The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes

We prove an asymptotic formula for the $p$-adic valuation of Hecke $L$-values of an imaginary quadratic field at an inert prime $p$ along the anticyclotomic ${\mathbb{Z}}_p$-tower. The key is determination of the $p$-adic valuation of generalized Gauss sums defined using Coates-Wiles homomorphism, and of local resolvents in ${\mathbb{Z}}_p$-extensions. This answers a question of Rubin.