Explicit constructions and arithmetic of local number fields

This paper is a survey of some of the results obtained by Saint Petersburg number theory school over recent decades. We consider explicit formulas for the Hilbert symbol in case of nonclassical formal modules and their applications in local arithmetic geometry and ramification theory. For topics studied by the school, but not included in this survey (due to the lack of space), references are given. In the section on explicit formulas, we present results for formal modules constructed using Honda formal group laws and polynomial formal group laws, and also the interpretation using p-adic integrals. Their implications concerning Galois modules are also described. We give the classification of formal group laws over local number fields as well. Further, we discuss constructions of complete discrete valuation fields’ extensions. Besides some results in structure theory of higher local fields are presented. The last section is concerned with ramification theory for higher local fields. Refs 139.Работа выполнена при финансовой поддержке РНФ (грант №16-11-10200)