p-adic Asai L-functions of Bianchi modular forms

Loeffler, David; Williams, Chris

doi:10.2140/ant.2020.14.1669

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

David Loeffler and Chris Williams

Vol. 14 (2020), No. 7, 1669–1710

DOI: 10.2140/ant.2020.14.1669

Abstract

The Asai (or twisted tensor) $L$ -function of a Bianchi modular form $Ψ$ is the $L$ -function attached to the tensor induction to $ℚ$ of its associated Galois representation. When $Ψ$ is ordinary at $p$ we construct a $p$ -adic analogue of this $L$ -function: that is, a $p$ -adic measure on $ℤ_{p}^{\times}$ 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.

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Keywords

Asai L-function, p-adic L-function, Bianchi modular form, Betti-Eisenstein classes, Iwasawa theory

Mathematical Subject Classification 2010

Primary: 11F67

Secondary: 11F41, 11F85, 11M41, 11S40

Milestones

Received: 30 July 2018

Revised: 16 September 2019

Accepted: 10 March 2020

Published: 18 August 2020

Authors

David Loeffler
Mathematics Institute University of Warwick Coventry United Kingdom

Chris Williams
Mathematics Institute University of Warwick Coventry United Kingdom

Vol. 14, No. 7, 2020