On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

TL;DR

This paper constructs a two-variable $p$-adic $L$-function for the symmetric square of a Coleman family associated with a supersingular cusp form, and proves a factorization formula relating it to Rankin--Selberg convolutions.

Contribution

It introduces a new $p$-adic $L$-function in the supersingular case and establishes a factorization formula extending previous results to this setting.

Findings

Constructed a 2-variable meromorphic $p$-adic $L$-function for the symmetric square of a Coleman family.

Proved a $p$-adic factorization formula relating the Rankin--Selberg $L$-function to symmetric square and Kubota-Leopoldt $L$-functions.

Extended the factorization results of Dasgupta to the supersingular case.

Abstract

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.