Speaker: Abbas Maarefparvar (IPM-Tehran)
Title: On Pólya groups of algebraic number fields: Some applications and generalizations.
Dates: Sept. 26, 2022, Monday 10:30-12:00 and Sept. 28, 2022, Wednesday 09:00-10:30
Pólya groups are subgroups of ideal class groups of number fields generated by all classes of Bhargava generalized factorial ideals, which more classicaly, can also be defined in terms of integer valued polynomials. Considering the action of Galois groups on class groups of Galois number fields, one can show that Pólya groups coincide with strongly ambiguous ideal class groups, and investigating on this subject can be seen as a generalization of Gauss's genus theory as Hilbert did for quadratic number fields. In this talk, I will present some well known results on Pólya groups of number fields in various degrees, and introduce some generalizations of this notion to relative case, namely the relative Pólya groups and Ostrowski qutients for finite extensions of number fields. In particular, I will talk about the “BRZ”, a four-term exact sequence obtained from some cohomological results of Brumer-Rosen and Zantema, and give some its applications. If time permits, I will talk also about the recent attempts for finding some analogies in elliptic curves.
Speaker: K. İlhan İkeda (FGC-İstanbul)
Title: Ono reciprocity law, absolute arithmetic and F1-geometry*
Date: September 28, 2022, Wednesday 11:00-12:30.
In this seminar talk we first introduce the reciprocity law of Ono for finite Galois extensions K/k of a global field k, which is a non-abelian generalization of the global Artin reciprocity law for finite abelian extensions E/k of k. We then plan to discuss further “functorial” properties of the Ono reciprocity law for K/k and, “passing to limits”, get the Ono reciprocity law for ksep/k. Finally, we plan to discuss the relationship between this theory with the absolute arithmetic and F1-geometry. * Joint work with Serkan Kızılavuz (Eskişehir Technical University).