The mock plectic regulator conjecture

Let KK be a quadratic imaginary field, let E/QE/\mathbb Q be an elliptic curve, let Tp(E)T_p(E) be its pp-adic Tate module, and let δ\delta_\infty and logEap\log_E^{a_p} be the maps defined in the source. Let Capricorn(τ)\text{Capricorn}(\tau) be the mock plectic invariant. Mock plectic regulator conjecture. If L(E/K,1)=0L(E/K,1)=0, then Capricorn(τ)\text{Capricorn}(\tau) lies in the image of

2E(K)Hf1(K,Tp(E))ZpKp,\bigwedge^2E(K)\longrightarrow H^1_f(K,T_p(E))\otimes_{\mathbb Z_p}K_p,

where

PQδ(P)logEap(Q)δ(Q)logEap(P).P\wedge Q\longmapsto\delta_\infty(P)\otimes\log_E^{a_p}(Q)-\delta_\infty(Q)\otimes\log_E^{a_p}(P).

This predicts that the mock plectic invariant is generated by a regulator built from global points over KK, linking the analytic invariant to the arithmetic of the Mordell–Weil group. The source does not state whether it has been proved or refuted.

Sources & referencesView supporting material

Primary source

Henri Darmon and Michele Fornea, “Mock plectic points”, arXiv:2310.16758 (2023).

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