Towards a finite-slope universal Rankin-Selberg p-adic L-function

This article studies the finite–slope analogue of Loeffler’s conjectural framework for Rankin–Selberg $p$-adic $L$-functions in universal deformation families. Starting from residual representations ${\bar{\rho }}_1,{\bar{\rho }}_2$ of tame level~$1$ satisfying Hypothesis~3.1 of~\cite{LoefflerUD}, we consider the half–ordinary Panchishkin family $(R,V,V^{+})$ of Example~3.17 of loc.\ cit., where the first factor varies in the ordinary Hida deformation and the second factor in the unrestricted universal deformation space.