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

TL;DR

This paper constructs a new $p$-adic Rankin-Selberg $L$-function for families of modular forms, extending previous work to more general deformation families and establishing interpolation and functional equations.

Contribution

It introduces a $p$-adic $L$-function for the product of an ordinary Hida family and a universal deformation family, without ordinarity at $p$, expanding the scope of $p$-adic $L$-functions.

Findings

Constructed a $p$-adic $L$-function on a 4-dimensional base space.

Proved the $p$-adic $L$-function interpolates all critical values.

Derived a functional equation for the $p$-adic $L$-function.

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.