Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 2

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

Pages: 597 – 639

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n2.a7


Luochen Zhao (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)


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.


infinite sum, totally real $p$-adic Hecke $L$-functions, Ferrero–Greenberg derivative formula, Brumer–Stark unit, Iwasawa invariants

2010 Mathematics Subject Classification

Primary 11S40. Secondary 11S80, 11Y35.

Received 31 July 2022

Received revised 15 February 2023

Accepted 27 February 2023

Published 7 April 2023