Overconvergent cohomology, $p$-adic $L$-functions and families for $\mathrm{GL}(2)$ over CM fields

TL;DR

This paper discusses the construction of $p$-adic $L$-functions using overconvergent cohomology for $ ext{GL}(2)$ over CM fields, demonstrating their variation in families and proving a related Artin formalism result.

Contribution

It provides an exposition of $p$-adic $L$-functions via overconvergent cohomology and constructs these functions for families of automorphic representations over CM fields, assuming the non-abelian Leopoldt conjecture.

Findings

Construction of $p$-adic $L$-functions for $ ext{GL}(2)$ over CM fields.

Proof of a $p$-adic Artin formalism result for base-change $L$-functions.

Compatibility with eigenvariety constructions and non-ordinary situations.

Abstract

The use of overconvergent cohomology in constructing $p$-adic $L$-functions, initiated by Stevens and Pollack--Stevens in the setting of classical modular forms, has now been established in a number of settings. The method is compatible with constructions of eigenvarieties by Ash--Stevens, Urban and Hansen, and is thus well-adapted to non-ordinary situations and variation in $p$-adic families. In this note, we give an exposition of the ideas behind the construction of $p$-adic $L$-functions via overconvergent cohomology. Conditional on the non-abelian Leopoldt conjecture, we illustrate them by constructing $p$-adic $L$-functions attached to families of base-change automorphic representations for $\mathrm{GL}(2)$ over CM fields. As a corollary, we prove a $p$-adic Artin formalism result for base-change $p$-adic $L$-functions.