MATHEMATICS G6657, Spring 2018


Algebraic number theory



            This year's course will be devoted primarily to class field theory.  Here is a tentative schedule:


1.  Adèles and idèles of number fields, including a proof of finiteness of class number and Dirichlet's unit theorem (2 weeks)


2.  Tate's thesis (functional equations of L-functions of Hecke characters), including the analytic class number formula (2-3 weeks)


3.   Cohomology of groups and Galois cohomology (2-3 weeks)


4.   Global class field theory and applications (4 weeks)


5.   Lubin-Tate formal groups and local class field theory (2 weeks)


If there is time, I will also cover the determination of the values at s = 1 of Dirichlet L-functions and the beginning of the theory of cyclotomic fields.


REFERENCES (incomplete list)


A. Weil, Basic Number Theory

J.W.S. Cassels and A. Fröhlich, Class Field Theory

E. Artin and J. Tate, Class Field Theory

S. Lang, Algebraic Number Theory

J.S. Milne, notes on class field theory

J.-P. Serre, Local Fields

J. Bernstein, S. Gelbart, An Introduction to the Langlands Program (especially S. Kudla's chapter on Tate's thesis)


For background reading, I also recommend Milne's notes on algebraic number theory, which provide a more complete introduction to algebraic number theory than my course notes (but with fewer applications to Diophantine problems).




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