The strong Stark conjecture for totally odd characters

TL;DR

This paper proves the $p$-part of the strong Stark conjecture for totally odd characters and establishes the minus $p$-part of the equivariant Tamagawa number conjecture for certain Galois CM-extensions, advancing number theory conjectures.

Contribution

It provides an unconditional proof of the strong Stark conjecture for totally odd characters and the minus $p$-part of the equivariant Tamagawa number conjecture under specific conditions.

Findings

Proved the $p$-part of the strong Stark conjecture for totally odd characters.

Established the minus $p$-part of the equivariant Tamagawa number conjecture for certain Galois CM-extensions.

Extended understanding of conjectures in algebraic number theory.

Abstract

We prove the $p$-part of the strong Stark conjecture for every totally odd character and every odd prime $p$. Let $L/K$ be a finite Galois CM-extension with Galois group $G$, which has an abelian Sylow $p$-subgroup for an odd prime $p$. We give an unconditional proof of the minus $p$-part of the equivariant Tamagawa number conjecture for the pair $(h^0(\mathrm{Spec}(L)), \mathbb{Z}[G])$ under certain restrictions on the ramification behavior in $L/K$.