Special values of L-functions and the arithmetic of Darmon points
-
Matteo Longo
,
Victor Rotger
Abstract.
Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over 
and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over without complex multiplication and 
is a ring class character such that 
, we prove a Gross–Zagier type formula relating Darmon points to a suitably defined algebraic part of 
; this generalizes results of Bertolini, Darmon and Dasgupta to the case of division quaternion algebras and arbitrary characters. Finally, as an application of this formula, assuming the rationality conjectures for Darmon points we obtain vanishing results for Selmer groups of E over extensions of K contained in narrow ring class fields when the analytic rank of E is zero, as predicted by the Birch and Swinnerton-Dyer conjecture.
© 2013 by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Masthead
- A vanishing theorem for characteristic classes of odd-dimensional manifold bundles
- Semi-homogeneous sheaves, Fourier–Mukai transforms and moduli of stable sheaves on abelian surfaces
- Equivariant algebraic cobordism
- Intersections of special cycles on the Shimura variety for
- Opening infinitely many nodes
- Stein neighborhoods of graphs of holomorphic mappings
- Special values of L-functions and the arithmetic of Darmon points
Articles in the same Issue
- Masthead
- A vanishing theorem for characteristic classes of odd-dimensional manifold bundles
- Semi-homogeneous sheaves, Fourier–Mukai transforms and moduli of stable sheaves on abelian surfaces
- Equivariant algebraic cobordism
- Intersections of special cycles on the Shimura variety for
- Opening infinitely many nodes
- Stein neighborhoods of graphs of holomorphic mappings
- Special values of L-functions and the arithmetic of Darmon points
