#### Vol. 303, No. 1, 2019

 Recent Issues Vol. 307: 1  2 Vol. 306: 1  2 Vol. 305: 1  2 Vol. 304: 1  2 Vol. 303: 1  2 Vol. 302: 1  2 Vol. 301: 1  2 Vol. 300: 1  2 Online Archive Volume: Issue:
 The Journal Editorial Board Subscriptions Officers Special Issues Submission Guidelines Submission Form Contacts ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Author Index To Appear Other MSP Journals
A mod-$p$ Artin–Tate conjecture, and generalizing the Herbrand–Ribet theorem

Vol. 303 (2019), No. 1, 299–316
DOI: 10.2140/pjm.2019.303.299
##### Abstract

We propose conjectures about the integrality properties of the values at $s=0$ of certain abelian $L$-functions of $ℚ$ and totally real number fields. We also propose a conjecture which generalizes the theorems of Herbrand and Ribet for values at $s=0$ of totally odd Artin $L$-functions of totally real number fields. Various calculations, some of which are familiar to experts, are made to provide examples.

##### Keywords
class groups, class number formula, Iwasawa theory, main conjecture, $L$-values, algebraicity of $L$-values, Herbrand–Ribet theorem
##### Mathematical Subject Classification 2010
Primary: 11F33, 11R23