Modular functions

[SLD]

So modular functions are really cool. They’re of the form of f(z)=(az+b)/(cz+d). These are supposed to be especially interesting if ad-bc=1. With applications to group theory and even quantum physics.

but I’ve been playing with a complex mapping app and noticed that they all seem to simply invert the complex plane with a Hole in the middle, although the details vary.


but is that true?

 

[SLD]

One thing I’ve noticed, if I increase my a in the above equation, it shifts the map to the right and makes it bigger. I need to play with the other parameters to see what they do.

 

[lpetrich]

So modular functions are really cool. They’re of the form of f(z)=(az+b)/(cz+d). These are supposed to be especially interesting if ad-bc=1. With applications to group theory and even quantum physics.

but I’ve been playing with a complex mapping app and noticed that they all seem to simply invert the complex plane with a Hole in the middle, although the details vary.

(LP: picture snipped for brevity)
but is that true?

Easy to verify.

\( \displaystyle{ f(z) = \frac{a z + b} {c z + d} = \frac{1}{c} \left( a + \frac{b c - a d}{c z + d} \right) } \)

 

[lpetrich]

If a*d = b*c, then the function degenerates into a constant function.

If one applies a modular function to another modular function, one obtains a new modular function.

\( \displaystyle{ f(A,z) = \frac{ a_{11} z + a_{12} }{ a_{21} z + a_{22} } } \)

for
\( \displaystyle{ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} } \)

Function composition:
\( f(A,f(B,z)) = f(A \cdot B,z) \)

where A.B is matrix multiplication: \( (A \cdot B)_{i,j} = \sum_k A_{i,k} B_{k,j} \)

Degeneration into a constant function: det(A) = 0 - the determinant of that matrix.

Determinants are a rather complicated sort of function: \( \det A = \sum \epsilon_{i1,i2,\dots,in} A_{1,i1} A_{2,i2} \cdots A_{n,in} \) for A being a n*n matrix, where \( \epsilon_{i1,i2,\dots,in} \) is 1 for an even permutation of 1, 2, ..., n and -1 for an odd permutation. An even permutation can be made with an even number of interchanges, an odd permutation an odd number. For the 2*2 case, it's easy: 1,2 is an even permutation because it has zero interchanges, and because zero is an even number, and 2,1 is an odd permutation because it has one interchange.

However, determinants have this nice composition rule: det(A.B) = det(A)*det(B).

 

[lpetrich]

Modular functions and group theory?

The set of all invertible matrices with size n over matrix multiplication forms a group, GL(n,X), the general linear group, where the matrices' elements are in set X with addition, subtraction, multiplication, and division defined for it. X is an algebraic field, like the rational numbers, the real numbers, the complex numbers, and the integers modulo some prime number.

Taking all of these matrices with determinant 1, one finds SL(n,X), the special linear group. Closure is easy to prove.

Looking back at modular functions, one can scale all of the parameters and get the same result, if one scales by the same number: a -> x*a, b -> x*b, c -> x*c, d -> x*d. So we group the matrices by sets related by multiplication by some number: scalar multiplication. For matrix A, {x*A for all nonzero x in X}. One then defines multiplication on these sets as {x*A} . {y*B} = {z*(A.B)}.The resulting group is PGL(n,X), the projective general linear group, and with determinant = 1, PSL(n,X), the projective special linear group. But in that group, the multipliers x are restricted to x^n = 1.

So for modular functions, a, b, c, d form the group PGL(2,X).