Riv. Mat. Univ. Parma, Vol. 7, No. 1, 2016

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 , UniversitÓ degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy
[2] : Dipartimento di Matematica, UniversitÓ 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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