2024-08-08
I made this toy so I could get more visual intuition for Bianchi modular forms, which are vector-valued automorphic forms for GL(2) over imaginary quadratic fields. The particular form on the page is weight 2, meaning it has 3 components. Each component is a function from hyperbolic 3-space to the complex numbers.
These plots are using the model of hyperbolic 2-space given by the upper-half space (in analogy with the upper-half plane used in the theory of classical modular forms). Each plot shows the complex value of the component in the standard way (argument is given by hue, magnitude by light/darkness). The plots are "looking down" at the copy of the complex plane sitting at the bottom of hyperbolic 3-space, at a particular height. This height can be controlled by the buttons.
The button labeled "0.3264..." is the plot of the form at height approximately 1/(88)^(1/4), where all three components appear to vanish in the middle of the fundamental domain.