$p$-adic Asai $L$-functions of Bianchi modular forms

Abstract

The Asai (or twisted tensor) $L$-function of a Bianchi modular form $\mathrm{\Psi }$ is the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its associated Galois representation. When $\mathrm{\Psi }$ 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.