Darmon's reciprocity conjecture for generalized Darmon points

Let FF be a totally real field, K/FK/F a quadratic extension, B/FB/F a quaternion algebra, q:KBq:K\hookrightarrow B an embedding, and let E/FE/F be a modular elliptic curve. For a suitable cycle Δb\Delta_b and modified differential ωφβ\omega_{\varphi}^{\beta}, let

Pbβ=Φ(Δbωφβ)E(C).P_b^{\beta}=\Phi\left(\int_{\Delta_b}\omega_{\varphi}^{\beta}\right)\in E(\mathbf C).

Darmon's reciprocity conjecture. The point PbβP_b^{\beta} lies in E(Kab)E(K^{\mathrm{ab}}) and

aAK×recK(a)Pbβ=β(a)PqA(a)bβ.\forall a\in\mathbf A_{K}^{\times}\qquad\mathrm{rec}_K(a)P_b^{\beta}=\beta(a_{\infty})P_{q_{\mathbf A}(a)b}^{\beta}.

This is the principal conjecture governing the algebraicity and class-field-theoretic reciprocity of Darmon's points; the supplied source gives no resolution.

Sources & referencesView supporting material

Primary source

Jerome Gartner, “Darmon's points and quaternionic Shimura varieties”, arXiv:1104.3338 (2011).

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