**Matteo Longo**^{[1]} and **Stefano Vigni**^{[2]}

*
Quaternionic Darmon points on abelian varieties
*

**Pages:** 39-70

**Received:** 29 December 2015

**Accepted in revised form:** 18 February 2016

**Mathematics Subject Classification (2010):** 14G05, 11G10.

**Keywords:** Darmon points, modular abelian varieties, \(p\)-adic logarithm, genus fields.

**Author address:**

[1] : Dipartimento di Matematica
, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy

[2] : Dipartimento di Matematica, Università degli Studi di Genova
, Via Dodecaneso 35
, 16146 Genova, Italy

The authors are partially supported by PRIN 2010-11 "Arithmetic Algebraic Geometry and Number Theory".

**Abstract:**
In the first part of the paper we prove formulas for the \(p\)-adic logarithm of quaternionic Darmon points
on modular abelian varieties over \(\mathbb{Q}\) 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.

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