AI Summary & Key Takeaways
AI InsightWe finally propose the definition of the conductor \frak f(E/K) of a metabelian extension E/K in the sense of Koch-de Shalit local class field theory, and compute \frak f\rm Artin(c\rho) under a suitable assumption.
📌 3 Key Takeaways
Abstract
Let K be a local field with finite residue class field and E a finite Galois extension over K. In this paper, we study the Artin conductor \frak f\rm Artin(c\rho) of a character c\rho associated to a representation r:\mbox{Gal}(E/K)\rightarrow GL(V) of \mbox{Gal}(E/K) with metabelian kernel \mbox{ker}(r). In order to do so, we first review the Artin character a\mbox{Gal}(E/K) of \mbox{Gal}(E/K) and review the metabelian local class field theory. We finally propose the definition of the conductor \frak f(E/K) of a metabelian extension E/K in the sense of Koch-de Shalit local class field theory, and compute \frak f\rm Artin(c\rho) under a suitable assumption.
Özet
Let K be a local field with finite residue class field and E a finite Galois extension over K. In this paper, we study the Artin conductor \frak f\rm Artin(c\rho) of a character c\rho associated to a representation r:\mbox{Gal}(E/K)\rightarrow GL(V) of \mbox{Gal}(E/K) with metabelian kernel \mbox{ker}(r). In order to do so, we first review the Artin character a\mbox{Gal}(E/K) of \mbox{Gal}(E/K) and review the metabelian local class field theory. We finally propose the definition of the conductor \frak f(E/K) of a metabelian extension E/K in the sense of Koch-de Shalit local class field theory, and compute \frak f\rm Artin(c\rho) under a suitable assumption.