Special values of L-functions and the arithmetic of Darmon points

TL;DR

This paper introduces Darmon points on Jacobians of Shimura curves, formulates rationality conjectures, proves a Gross-Zagier type formula relating these points to special L-values, and applies it to Selmer group vanishing results.

Contribution

It extends the theory of Darmon points to quaternion algebra cases, formulates new rationality conjectures, and proves a Gross-Zagier type formula linking Darmon points to L-values.

Findings

Established a Gross-Zagier type formula for Darmon points and L-values.

Proved results relating Darmon points to algebraic parts of L-functions.

Derived vanishing results for Selmer groups under certain conditions.

Abstract

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over Q without complex multiplication and \chi is a ring class character such that L(E/K,\chi,1) is not 0 we prove a Gross-Zagier type formula relating Darmon points to a suitably defined algebraic part of L(E/K,\chi,1); 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.