Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Abstract: Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for $\mathrm{GL}2/K$, proving an \'etaleness 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 $k \geq 2$ without CM by $K$. Suppose $f$ has finite slope at $p$ and its base-change $f{/K}$ to $K$ is $p$-regular. Then: (1) We construct a two-variable $p$-adic $L$-function attached to $f_{/K}$ 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. In an appendix, Carl Wang-Erickson describes a base-change deformation functor and gives a characterisation of its Zariski tangent space.