P-adic Asai L-functions of Bianchi modular forms

TL;DR

This paper constructs a p-adic analogue of the Asai L-function for Bianchi modular forms, interpolating critical values and extending the understanding of p-adic L-functions in the context of automorphic forms and Galois representations.

Contribution

It introduces a novel p-adic measure for the Asai L-function of Bianchi modular forms, especially when the form is ordinary at p, using techniques similar to Euler system constructions.

Findings

Constructed a p-adic measure interpolating critical Asai L-values.

Extended methods from Hilbert modular forms to Bianchi modular forms.

Provides tools for studying p-adic properties of automorphic L-functions.

Abstract

The Asai (or twisted tensor) $L$-function of a Bianchi modular form $\Psi$ is the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its associated Galois representation. In this paper, when $\Psi$ is ordinary at $p$ we construct a $p$-adic analogue of this $L$-function: that is, a $p$-adic measure on $\mathbb{Z}_p^\times$ that interpolates the critical values of the Asai $L$-function twisted by Dirichlet characters of $p$-power conductor. The construction uses techniques analogous to those used by Lei, Zerbes and the first author in order to construct an Euler system attached to the Asai representation of a quadratic Hilbert modular form.