Visualizing Automorphic Functions

Frank A. Farris and Jeffrey Hoffstein

We use Farris' methods for visualizing complex-valued functions in the plane.

Given any complex number s, with Re(s)>1, we start with the function ys (in the upper half plane) and average it over the action of the group SL(2,Z). Because that group is infinite, we can't possibly use every element, but the convergence properties for these functions are nice enough that the error isn't too bad when we only include some hundreds of terms. We are confident that these pictures look reasonable away from the real axis.

You can click on any of the images to enlarge them to big 800 x 600 images in 24-bit color.

A good place to start is with s = 2 + 5 i . We have multiplied the function by a constant (25) to scale it into a range of values that looks pretty.

A slightly more oscillatory function has s = 1.5 + 4 i:

Increasing the real part of s to 3 makes a function that approaches infinity a bit more rapidly as y grows. This one is also more oscillatory, as s = 3 + 12 i :

The most oscillatory case we computed becomes rather wild. This one has s = 1.1 + 15 i:

Finally, our most recent efforts involve averaging functions like yssin(2 pi k x) over the group. Siman Wong provided some nice code that helped with this.

Here is a close-up of a different function based on sines.

The next step is to depict a linear combination of functions like these. Here is one example:

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