarmon points in the group of connected components of a quotient ofthe Jacobian of a suitable Shimura curve obtained by raising the level.As a corollary, if one assumes that local Darmon points are globaland satisfy a suitable reciprocity law, the machinery of Kolyvagincan be used toprove the vanishing of the Selmer group of $E/H$.This is joint work with Matteo Longo and Stefano Vigni. |