Quaternionic Darmon points on abelian varieties

vigni@dima.unige.it In the first part ofthe paper we prove formulasfor the p-Adic logarithm of quaternionic Darmon points on modular abelian varieties over Q with toric reduction at p. These formulas are amenable to explicit computations and are the first to treat Stark-Heegner type points on higher-dimensional abelian varieties. In the second part of the paper we explain how these formulas, together with a mild generalization of results of Bertolini and Darmon on Hida families of modular forms and rational points, can be used to obtain rationality results over genus fields of real quadratic fields for Darmon points on abelian varieties.