Vol. 303, No. 1, 2019

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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 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.

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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
Authors
Dipendra Prasad
Indian Institute of Technology Bombay
Mumbai
India
School of Mathematics
Tata Institute of Fundamental Research
Mumbai
India
Laboratory of Modern Algebra and Applications
Saint Petersburg State University
Saint Petersburg
Russia