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?
poles must be isolated, unless you have an essential singularity which is not meromorphic anymore. The fundamental domain is compact so it only allows finite many isolated points thus poles
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?
The fundamental domain has area π/3 so 12 of them cover the Riemann sphere of area 4π? But we're on the half plane and it has an infinite # of fundamental domains. Also one was glued to form a disk then compacted by the point at infinity making the sphere of 4π?
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' ?
the more I watch this series, the more I think the very existence of (non-trivial) modular forms is a miracle
@@ericvosselmans5657 it's never too late! :D
This is mind-bending, neuron-jangling material.
This was related to my undergrad thesis... Time to walk down the memory lane
amazing videos, keep it up 👍
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?
poles must be isolated, unless you have an essential singularity which is not meromorphic anymore. The fundamental domain is compact so it only allows finite many isolated points thus poles
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?
The fundamental domain has area π/3 so 12 of them cover the Riemann sphere of area 4π? But we're on the half plane and it has an infinite # of fundamental domains. Also one was glued to form a disk then compacted by the point at infinity making the sphere of 4π?
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' ?
Amazing...
yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Who think he looks like the actor Jonathan Pryce?
First
badly explained