Darmon-style norm and Mazur–Tate height conjecture for Rubin lattices

Let $A$ be an abelian variety, let $F/k$ be a finite Galois extension with group $G$, let $J\leq G$ with $E=F^J$, and let $p$ be a prime. Let $A^t$ be the dual abelian variety, let $U^t_{F/E,p}$ be the relevant semi-local module, let $I_p(J)$ be the augmentation ideal of $\mathbb Z_p[J]$, and let $\bigcap_{\mathbb Z_p[G]}^rM$ denote the $r$th Rubin lattice. For a $p$-separable choice of points $(\mathcal Y,\mathcal Y')$ of ranks $(a,a')$, let $\eta_{\mathcal Y,x_\bullet}$ and $\eta_{\mathcal Y',x_\bullet^J}$ be the associated elements, let $\mathcal N_J$ be the norm operator, let ${\rm ht}_{\mathcal Y'}^{\rm MT}$ be the combined Mazur–Tate height map, and let $\nu_J$ be the canonical injection. Darmon-style norm–height conjecture. If neither $A(F)$ nor $A^t(F)$ has a point of order $p$, then

$$\eta_{\mathcal Y,x_\bullet}\in\bigcap_{\mathbb Z_p[G]}^aA^t(F)_p$$

and

$$\eta_{\mathcal Y',x_\bullet^J}\in\bigcap_{\mathbb Z_p[G/J]}^{a'}U^t_{F/E,p},$$

and, in

$$\left(\bigcap_{\mathbb Z_p[G]}^aA^t(F)_p\right)\otimes_{\mathbb Z_p}I_p(J)^{a'-a}/I_p(J)^{1+a'-a},$$

one has

$$\mathcal N_J(\eta_{\mathcal Y,x_\bullet})=(-1)^{a(a'-a)}\nu_J\bigl({\rm ht}_{\mathcal Y'}^{\rm MT}(\eta_{\mathcal Y',x_\bullet^J})\bigr).$$

This is a refined $p$-adic congruence relating special $L$-value elements to Mazur–Tate heights and norm operators. The source calls it the central conjecture of the section but supplies no resolution.