Abstract Let K be an imaginary quadratic field. In this article, we study the eigenvariety for $$\mathrm {GL}_2/K$$ GL 2 / 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; let f be a p -stabilised newform of weight $$k \ge 2$$ k ≥ 2 without CM by K . Suppose f has finite slope at p and its base-change $$f_{/K}$$ f / K to K is p -regular. Then: (1) We construct a two-variable p -adic L -function attached to $$f_{/K}$$ f / K under assumptions on f that conjecturally always hold, in particular with no non-critical assumption on f / K . (2) We construct three-variable p -adic L -functions over the eigenvariety interpolating the p -adic L -functions of classical base-change Bianchi cusp forms. (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.

Daniel Barrera Salazar

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