On $p$-adic Asai $L$-functions of Bianchi modular forms at non-ordinary primes and their decomposition into bounded $p$-adic $L$-functions

Abstract

Let $p$ be an odd prime integer, $F/\mathbb{Q}$ be an imaginary quadratic field, and $\Psi$ be a small slope cuspidal Bianchi modular form over $F$ which is non-ordinary at $p$. In this article, we first construct a $p$-adic distribution $L^{\mathrm{As}}_{p}(\Psi)$ that interpolates the twisted critical $L$-values of Asai (or twisted tensor) $L$-function of $\Psi$, generalizing the works of Loeffler--Williams from the ordinary case to the non-ordinary case. To obtain this distribution, we construct some polynomials using Asai--Eisenstein elements: the Betti analogue of the Euler system machinery, developed by Loeffler--Williams. We use some techniques analogous to those of Loeffler--Zerbes for interpolating the twists of Beilinson--Flach elements arising in the Euler system associated with Rankin--Selberg convolutions of elliptic modular forms. We also use the interpolation method developed by Amice--V\'elu, Perrin-Riou, and B\"uy\"ukboduk--Lei in the construction. Furthermore, under some assumptions, we decompose these unbounded $p$-adic distributions into the linear combination of bounded measures as done by Pollack, Sprung, and Lei--Loeffler--Zerbes in the elliptic modular forms case.