Math 621 COURSE INFORMATION  Spring 2010
Faculty: Ted Chinburg
Office Hours: By appointment. I'll organize some Skype office hours in the evening as well.
 TEXTS:

Basic Texts
 (Primary Text 1) Lang, S.: ``Algebraic number theory,"
second edition, SpringerVerlag, U.S.A., (1994), ISBN ISBN 3540942254.
This is a comprehensive text which includes class field theory as well
as various topics from analytic number theory such as the BrauerSiegel Theorem
and Weil's explicit formulas. This book is best after having already seen a
lower level introduction to the theory.
 (Secondary Text) Lorenzini, D.: ``An Invitation to Arithmetic Geometry," Graduate Studies in Mathematics, Vol. 9,
Amer. Math. Soc., U.S.A. (1996), ISBN 0821802674.
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)
 The Gigapedia website
This is an extremely useful site. After registering, search for books using the gigapedia option in
the upper right corner of the home web page. Click on icons of books you would like to download.
Then clink on the ``links" button for the associated book to find mirror sites from which you can
download .pdf or .djvu files of the book.
 Janusz, G.: ``Algebraic number fields," second edition, Graduate Studies
in Mathematics, Vol. 7, American Mathematical Society, U. S. A. (1996), ISBN 0821804294.
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.
 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.
 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.
 Serre, JeanPierre, ``Local Fields,"Graduate Texts in Mathematics, Springer, U.S.A. (1991),
ISBN 3540904247.
This is the standard reference for the theory of local fields and for the theory of group cohomology.
 Serre, JeanPierre, ``A course in arithmetic," SpringerVerlag, 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.
 Shimura, G.: ``Introduction to the Arithmetic Theory of Automorphic Functions," Princeton
University Press, U.S.A. (1971), ISBN13: 9780691080925.
This is a high level introduction to the theory of modular forms, including the theory of complex multiplication.
 Milne, J. S., ``Algebraic Number Theory," from his web page
This page has links to a large number of wellwritten books and course notes on topics in number theory and arithmetic geometry
 Niven, I., ``Irrational Numbers," Carus Mathematical Monographs, Mathematical Association of
America, U. S. A. (2005), ISBN 0883850389.
This is a nice introduction to the theory of irrational and transcendental numbers. It includes a proof of the
GelfondSchneider theorem.
.
 APPROXIMATE SYLLABUS

 Algebraic numbers, algebraic integers and transcendental numbers.
 Dedekind rings, ideals, class numbers, function field analogs.
 Completions, local fields, ramification theory.
 Differents and discriminants. Geometric analogs.
 Geometry of numbers, finiteness theorems, Riemann Roch theorems.
 Cyclotomic fields.
 Ideles and Adeles.
 Zeta and Lfunctions, density theorems.
 Local and Global class field theory. Cohomology of groups.
 Functional equations of Lseries.
 Weil's explicit formulas and the BrauerSiegel theorem.
 Introduction to modular forms and the theory of Galois representations.
 Elliptic curves and Fermat's Last Theorem.
 COURSE ORGANIZATION

I will post reading and homework assignments on the web.
 GRADING:

Homework 100% of grade
Last updated: 1/3/2010
Send email comments to: ted@math.upenn.edu