Modular forms, the 'fifth fundamental operation' of math (2023)

Two kinds of transformations copy the fundamental domain to the right and left, as well as to a series of ever-shrinking semicircles along the horizontal axis. These copies fill the entire upper half of the complex plane. A modular form relates the copies to each other in a very particular way. That’s where its symmetries enter the picture. If you can move from a point in one copy to a point in another through the first kind of transformation — by shifting one unit to the left or right — then the modular form assigns the same value to those two points. Just as the values of the cosine function repeat in intervals of $latex 2\pi$, a modular form is periodic in one-unit intervals. Meanwhile, you can get from a point in one copy to a point in another through the second type of transformation — by reflecting over the boundary of the circle with radius 1 centered at the origin. In this case, the modular form doesn’t necessarily assign those points the same value. However, the values at t ... Read full article.