Abstract
Let K be an imaginary quadratic field. In this article, we study the eigenvariety for mathrm {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 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.
Highlights
Let f ∈ Sk+2( 1(N )) be a classical eigenform of weight k + 2 ≥ 2 and level N divisible by p
(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
Previous constructions of p-adic L-functions attached to Bianchi modular forms have focused exclusively on the case of ‘non-critical slope’, namely, under the hypothesis that the slope—the p-adic valuation of the Up-eigenvalue αp( f )—is ‘sufficiently small’
Summary
Let f ∈ Sk+2( 1(N )) be a classical eigenform of weight k + 2 ≥ 2 and level N divisible by p (which we may always assume after possibly p-stabilising from primeto- p level). Suppose Up f = 0, i.e. f has finite slope.
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