Description
In this project, I offer, based on the first term of TAG IV (modular forms) several to[ics in the theory of modular forms. Among these are- Congruences.
Ramanujan found a spectacular congruence of his
mysterious
tau-function (the Fourier coefficients of the
discriminant function) with a divisor sums function, a
much more elementary function:
- Modular Forms and Algebraic Number Theory:
Let K be an imaginary quadratric field. Then to
every ideal class of K one can assciate an anamytic
object, it's so-called L-function
which encodes important information about K but is
also closely related to modfular forms. 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.
Please note that on term 2 I will be away on research leave and the supervision will happen remotely during that period.
Resources
Depend on the chosen topic. For further questions feel free to see me.Prerequisites
- Topics in Algebra and Geometry IV.
- For some of the topics Number Theory III and/or
Representation Theory IV will be helpful.
email: J Funke