Introduction to Modular Forms
Boston College, Fall 2011
Modular forms are classical analytic objects which were the center of much attention early in the last century. For some time their interest appeared to have diminished, but then remarkable connections with a huge range of other areas in pure mathematics were discovered. Currently, modular forms and their close relatives, the automorphic forms, are present in almost every area of modern number theory. The most celebrated is perhaps the role they played in the proof of Fermat's last theorem, through the conjecture of ShimuraTaniyamaWeil that elliptic curves are modular. Modular forms also play an important role in the Langlands program and in recent advancement towards the quantum unique ergodicity conjecture.
The aim of this course is to cover the classical theory of modular forms and point out some classical applications to number theory. In particular, some of the topics that will be covered are
 Elliptic functions
 Eisenstein series
 The modular group and its subgroups
 Basic properties of modular forms
 Hecke operators
 Dirichlet series attached to modular forms
 Theta functions and quadratic forms
 Maass waveforms
Prerequisites
The course is intended for the second year graduate students; first year graduate students are also welcome.
I will assume knowledge of basic courses in complex analysis, linear algebra, and group theory.
Bibliography
 H. Iwaniec, Topics in classical automorphic forms
 N. Koblitz, Introduction to elliptic curves and modular forms
 J.P. Serre, A Course in Arithmetic
Schedule
MWF 910
Room: Gasson 208
Homework
There will be periodic homework assignments posted here.
Assignments

Assignment 1, due date: Sep. 28, 2011.

Assignment 2, due date: Oct. 12, 2011.

Assignment 3, due date: Nov. 09, 2011.

Assignment 4, due date: Dec. 14, 2011.
Contact me at: dubi.kelmer@bc.edu, Office : Carney 268, tel: (617) 5526588
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