Vol. 303, No. 1, 2019

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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.

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class groups, class number formula, Iwasawa theory, main conjecture, $L$-values, algebraicity of $L$-values, Herbrand–Ribet theorem