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

Abstract

Let $p\geq 3$ be a prime number and $K$ be a quadratic imaginary field in which $p$ splits as $\mathfrak{p}\overline{\mathfrak{p}}$. Let $\mathcal{F}$ be a cuspidal Bianchi eigenform over $K$ of weight $(k,k)$, where $k\geq 0$ is an integer, level $\mathfrak{m}$ coprime to $p$, and non-ordinary at both of the primes above $p$. We assume $\mathcal{F}$ has trivial nebentypus. For $\mathfrak{q}\in\{\mathfrak{p}, \overline{\mathfrak{p}}\}$, let $a_{\mathfrak{q}}$ be the $T_{\mathfrak{q}}$ Hecke eigenvalue of $\mathcal{F}$ and let $α_{\mathfrak{q}},β_{\mathfrak{q}}$ be the roots of polynomial $X^{2} -a_{\mathfrak{q}}X+ p^{k+1}$. Then we have four $p$-stabilizations of $\mathcal{F}$: $\mathcal{F}^{α_{\mathfrak{p}},α_{\overline{\mathfrak{p}}}}, \mathcal{F}^{α_{\mathfrak{p}},β_{\overline{\mathfrak{p}}}}, \mathcal{F}^{β_{\mathfrak{p}},α_{\overline{\mathfrak{p}}}},$ and $ \mathcal{F}^{β_{\mathfrak{p}},β_{\overline{\mathfrak{p}}}}$ which are Bianchi cuspforms of level $p\mathfrak{m}$. By the works of Williams, to each $p$-stabilization $\mathcal{F}^{*,\dagger}$, we can attach a locally analytic distribution $L_{p}(\mathcal{F}^{*,\dagger})$ over the ray class group $\text{Cl}(K,p^{\infty})$. On viewing $L_{p}(\mathcal{F}^{*,\dagger})$ as a two-variable power series with coefficients in some $p$-adic field having unbounded denominators satisfying certain growth conditions, we decompose this power series into a linear combination of power series with bounded coefficients in the spirit of Pollack, Sprung, and Lei--Loeffler--Zerbes.