Exceptional case of the non semi-simple Real Abelian Main Conjecture

TL;DR

This paper investigates a special case of the real abelian Main Conjecture in number theory, focusing on non semi-simple cases and extending previous work on cyclic fields and inert primes.

Contribution

It explores the non linearly disjoint case of the real abelian Main Conjecture, providing new insights beyond semi-simple scenarios.

Findings

Analysis of non semi-simple cases of the conjecture

Extension of criteria for capitulation of class groups

New conditions for the conjecture in non disjoint fields

Abstract

In the papers: "The Chevalley--Herbrand formula and the real abelian Main Conjecture (New criterion using capitulation of the class group),J. Number Theory 248 (2023)" and "On the real abelian main conjecture in the non semi-simple case, arXiv (2023)", we consider real cyclic fields $K$ and primes $\ell \equiv 1 \pmod {2p^N}$ totally inert in $K$, implying implicitly, $K \cap \mathbb{Q}(\mu_{p^\infty}^{})^+ = \mathbb{Q}$. In this complementary work, we examine the non linearly disjoint case.