Modular forms: Hecke operators

This lecture is part of an online graduate course on modular forms.
We introduce Hecke operators for modular functions in three different ways.
For the other lectures in the course see • Modular forms

[@timduncankobebryant]

These modular function is hecka fun

[@epsilonmath6774]

Great lecture. I hope you are well and keep up the good work!

[@caspermadlener4191]

Just figured out that Hecke operators do more than what is shown in this video!
So, the video showed that
j(z/2)+j((z+1)/2)+j(2z) is a modular function, but you can also do other constructions with them.
Just take the product of the three fake modular forms!
Since every symmetric expression with them becomes a modular function, you can easily calculate the polynomial with these roots!
This means that you can evaluate j(z/2), simply by solving a third degree polynomial with coefficients polynomials j(z). Cool, right?
The fact that you have to solve a polynomial in j(z) actually implies cool Galois theory... because you basically get a Galois extension of SL(2,p) of the domain (ring) of modular functions, which is equivalent to the domain of polynomials, and proofs that for every prime p, the domain of polynomails has an extension isomorfic with SL(2,p)
Glad I figured this out. Definitely going to try to do more algebra with this!

[@SG-kj2uy]

Hecke! Hecke! Hecke!

[@migarsormrapophis2755]

yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee