Math 702 COURSE INFORMATION - Spring 2018

Faculty: Ted Chinburg
Office Hours: By appointment.

COURSE PLAN

The overall goal of the course is to discuss the application of etale cohomology to cryptography, Iwasawa theory, and arithmetic Chern Simons theory. This course will be similar to a working seminar. Three primary goals are:

  1. Construct cryptographically useful multilinear maps using etale cohomology. There was a breakthrough in October at a meeting at AIM about how a large group of people can establish a common secret. I will talk about cryptographic multilinear maps and their uses, and some particular approaches to building them using elliptic curves.

  2. Apply duality theorems in etale and Galois cohomology to study the structure of Iwasawa modules. The new direction in this topic has to do with understanding higher order terms in the natural growth rates of numerical invariants of number fields in towers.

  3. Apply etale cohomology to study arithmetic analogs of Chern Simons theory and topological quantum field theory. This is a very new subject suggested by M. Kim in 2015.

TEXTS:

Basic Texts

  1. (Primary Text 1) Milne, J.: ``Etale cohomology," Princeton University Press, U.S.A. (1980).

    This book gives an accessible introduction to the topic at a somewhat more formal level than the book ``Lectures on Etale cohomology" listed below.

  2. (Primary Text 1) Milne, J.: ``Lectures on etale cohomology"

    This is a more informal approach to the reference mentioned above.

  3. (Primary Text 1) Hartshorne R..: ``Algebraic geometry," Springer Graduate Texts in Mathematics, 1997 edition.

    This a good all around reference for the subject.

  4. (Primary Text 1) Silverman, J.: ``The arithmetic of elliptic curves," 2nd edition, Springer (2016)

    This gives a solid introduction to all aspects of the theory of elliptic curves.

  5. (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

  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: 1/10/18
Send e-mail comments to: ted@math.upenn.edu