An important tool to study automorphic representations in the framework of the Langlands program is to produce $p$-adic variation. Such variation is captured geometrically in the study of certain rigid analytic spaces, called eigenvarieties.
In this talk, we first introduce Bianchi modular forms, that is, automorphic forms for $\mathrm{GL}_2$ over an imaginary quadratic field, and then discuss its contribution to the cohomology of the Bianchi threefold. Further, we present the Bianchi eigenvariety and state our result about its geometry at a special non-cuspidal point. Time permitting, we will give some ideas about the proof. This is a joint work in progress with Daniel Barrera (Universidad de Santiago de Chile).