P-adic Rankin-Selberg L-functions in universal deformation families and functional equations

TL;DR

The paper constructs a p-adic Rankin–Selberg L-function for the product of a Hida family and a universal deformation family, defined over a four-parameter base and extending beyond the ordinary eigenvariety. It proves an interpolation formula that recovers critical complex L-values with explicit local factors, and develops a functional equation by introducing universal gamma-factors in the Helm–Moss framework to handle ramified primes. A dual p-adic L-function is defined to relate to the original via a p-adic functional equation, tying the interpolation to the complex functional equation. Overall, the work broadens the landscape of p-adic L-functions to big parabolic eigenvarieties and four-parameter deformation spaces, with potential implications for Iwasawa theory and extensions to Hilbert modular forms.

Abstract

We construct a $p$-adic Rankin-Selberg $L$-function associated to the product of two families of modular forms, where the first is an ordinary (Hida) family, and the second an arbitrary universal-deformation family (without any ordinarity condition at $p$). This gives a function on a 4-dimensional base space - strictly larger than the ordinary eigenvariety, which is 3-dimensional in this case. We prove our $p$-adic $L$-function interpolates all critical values of the Rankin-Selberg $L$-functions for the classical specialisations of our family, and derive a functional equation for our $p$-adic $L$-function.

Theorems & Definitions (51)

• Theorem A: Theorem \ref{['thm:interpolation formula']}
• Theorem B: Theorem \ref{['thm:p-adicfunctionalequation']}
• Theorem 2.1
• proof
• Theorem 2.2: boe03
• Definition 2.3: classical and nearly classical points
• Definition 3.1
• Proposition 3.2: Gouvea, Loeffler
• proof
• Definition 3.3