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.
Mode of Operation and Evidence of Learning
The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.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 contact me.Co/Pre-requisites
- Algebra II, Complex Analysis II
- Topics in Algebra and Geometry IV (essential)
- Number Theory III would be very helpful. Depending on
the exact direction Representation Theory IV wouldn't
hurt (but is not essential).
email: J Funke