Textbooks, Talks, Videos, Lecture Notes



Textbooks and surveys


I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions,  Second extended edition , AMS 2002, 341 pp.

Invitation to higher local fields, I. Fesenko, M. Kurihara (eds.) Geometry and Topology Monographs vol 3, Warwick 2000 


Files of some talks


90 minutes talk on Adelic geometry and analysis on regular models of elliptic curves over global fields and their zeta functions

90 minutes talk on Inter-universal Teichmüller theory of Shinichi Mochizuki

Reciprocity and IUT, talk at RIMS workshop on IUT Summit, July 2016

Adelic lifts of geometry and arithmetic of surfaces and their BSD interaction, talk at EDGE workshop, June 2017


Videos


New directions and perspectives in two dimensional number theory I

New directions and perspectives in two dimensional number theory II

New directions and perspectives in two dimensional number theory III

New directions and perspectives in two dimensional number theory IV



Lecture Notes


Introduction to algebraic number theory

This course (36 hours) is a relatively elementary course which requires minimal prerequisites from commutative algebra for its understanding. Its first part (modules over principal ideal domains, Noetherian modules) follows to a certain extent the book of P. Samuel "Algebraic Theory of Numbers". Then integrality over rings, algebraic extensions of fields, field isomorphisms, norms and traces are discussed in the second part. In the main third part Dedekind rings, factorization in Dedekind rings, norms of ideals, splitting of prime ideals in field extensions, finiteness of the ideal class group and Dirichlet's theorem on units are treated. 


Homological algebra

This course (25 hours) presents categories, functors, chain complexes, homologies, free, projective and injective objects in the category of modules over a ring, projective and injective resolutions, derived functors, Tor and Ext, cohomologies of modules over a finite group, restriction and corestriction.


Lectures on complete discrete valuation fields

This is a very short introduction to local fields and local class field theory. This course (36 hours) presents basic features of local fields and the local reciprocity map.