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

TL;DR

This paper establishes an asymptotic formula for the $p$-adic valuation of Hecke $L$-values in imaginary quadratic fields at inert primes, using generalized Gauss sums and local resolvents, answering a question posed by Rubin.

Contribution

It introduces a novel method to determine the $p$-adic valuation of local resolvents and generalized Gauss sums in the context of anticyclotomic $Z_p$-extensions, advancing understanding of $L$-values.

Findings

Derived an asymptotic formula for $p$-adic valuations of Hecke $L$-values.

Determined the $p$-adic valuation of generalized Gauss sums using Coates-Wiles homomorphism.

Solved a problem posed by Rubin regarding local resolvents in $Z_p$-extensions.

Abstract

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.