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

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

The Rationality of Quaternionic Darmon Points Over Genus Fields of Real Quadratic Fields

“…In this paper we prove the analogue of the results of [4] for projections to elliptic curves of the quaternionic Darmon points introduced in [20]. In other words, we show that linear combinations with genus character coefficients of Darmon points on elliptic curves are rational over the fields that were predicted in [20].…”

“…In this paper we prove the analogue of the results of [4] for projections to elliptic curves of the quaternionic Darmon points introduced in [20]. In other words, we show that linear combinations with genus character coefficients of Darmon points on elliptic curves are rational over the fields that were predicted in [20]. In light of this feature, quaternionic Darmon points represent the first instance of a systematic supply of points of Stark-Heegner type other than Darmon's original ones for which explicit rationality results are known.…”

Diagonal cycles and Euler systems II: The Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin $L$-functions

“…The approach followed in this article makes no use of CM points, which accounts for why it extends unconditionally to more general settings, including the case where K is a real quadratic field. The intimate connection between Corollary A1 for K real quadratic and the theory of Stark-Heegner points initiated in [Da] is suggested by the articles [BDD07] and [LRV13], which give a conditional proof of Corollary A1 resting on the algebraicity of Stark-Heegner points, essentially by replacing CM points with their conjectural real quadratic counterparts in the proof of [BD99, Thm.1.2]. Stark-Heegner points are still poorly understood, but the hope that the structures developed in the proof of Corollary A1 might shed light on their algebraicity was an important motivation for the present work.…”

Overconvergent cohomology and quaternionic Darmon points

“…where δ is the connecting homomorphism. The group H 1 (Γ, Z) is isomorphic to the abelianization of Γ, which is finite (see, for example, [15,Section 2]). If we let e Γ denote its exponent, then e Γ [γ ψ ⊗ τ ψ ] has a preimage [c ψ ] ∈ H 1 (Γ, Div 0 H p ), and this is the homology class attached to ψ we were looking for.…”