# Non-cuspidal Bianchi modular forms and Katz $p$-adic $L$-functions

## TL;DR

This paper constructs $p$-adic $L$-functions for non-cuspidal Bianchi modular forms over imaginary quadratic fields, introducing new notions and demonstrating factorization properties when $p$ splits.

## Contribution

It introduces the concepts of $C$-cuspidality and partial Bianchi modular symbols to define $p$-adic $L$-functions for non-cuspidal forms, extending existing theories.

## Key findings

- Constructed $p$-adic $L$-functions for non-cuspidal Bianchi forms.
- Proved factorization of $p$-adic $L$-functions into Katz $p$-adic $L$-functions when $p$ splits.
- Extended the theory of $p$-adic $L$-functions to non-cuspidal automorphic forms.

## Abstract

Let $K$ be an imaginary quadratic field. In this article, we construct $p$-adic $L$-functions of non-cuspidal Bianchi modular forms by introducing the notions of $C$-cuspidality and partial Bianchi modular symbols. When $p$ splits in $K$, we focus on $p$-adic $L$-functions of non-cuspidal base change Bianchi modular forms, showing that they factor as products of two Katz $p$-adic $L$-functions.