MATHEMATICS G6657, Spring 2018: Algebraic number theory





            There will be no classes on Wednesday, January 17. 


            Instead, there will be a three-hour review of     basic algebraic number theory, and an introduction to adèles, on the morning of Friday, January 19, from 9-12 am, in room 507.



            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

D. Cox, Primes of the form x2 + ny2

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).


Grades will be based on a take home final; undergraduates attending the class will also be expected to turn in regular homework assignments (roughly every two weeks).



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