In the 1920s and \u201930s, the German mathematician Erich Hecke developed a deeper theory around modular forms. Crucially, he realized that they exist in certain spaces \u2014 spaces with specific dimensions and other properties. He figured out how to describe these spaces concretely and use them to relate different modular forms to one another.<\/p>\n

This realization has driven a lot of 20th- and 21st-century mathematics.<\/p>\n

To understand how, first consider an old question: How many ways can you write a given integer as the sum of four squares? There is only one way to write zero, for instance, while there are eight ways to express 1, 24 ways to express 2, and 32 ways to express 3. To study this sequence \u2014 1, 8, 24, 32 and so on \u2014 mathematicians encoded it in an infinite sum called a generating function:<\/p>\n

$latex 1 + 8q + {{24q}^2} + {{32q}^3} + {{24q}^4} + {{48q}^5} + \u2026$<\/p>\n

There wasn\u2019t necessarily a way to know what the coefficient of, say, $latex q^{174}$ should be \u2014 that was precisely the question they were trying to answer. But by converting the sequence into a generating function, mathematicians could apply tools from calculus and other fields to infer information about it. They might, for instance, be able to come up with a way to approximate the value of any coefficient.<\/p>\n

But it turns out that if the generating function is a modular form, you can do much better: You can get your hands on an exact formula for every coefficient.<\/p>\n