# A finiteness result for $p$-adic families of Bianchi modular forms

## TL;DR

This paper investigates the conditions under which $p$-adic families of Bianchi modular forms can interpolate infinitely many classical forms, establishing finiteness results and computational methods for specific cases.

## Contribution

It proves a finiteness result for $p$-adic families of Bianchi modular forms and provides computational techniques to identify when such families are finite.

## Key findings

- Families interpolating a Zariski-dense set of forms are highly restricted.
- Concrete examples of finite families of Bianchi modular forms are constructed.
- A computational method to determine the finiteness of $p$-adic families is developed.

## Abstract

We study $p$-adic families of cohomological automorphic forms for ${\mathrm{GL}}(2)$ over imaginary quadratic fields and prove that families interpolating a Zariski-dense set of classical cuspidal automorphic forms only occur under very restrictive conditions. We show how to computationally determine when this is not the case and establish concrete examples of families interpolating only finitely many Bianchi modular forms.