# A mod-p Artin-Tate conjecture, and generalized Herbrand-Ribet

## TL;DR

This paper explores a mod-p version of the Artin-Tate conjecture and extends the Herbrand-Ribet theorem to analyze the p-part of class groups in CM Galois extensions, linking algebraic L-values and representation theory.

## Contribution

It formulates a mod-p Artin-Tate conjecture and generalizes Herbrand-Ribet results to broader Galois extensions, connecting class number factors with representation theory.

## Key findings

- Proposes a mod-p Artin-Tate conjecture relating L-values and class groups.
- Extends Herbrand-Ribet theorem to CM Galois extensions.
- Highlights the role of roots of unity in class number poles.

## Abstract

Following the natural instinct that when a group operates on a number field then every term in the class number formula should factorize `compatibly' according to the representation theory (both complex and modular) of the group, we are led to some natural questions about the $p$-part of the classgroup of any CM Galois extension of $\Q$ as a module for $\Gal(K/Q)$, in the spirit of Herbrand-Ribet's theorem on the $p$-component of the class number of $Q(\zeta_p)$. In trying to formulate these questions, we are naturally led to consider $L(0,\rho)$, for $\rho$ an Artin representation, in situations where this is known to be nonzero and algebraic, and it is important for us to understand if this is $p$-integral for a prime $\p$ of the ring of algebraic integers $\bar{Z}$ in $C$, that we call {\it mod-$p$ Artin-Tate conjecture}. The most minor term in the class number formula, the number of roots of unity, plays an important role for us --- it being the only term in the denominator, is responsible for all poles!