The Hilbert–Samuel cycle formula for Bernstein eigenvarieties

Let $L$ be the relevant local field, let $x=(\rho,\underline{x},\chi)$ be a point of $X_{\Omega,\mathbf h}(\overline\rho)$ with $\rho$ de Rham and $(\underline{x},\chi)$ generic, and let $w_x\in\mathscr W^P_{\min,L}$ and $w_y\in\mathscr W^P_{\max,L}$ be the Weyl-group elements attached to $x$ and to the associated point $y$ of the parabolic variety. Let $\mathfrak C_w$ be the cycle defined from the cycles $\mathfrak Z_{w'}$ and coefficients $a_{w,w'}$, and let $b_{w_xw_{0,L},w}$ be the corresponding coefficient. Cycle formula conjecture. The cycle of the completed local ring of the weight fibre at $x$ is

$$[\operatorname{Spec}\widehat{\mathcal O}_{X_{\Omega,\mathbf h}(\overline\rho)_{\operatorname{wt}(\chi)},x}]=\sum_{\substack{w\in\mathscr W_{L_P,L}\backslash\mathscr W_L\\\\ w_y\leq w^{\max}\leq w_xw_{0,L}}}b_{w_xw_{0,L},w}\mathfrak C_w$$

inside $Z^{[L:\mathbb Q_p]n(n+1)/2}(\operatorname{Spec}\widehat{\mathcal O}_{\mathfrak X_{\overline\rho,\rho}})$. The formula is presented as a consequence of the companion-constituent conjecture and the preceding nonvanishing lemma; the source gives no resolution of that underlying conjectural formula.