A mod-$p$ Artin–Tate conjecture, and generalizing the Herbrand–Ribet theorem

Dipendra Prasad

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 $\mathbb{Q}$ 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

Milestones

Received: 10 March 2018

Revised: 5 April 2019

Accepted: 6 April 2019

Published: 21 December 2019