On $p$-adic adjoint $L$-functions for Bianchi cuspforms: the $p$-split case

TL;DR

This paper constructs a $p$-adic adjoint L-function for Bianchi cuspforms using a Hecke-equivariant pairing on overconvergent cohomology, revealing its relation to eigenvariety ramification and special L-values.

Contribution

It introduces a new $p$-adic adjoint L-function for Bianchi cuspforms and demonstrates its ability to detect ramification and relate to classical L-values.

Findings

The pairing detects the ramification locus of the eigenvariety.

Non-vanishing of the $p$-adic L-function at certain points.

A formula relating the pairing to classical adjoint L-values.

Abstract

We construct a Hecke-equivariant pairing on the overconvergent cohomology of Bianchi threefolds. Applying the strategy of Kim and Bella\"iche, we use this pairing to construct $p$-adic adjoint $L$-functions for Bianchi cuspforms and show that it detects the ramification locus of the cuspidal Bianchi eigenvariety over the weight space. Combining results of Barrera Salazar--Williams, we show a non-vanishing result of this $p$-adic adjoint $L$-function at certain points. Finally, we obtain a formula relating this pairing with the adjoint $L$-values of the corresponding cuspidal Bianchi eigenforms (of level 1).