Equivariant epsilon constant conjectures for weakly ramified extensions

TL;DR

This paper proves Breuning's local epsilon constant conjecture for certain weakly ramified abelian extensions and demonstrates its implications for global epsilon constant and Omega(2)-conjectures in number theory.

Contribution

It establishes the validity of Breuning's local epsilon constant conjecture for specific weakly ramified abelian extensions, linking local and global conjectures.

Findings

Proves Breuning's conjecture for weakly ramified abelian extensions with cyclic ramification group.

Confirms the global epsilon constant conjecture for certain infinite families of number fields.

Validates Chinburg's Omega(2)-conjecture in specific cases.

Abstract

We study the local epsilon constant conjecture as formulated by Breuning. This conjecture fits into the general framework of the equivariant Tamagawa number conjecture (ETNC) and should be interpreted as a consequence of the expected compatibility of the ETNC with the functional equation of Artin-L-functions. Let K / Q_p be unramified. Under some mild technical assumption we prove Breuning's conjecture for weakly ramified abelian extensions N / K with cyclic ramification group. As a consequence of Breuning's local-global principle we obtain the validity of the global epsilon constant conjecture as formulated by Bley and Burns and of Chinburg's Omega(2)-conjecture for certain infinite families F / E of weakly and wildly ramified extensions of number fields.