Description
In this project, I offer, based on the second term of the
Elliptic Functions Course last year, several topics in the
theory of modular forms. Among these are
- Rankin-Selberg Method.
This method was originally developed by Rankin and
Selberg to improve Hecke's bound on the growth of the
Ramanujan tau-function, the Fourier coefficients of the
discriminant function. It is a beautiful fusion of ideas
and techniques from theory of modular forms and analytic
number theory.
- Theta Series and Eisenstein Series: the Siegel-Weil
Formula. We have seen that a theta series is
typically not an Eisenstein series (essentially, only
when it was forced to be one since the underlying space
was one-dimensional). However, a certain linear
combination of theta series is an Eisenstein series.
This is the theme of the celebrated Siegel Maass
formula. It yields beautiful arithmetic formulas such as
a formula for the sum of 3 squares.
- Modular Forms and the Partition Function: The
partition function is one of the most classical objects
in combinatorial number theory. It has been studied
extensively, in particular by Ramanujan. Recently, the
theory of modular forms has yielded spectacular
progress.
- Modular Forms, Theta Series and Algebraic Number
Theory: Let K be an imaginary quadratric
field. Then every ideal class of K gives rises to a
positive definite quadratic form which gives rise to a
theta series. In this way we can study the
arithmetic of K using modular forms. The connection goes
actually much deeper and is very conceptual. It is an
instant of the celebrated Langlands Programme; one of
the most fundamental theme in mathematics of our time.
Depending on interest we can also look at other topics in
the theory of modular forms, such as its connections to
elliptic curves.
Resources
Depend on the chosen topic. For further questions feel free
to see me.
Prerequisites
- Elliptic functions III.
- For some of the topics Number Theory III and/or
Representation Theory IV will be helpful.
email: J Funke
|