Number Theory Seminar

An important tool for linking arithmetic data and $L$-functions is the use of $p$-adic methods, and in particular, the construction and study of $ p$-adic $L$-functions. These functions are proven to be useful in connecting properties of arithmetic objects with special $L$-values, with applications to interesting problems such as the Birch and Swinnerton-Dyer conjecture. For a prime $p$, $p$-adic $L$-functions can be viewed as a power series with coefficients in a local field or in the ring of integers of a local field. These functions satisfy certain growth properties and interpolate special values of complex $L$-functions. In this talk, I will present the construction of $p$-adic Asai $L$-functions with unbounded coefficients attached to cuspidal Bianchi modular forms at non-ordinary primes. I will then discuss the Pollack, Sprung, and Lei--Loeffler--Zerbes style decomposition of these $p$-adic Asai $L$-functions with unbounded coefficients into signed $p$-adic Asai $L$-functions with bounded coefficients. The talk will begin with a brief review of $p$-adic L-functions, Bianchi modular forms, and their Asai $L$-functions.