Hecke L-values, definite Shimura sets and Mod non-vanishing

Let be a self-dual Hecke character over an imaginary quadratic field K of infinity type (1, 0). Let and p be primes which are coprime to 6N₊/ₐ (cond () ). We determine the -adic valuation of Hecke L-values L (1, ) /K as varies over p-power order anticyclotomic characters over K. As an application, for p inert in K, we prove the vanishing of the -invariant of Rubin's p-adic L-function, leading to the first results on the -invariant of imaginary quadratic fields at non-split primes. Our approach and results complement the work of Hida and Finis. The approach is rooted in the arithmetic of a CM form on a definite Shimura set. The application to Rubin's p-adic L-function also relies on the proof of his conjecture. Along the way, we present an automorphic view on Rubin's theory.