On 𝑝-adic 𝐿-functions Arising From Bianchi Modular Forms

Abstract

We study and construct 𝑝-adic 𝐿-functions of Bianchi modular forms, i.e., automorphic forms for GL₂ over quadratic imaginary fields, at non-ordinary primes in three different scenarios.

In the first part, we construct signed two-variable 𝑝-adic 𝐿-functions with bounded coefficients from 𝑝-adic 𝐿-functions with unbounded coefficients for cuspidal Bianchi modular forms of parallel weight constructed by Williams. This construction extends the works of Pollack, Sprung, and Lei-Loeffler-Zerbes from the elliptic modular forms setting to the Bianchi modular forms setting. Additionally, we extend the results to 𝑝-adic 𝐿-functions coming from non-parallel weight 𝐶-cuspidal Bianchi modular forms constructed by Palacios. We construct logarithmic matrices using Wach modules basis constructed by Berger-Li-Zhu for the decomposition of 𝑝-adic 𝐿-functions with unbounded coefficients. We use Perrin-Riou's exponential map and the 𝑝-adic regulator to prove certain properties of logarithmic matrices.

In the second part, we construct a 𝑝-adic Asai 𝐿-function, associated to a 𝑝-non-ordinary Bianchi modular form, which interpolates special complex 𝐿-values of the Asai 𝐿-function of that Bianchi modular form. This 𝑝-adic 𝐿-function has unbounded coefficients. We use modular symbols and some special cohomological elements, called Asai-Eisenstein elements, to construct polynomials. These polynomials satisfy some growth conditions, norm properties, and congruence relations. After taking the limit of these polynomials, we obtain the 𝑝-adic Asai 𝐿-function with unbounded coefficients. Moreover, we also construct signed 𝑝-adic Asai 𝐿-functions with bounded coefficients under some assumptions.

In the third part, we construct a two-variable 𝑝-adic Asai 𝐿-function over the eigenvariety interpolating 𝑝-adic Asai 𝐿-functions of non-critical small-slope base-change Bianchi modular forms of parallel weight 0. To construct this 𝑝-adic 𝐿-function, we construct polynomials using a certain overconvergent modular symbol coming from a parallel eigenvariety associated with Bianchi modular forms and Asai-Eisenstein elements over an affinoid in a weight space. Their specialization at the weight (0,0) Bianchi modular form ℱ gives the 𝑝-adic Asai 𝐿 function associated to ℱ, which is constructed in the second part.