北大数論セミナー：

Time： 15:00 – 16:00

Place：理学部3号館3-307室　　　

Speaker：Sudipa Das 氏（Harish-Chandra Research Institute）

Title： On the tame realisable classes of the $k$-th root of
the inverse different

Abstract：Let $F$ be a number field and $G$ be a finite group. Motivated by the classical Stick-elberger theorem, McCulloh proved that the realizable classes arising from the rings of integers of tame abelian $G$-Galois algebras form a subgroup of the locally free class group $\mathrm{Cl}(\mathcal{O}_F G)$. In 2018, Agboola and McCulloh extended this framework to the non-abelian setting.
In this talk, we employ relative algebraic $K$-theory together with the framework
developed by Agboola and McCulloh to study the tame realizable classes arising from the $k$-th roots of the inverse different of tame $G$-Galois algebras. Assuming that $F$ satisfies certain mild hypotheses and that $G$ is a finite group of odd order satisfying the necessary conditions for the existence of the $k$-th root of the inverse different, we
prove that these realizable classes also form a subgroup of the locally free class group $\mathrm{Cl}(\mathcal{O}_F G)$. This part is a joint work with Dr. Aprameyo Pal.
In the final part of the talk, we describe the relationship between these classes and
those arising from rings of integers via Adams operators and discuss some consequences of this connection.

Organizers：朝倉 政典，蔡 園青，安田 正大