Sum expressions for $p$-adic Hecke $L$-functions of totally real fields

TL;DR

This paper derives explicit sum formulas for $p$-adic Hecke $L$-functions of totally real fields and extends the Ferrero-Greenberg formula to this context, assuming a totally real Heegner hypothesis.

Contribution

It provides explicit sum expressions for $p$-adic Hecke $L$-functions of totally real fields and generalizes the Ferrero-Greenberg formula.

Findings

Explicit formulas for periods of $p$-adic measures.

Sum expressions for $p$-adic Hecke $L$-functions.

Extension of Ferrero-Greenberg formula.

Abstract

As a continuation of previous work, we establish sum expressions for $p$-adic Hecke $L$-functions of totally real fields in the sense of Delbourgo, assuming a totally real analog of Heegner hypothesis. This is done by finding explicit formulas of the periods of the corresponding $p$-adic measures. As an application, we extend the Ferrero-Greenberg formula of derivatives of $p$-adic $L$-functions to this setting.