Math 678

Math 678: Modular Forms

Fall 2012, Section 1

Time:

MWF 10:10 a.m- 11:00 am.

Place:

East Hall 3866

Instructor:

Jeffrey Lagarias, 3086 East Hall, 763-1186, lagarias@umich.edu

Office hours:

MWF 12:00-1:00pm [Tentative] TBA.
(Or by appointment: call or email me)

Course homepage:
http://www.math.lsa.umich.edu/~lagarias/ Public/html/m678fa12.html

Textbook:

Course notes to be developed.

Reference Books :

These texts cover more ground than the course can possibly cover.

(1) Igor Dolgachev,
Modular Forms
Course Notes 1997--1998, 147pp
[Emphasis on theta function viewpoint]

pdf file

(2) James Milne,
Modular Functions and Modular Forms (Elliptic Modular Curves),
138pp, available on Milne website
[Modular forms and algebraic curves]

(3) Fred Diamond and Jerry Schurman
A First Course in Modular Forms,
Springer-Verlag: GTM 228, (2005)
[Arithmetic modular forms, aimed at Wiles-Taylor FLT Proof]
[Copies available inexpensively through UM Library system]

(4) Joseph Lehner,
Discontinuous Groups and Automorphic Functions ,
Math. Surveys No. 8, Amer. Math. Soc. 1964
[Classical and careful, detailed treatment of subgroups of modular groups]

(5) Toshitsune Miyake,
Modular Forms,
Springer-Verlag 1989

(6) H. Maass,
Lectures on Modular Functions of One Complex Variable
Tata Institute Lecture Notes: Bombay 1964 (revised 1983)
[Treats non-holomorphic modular forms]

(7) Henri Cohen,
Modular Forms: A Classical Approach,
Book in preparation, pdf file, 614 pages.
[Has emphasis on being able to compute things. Complex variables orientation.]

(8) J.-P. Serre,
A Course in Arithmetic
Springer-Verlag, NY 1973.
[A beautiful book. See Chap. VII]

On the automorphic side:

(A1) Daniel Bump,
Automorphic Forms and Representations,
Cambridge University Press, 1997
[Good general introduction via number theory and Langlands program.]

(A2) Henryk Iwaniec,
Topics in Classical Automorphic Forms
AMS: Providence 1997.

(A3) Henryk Iwaniec,
Spectral Theory of Automorphic Forms, Second Edition
AMS, Providence 2002. [Spectral theory of Laplacian, trace formula.]

(A4) Armand Borel,
Automorphic Forms on SL(2, R)
Cambridge Univ Press, 1997 [From viewpoint of general reductive group.]

Course description:

From blurb: ``Modular forms involve a wonderful overlap of arithmetic, algebra, analysis and geometry. This is a basic course on modular forms, expected to take an analytic viewpoint
but covering algebraic aspects. It will cover the modular group, classical modular forms (holomorphic and non-holomorphic) Eisenstein series, and may cover related spectral theory
for SL(2, R). This will include Hecke operators and the connection with Dirichlet series with Euler products. It will also cover various aspects of theta functions, quadratic forms
and associated theory. Applications may include theory of partitions, representations by quadratic forms, connections to elliptic curves, with and without complex multiplication."

Prerequisites:

Math 575/675 is helpfu1.
Some comfort with complex analysis is helpful.

Grading:

Grades: These will be based on problem sets. There will be approximately 7 or 8 problem sets.

Collaboration on the homework is permitted, but each person is responsible for writing up her/his own solutions. All external sources used in getting solutions should be explicitly credited, individually or at end of homework.