Modular forms: Modular functions

This lecture is part of an online graduate course on modular forms.

We classify all meromorphic modular functions, showing that they are all rational functions of the elliptic modular function j.

As an application of j we use it to prove Picard's theorem that a non-constant meromorphic function can omit at most one value.

[officialEricBG]

the more I watch this series, the more I think the very existence of (non-trivial) modular forms is a miracle

[criskity]

This is mind-bending, neuron-jangling material.

[unkennyvalley]

This was related to my undergrad thesis... Time to walk down the memory lane

[jeffreyhowarth]

Category algebra generalize the notion of group algebras . Stack is a generalization of schemes. Automorphic forms are a generalization of periodic functions. Tensors are a generalization of scalers. Any thoughts on the word 'generalization' ?

[koenbres]

At 5:12, can't f have an infinite number of poles in the upper half plane going upwards? In this case step (1) would give an infinite product. Or would this imply that f is not meromorphic at i\infty?

[bobbobson]

I believe j(t) is related by the expression A(t)/B(t)^3 = 1-720/j(t) where B(t) is the theta function over the lattice E8 and A(t) is the theta function over the 12 dimensional Leech Lattice. I think this is a neat way to describe a modular function?

[truthteller]

Who think he looks like the actor Jonathan Pryce?