Crucially, he realized that they exist in certain spaces - spaces with specific dimensions and other properties. In the 1920s and ’30s, the German mathematician Erich Hecke developed a deeper theory around modular forms. And there’s a reason why that’s the case. “I mean, carving up the upper half-plane and putting numbers on various places - who cares?” “That doesn’t necessarily sound very exciting,” said John Voight, a mathematician at Dartmouth College. You can combine these transformations in infinitely many ways, which gives you the infinitely many symmetry conditions that the modular form must satisfy. However, the values at the two points relate to each other in a regular way that also gives rise to symmetry. In this case, the modular form doesn’t necessarily assign those points the same value. 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. If you know how the function behaves there, you’ll know what it does everywhere else. This region looks like a strip going up from the horizontal axis with a semicircular hole cut out of its bottom. To do so, it helps to try to simplify the way we look at these complicated functions.īecause of the modular form’s symmetries, you can compute the entire function based on just a narrow sliver of inputs, located in a region of the plane called the fundamental domain. But it can be tough to make sense of the actual function those colors and contour lines represent. Modular forms exhibit a bewildering variety of such symmetries. This is one symmetry of a complex-valued function. More generally, the graph looks the same when you flip any point across the center (or origin). It also has more contour lines, because the outputs grow in size more quickly. The graph of $latex f(z) = z^2$ runs through the colors twice, because squaring a complex number doubles its angle. In general, a shape is said to have symmetry when there is some transformation that leaves it the same. To understand a modular form, it helps to first think about more familiar symmetries. ![]() And it’s what now makes them crucial to the ongoing development of a “mathematical theory of everything” called the Langlands program. It’s what made them central to more recent work on sphere packing. It’s what made them key players in the landmark 1994 proof of Fermat’s Last Theorem. The properties that come with those symmetries make modular forms immensely powerful. They are often described as functions that satisfy symmetries so striking and elaborate that they shouldn’t be possible. Every week, new papers extend their reach into number theory, geometry, combinatorics, topology, cryptography and even string theory. But “there are probably fewer areas of math where they don’t have applications than where they do,” said Don Zagier, a mathematician at the Max Planck Institute for Mathematics in Bonn, Germany. Modular forms are much more complicated and enigmatic functions, and students don’t typically encounter them until graduate school. Part of the joke, of course, is that one of those is not like the others. “Addition, subtraction, multiplication, division and modular forms.” (x,y)\rightarrow (−y,−x)\).“There are five fundamental operations in mathematics,” the German mathematician Martin Eichler supposedly said.
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