Re:Modular forms, the ‘fifth fundamental operation’ of math
by joe__f on Friday September 22, 2023 @06:12AM
(#37610137)
Oh yeah that's a nice observation!
Protective space in 2D is the same as the Riemann sphere, which is the complex plane with one point added at infinity.
I forgot to add to my post that, SL(2) is the symmetry group of the Riemann sphere. So since you often use this as your space in complex analysis, this symmetry transformation is everywhere. So that's one reason why you might expect to see modular forms in lots of places. (At least, that's what I understood, maybe a mathematician would tell you differently.)
let f(ax, ay) = a^k f(x,y) be a homogeneous function in the plane x,y for a scalar a, then f can also be written as:
f(x,y) = f(y*x/y, y) = y^k f(x/y,1)
and the symmetry by a matrix transformation in the plane
x-> ax + by and y-> cx+dy then transforms the function f as:
f(ax+by, cx+dy) = (cx+dy)^k f((ax+by)/(cx+dy),1)
now introduce z=x/y and f(x/y,1)=F(z) then
F((az+b)/(cz+d)) = (cz+d)^k F(z)
Projective spaces are cool ;-)
just my 2 ct