Math 720 COURSE INFORMATION - Spring 2018

COURSE PLAN

In this course I will give an overview of Iwasawa theory and of some aspects of the theory of modular forms and Galois representations. The eventual goal of the course is to describe some current research topics having to do with higher codimension Iwasawa theory.

Iwasawa theory began as the study of the rates of growth of classical invariants of number fields in towers, e.g. of the order of the p-parts of their ideal class groups. The classical invariants have now been vastly generalized by the theory of Selmer groups and Panchisckin conditions. The deepest conjectures in the subject link algebraically defined growth rates to analytic objects, e.g. to p-adic L-functions. These links have been established in two basic ways: via the theory of modular forms and Galois representations, and via the theory of Euler systems.

Higher codimension Iwasawa theory has been a new development in the subject over the last 5 years. The main idea is that rather than studying the linear terms in the growth rates, as in classical Main Conjectures, one should study the leading terms. I will eventually describe was is now known and some natural ideas to pursue now.

TEXTS:

Basic Texts

1. (Primary Text 1) Washington, L., "Introduction to cyclotomic fields," 2nd ed. Graduate texts in mathematics ; 83, Springer, 1996.
   
   This book gives an accessible introduction to Iwasawa theory.
   
   
2. (Primary Text 2) Hida, H., "Hilbert modular forms and Iwasawa theory," Oxford mathematical monographs. Oxford : Clarendon, 2006.
   
   This book describes the use of modular forms and their associated Galois representations in Iwasawa theory.
   
   
3. (Primary Text 3) Nekovar, Jan. "Selmer complexes," Asterisque ; 310. Societe mathematique de France, 2006.
   
   This book includes a general theory of Selmer groups and complexes as well as an account of various duality theorems which are useful in this theory.
   
   
4. (Primary Text 4) Rubin, K.: ``Euler systems," Annals of Mathematics Studies, Princeton University Press, 2000.
   
   This gives an introduction to Euler systems. These provide a way to prove upper bounds on Selmer groups.
   
   
5. (Primary Text 5)Serre, J.P..: ``Number Theory " (1976 Harvard course); Notes by Jim Weisinger.
   
   This is a terrific reference for the state of the art at the time.
   
   

Supplemental Texts

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. 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.
   
   
3. 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.
   
   
4. 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.
   
   
5. 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.
   
   
6. 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.
   
   
7. 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.
   
   
8. 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
   
   

Last updated: 8/21/18
Send e-mail comments to: ted@math.upenn.edu