Quaternionic Darmon points on abelian varieties. (English) Zbl 1378.14024

Longo, Matteo; Vigni, Stefano

Riv. Mat. Univ. Parma (N.S.) 7, No. 1, 39-70 (2016).

Summary: 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 M. Bertolini and H. Darmon [Invent. Math. 168, No. 2, 371–431 (2007; Zbl 1129.11025); Ann. Math. (2) 170, No. 1, 343–369 (2009; Zbl 1203.11045)] 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.

MSC:

14G05	Rational points
11G10	Abelian varieties of dimension \(> 1\)

Keywords:

Darmon points; modular abelian varieties; \(p\)-adic logarithm; genus fields

Citations:

Zbl 1129.11025; Zbl 1203.11045