## Math 678: Modular Forms

**Fall 2017, Section 1 **

### Time:

TR 2:30 pm- 4:00 pm.
### Place:

3866 East Hall
### Instructor:

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

### Office hours:

[Tentative]
TBA.

(Or by appointment: call or email me)
**Course homepage:**

http://www.math.lsa.umich.edu/~lagarias/
Public/html/m678-17fa.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.

**Homework Assignments:**