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 $$\mathrm {GL}_2/K$$ 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.

A p-adic L-function for non-critical adjoint L-values

“…(1) We make no attempt to control the p-adic error terms c n -see the discussion following Definition 5.3 -but simply remark that they are of a similar nature to those appearing in the p-adic L-function of [GS93]. In the Bianchi setting, the three-variable p-adic Lfunction constructed in [BSW18] involves p-adic error terms defined using H 1 of the Bianchi threefold, whereas our p-adic error terms are defined using H 2 . (2) For a Hida family with general tame character, a similar construction should yield a p-adic L-function whose specializations on (a translate of) A r can be determined.…”

Arithmetic of p-irregular modular forms: families and p-adic L-functions

“…Working with families in this setting is much harder, due to the presence of cuspidal Hecke eigensystems in H 2 c . Accordingly, in this case we make use of the technical machinery developed in [BSW18], in which these families were carefully studied. We prove: Theorem B.…”

“…We consider variation of the overconvergent cohomology (in degree 1) over the Coleman-Mazur and Bianchi eigencurves, as studied in [Bel12] (classical case) and [BSW18] (Bianchi case). In each case, we show that the local ring of the eigencurve is Gorenstein at the p-irregular form f , and use this to deduce the existence of a family of overconvergent eigenclasses in the cohomology, interpolating the classes Φ g as g varies in a Coleman family.…”

“…We comment briefly on why new arguments are required in the p-irregular case. The previous most general constructions of multi-variable p-adic L-functions, in [Bel12] and [BSW18], treat two cases separately, using the arithmetic of f to deduce results about the structure of the eigenvariety.…”

“…This interpolates the critical L-values Λ(f, χ F Q χ, j +1) for χ as above. To prove Theorem C, we use the strategy of [BSW18]; indeed, this factorisation holds at a Zariski-dense set of classical points in the eigencurve, as can be seen from the interpolation formula and growth properties of the respective p-adic L-functions, and this interpolates to give the factorisation at the level of two-variable p-adic L-functions. We prove the theorem by specialising to the form f .…”