Abstract of the appendix: Under a mild hypothesis, we show that when a group G is presented as a semi-direct product of an index 2 subgroup H and an order 2 subgroup, the condition that a representation of H extends to a representation of G is a Zariski-closed condition.
Abstract of the main article: Let K be an imaginary quadratic field. In this article, we study the eigenvariety for GL2/K, proving an étaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this. (1) We construct three-variable p-adic L-functions over the eigenvariety interpolating the (two-variable) p-adic L-functions of classical Bianchi cusp forms in families. (2) Let f be a p-stabilised newform of weight k at least 2 without CM by K. We construct a two-variable p-adic L-function attached to the base-change of f to K under assumptions on f that we conjecture always hold, in particular making no assumption on the slope of f. (3) We prove that these base-change p-adic L-functions satisfy a p-adic Artin formalism result, that is, they factorise in the same way as the classical L-function under Artin formalism.
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