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

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.

arXiv comment22 pages, v2: minor edits and corrected some sign or Galois conjugate errors in Table 2, adapting the proof of Prop 4.5 accordingly. To appear in Journal of Number Theory