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

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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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
Received: 10 March 2018
Revised: 5 April 2019
Accepted: 6 April 2019
Published: 21 December 2019
Dipendra Prasad
Indian Institute of Technology Bombay
School of Mathematics
Tata Institute of Fundamental Research
Laboratory of Modern Algebra and Applications
Saint Petersburg State University
Saint Petersburg