Math 702 COURSE INFORMATION - Fall 2012

TEXTS:

Basic Texts

1. (Primary Text 1) Lang, S.: ``Algebraic number theory," second edition, Springer-Verlag, U.S.A., (1994), ISBN ISBN 3-540-94225-4.
   
   This is a comprehensive text which includes class field theory as well as various topics from analytic number theory such as the Brauer-Siegel Theorem and Weil's explicit formulas. This book is best after having already seen a lower level introduction to the theory.
   
   
2. (Secondary Text) Lorenzini, D.: ``An Invitation to Arithmetic Geometry," Graduate Studies in Mathematics, Vol. 9, Amer. Math. Soc., U.S.A. (1996), ISBN 0-8218-0267-4.
   
   This book covers the beginning of the theory for number fields and for function fields of curves at the same time.
   
   

Supplemental Texts (not required)

1. Janusz, G.: ``Algebraic number fields," second edition, Graduate Studies in Mathematics, Vol. 7, American Mathematical Society, U. S. A. (1996), ISBN 0-8218-0429-4.
   
   This is a comprehensive text, in that it covers class field theory. It takes a more concrete approach than Lang, but does not cover as many topics. it also has useful exercises.
   
   
2. Frohlich, A. and Taylor, M. J., ``Algebraic number theory," Cambridge studies in advanced mathematics, Vol. 27, Cambridge Univ. Press, Cambridge (paperback 1994, hardcover 1991), ISBN 0521438349 (paperback), 052136664 (hardback).
   
   This book covers the beginning of the theory, in that it does not include classfield theory. There is a greater emphasis on module theory than in other books, and it has some discussion of other topics such as elliptic curves. There are many useful exercises toward the back of the book.
   
   
3. Samuel, P.: ``Algebraic Theory of Numbers," Hermann Publishers and Kershaw Publishing company, (1971).
   
   This an elegant and concise summary of the beginning of the theory.
   
   
4. Serre, Jean-Pierre, ``Local Fields,"Graduate Texts in Mathematics, Springer, U.S.A. (1991), ISBN 3-540-904247.
   
   This is the standard reference for the theory of local fields and for the theory of group cohomology.
   
   
5. Serre, Jean-Pierre, ``A course in arithmetic," Springer-Verlag, New York Heidelberg Berlin.
   
   This is an excellent introduction to various topics not usually covered in books on algebraic number theory. These include the theory of quadratic forms in many variables and an introduction to the theory of modular forms.
   
   
6. Shimura, G.: ``Introduction to the Arithmetic Theory of Automorphic Functions," Princeton University Press, U.S.A. (1971), ISBN13: 978-0-691-08092-5.
   
   This is a high level introduction to the theory of modular forms, including the theory of complex multiplication.
   
   
7. Milne, J. S., ``Algebraic Number Theory," from his web page
   
   This page has links to a large number of well-written books and course notes on topics in number theory and arithmetic geometry
   
   
8. Niven, I., ``Irrational Numbers," Carus Mathematical Monographs, Mathematical Association of America, U. S. A. (2005), ISBN 0-88385-038-9.
   
   This is a nice introduction to the theory of irrational and transcendental numbers. It includes a proof of the Gelfond-Schneider theorem. .

APPROXIMATE SYLLABUS

1. Algebraic numbers, algebraic integers and transcendental numbers.
2. Dedekind rings, ideals, class numbers, function field analogs.
3. Completions, local fields, ramification theory.
4. Differents and discriminants. Geometric analogs.
5. Geometry of numbers, finiteness theorems, Riemann Roch theorems.
6. Cyclotomic fields.
7. Ideles and Adeles.
8. Zeta and L-functions, density theorems.
9. Local and Global class field theory. Cohomology of groups.
10. Functional equations of L-series.
11. Weil's explicit formulas and the Brauer-Siegel theorem.
12. Introduction to modular forms and the theory of Galois representations.
13. Elliptic curves and Fermat's Last Theorem.

COURSE ORGANIZATION

I will post reading and homework assignments on the web.

GRADING:

               Homework                                 100% of grade