The Darmon-point rationality, reciprocity and Gross–Zagier conjecture

Let $K_S^{\mathrm{ab}}$ be the maximal abelian extension of $K$ unramified and totally split at the primes in $S$, let $\mathcal G=\operatorname{Gal}(K_S^{\mathrm{ab}}/K)$, and let $P_\lambda(\chi,f)$ be the Darmon point constructed from a character $\chi\in C(\mathcal G,\overline{\mathbb Q})$ and an automorphic form $f$. Let $\lambda$ be the relevant component, let $A_\Pi$ be the associated abelian variety, and let $\langle\,\ \rangle_{NT}$ denote its Néron–Tate height pairing.

The Darmon-point conjecture. The following properties should hold: (1) $P_\lambda(\chi,f)$ belongs to $A_\Pi(K_S^{\mathrm{ac}})\otimes_{L_\pi}\overline{\mathbb Q}$; (2) for every $\sigma\in\mathcal G$,

$$P_\lambda(\chi,f)^\sigma=P_\lambda(\chi^\sigma,f),\qquad \chi^\sigma(\gamma)=\chi(\sigma^{-1}\gamma);$$

(3) for every finite character $\chi\in C(\mathcal G,\overline{\mathbb Q})$ satisfying $\chi\circ\rho_K|_{K^\times/K^+}=\lambda$,

$$\langle P_\lambda(\chi,f),P_\lambda(\chi,g)\rangle_{NT}=c\,L'(1/2,\pi,\chi)\prod_{u\notin S}\alpha_{\pi_u,\chi_u}(f_u\otimes g_u),$$

where $c$ is a positive constant depending on the choices and $\alpha_{\pi_u,\chi_u}$ is the local factor specified in the source.

These assertions would provide algebraicity and a Galois reciprocity law for Darmon points, together with a Gross–Zagier-type formula relating their heights to derivatives of automorphic $L$-functions. The supplied text gives no resolution status beyond stating the conjectural properties.