Finite-slope universal Rankin–Selberg $p$-adic $L$-function conjecture

Let $U$ be the affinoid neighbourhood of a finite-slope classical point $x_0$ on the Rankin–Selberg eigenvariety, with $U$ smooth, the weight map finite étale, and classical cuspidal points Zariski dense. Let $\mathscr{W}$ be the $p$-adic cyclotomic weight space. For each classical point $(x,\kappa)\in U\times\mathscr{W}$, write $(f_x,g_x)$ for the corresponding pair and let $s=\kappa(x)$ be an integer critical value of $L(f_x\otimes g_x,s)$ in Deligne's sense. Finite-slope universal Rankin–Selberg $p$-adic $L$-function conjecture. There exists a rigid-analytic function

$$L_p^{\mathrm{fs}}\in\mathcal{O}(U\times\mathscr{W})$$

such that

$$L_p^{\mathrm{fs}}(x,\kappa)=\frac{E_p(f_x,g_x,s)}{\Omega_p(f_x,g_x,\pm)}\cdot\frac{L^{(p)}(f_x\otimes g_x,s)}{(2\pi i)^{2s}},$$

where $E_p(f_x,g_x,s)$ is the explicit local Euler factor at $p$, $\Omega_p(f_x,g_x,\pm)$ is a $p$-adic period depending analytically on $x$ and the choice of sign, and $L^{(p)}(f_x\otimes g_x,s)$ is the Rankin–Selberg $L$-function with its Euler factor at $p$ omitted. This is the finite-slope analogue of the universal Rankin–Selberg $p$-adic $L$-function; the conjecture asserts analytic variation and interpolation beyond the ordinary setting, and its resolution is not supplied here.