**Fall 2012, Section 1 **

(Or by appointment: call or email me)

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

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

(1) Igor Dolgachev,

Course Notes 1997--1998, 147pp

[Emphasis on theta function viewpoint]

(2) James Milne,

138pp, available on Milne website

[Modular forms and algebraic curves]

(3) Fred Diamond and Jerry Schurman

Springer-Verlag: GTM 228, (2005)

[Arithmetic modular forms, aimed at Wiles-Taylor FLT Proof]

[Copies available inexpensively through UM Library system]

(4) Joseph Lehner,

Math. Surveys No. 8, Amer. Math. Soc. 1964

[Classical and careful, detailed treatment of subgroups of modular groups]

(5) Toshitsune Miyake,

Springer-Verlag 1989

(6) H. Maass,

Tata Institute Lecture Notes: Bombay 1964 (revised 1983)

[Treats non-holomorphic modular forms]

(7) Henri Cohen,

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.]

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."

Some comfort with complex analysis is helpful.

**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.

Here is a current

**Homework Assignments:**