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

TL;DR

This paper develops a finite-slope analogue of Loeffler's conjectural framework for Rankin--Selberg p-adic L-functions within universal deformation families, extending the theory to new p-adic contexts.

Contribution

It introduces a finite-slope version of Loeffler's conjecture for Rankin--Selberg p-adic L-functions in universal deformation settings.

Findings

Constructs a finite-slope analogue of Loeffler's framework.

Analyzes residual Galois representations in deformation families.

Extends p-adic L-function theory to finite-slope cases.

Abstract

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.