Mathematicians find surprising order in modular forms
A paper published in the Proceedings of the National Academy of Sciences (PNAS) uncovers a surprising pattern in one of math’s oldest areas: number theory.
Number theory is the study of whole numbers, often starting with simple-looking questions. For example, the equation y² = x³ + 1 has a solution at (2,3). But finding all the solutions is harder than it looks.
“Even when these questions look simple, they can often be extremely difficult to answer because the usual tools of mathematics don’t easily apply,” said Preston Wake, associate professor of mathematics at Michigan State University and co-author of the study.
To tackle such problems, researchers turn to advanced tools called modular forms. These functions are complicated, but they allow mathematicians to reframe number theory problems in ways that can be solved using geometry, algebra and calculus.
“It may seem strange to go from something simple like y² = x³ + 1 to something as complex as modular forms, but this complexity is kind of the point,” Wake said. “If you can change a number theory problem into a problem about modular forms, then suddenly you have a whole new range of tools to use.”
Wake teamed with Jaclyn Lang of Temple University to publish “The Eisenstein ideal at prime-square level has constant rank.” The study focuses on special connections between modular forms called Eisenstein congruences. These connections link a complicated modular form with a simpler one, offering a shortcut to understanding.
In earlier research, Wake and a colleague found that the number of Eisenstein congruences could change in irregular and unpredictable ways. But the new paper shows the opposite: in one setting, the number of congruences is constant and can be described exactly.
“In a previous paper, we showed that the answer to this kind of question could be very irregular,” Wake said. “In this paper, we show that in a slightly different situation, the answer is remarkably regular and can be described precisely.”
Number theory often inspires techniques that later spread into other areas of math and science. The discovery shows that even in one of the most complex areas of mathematics, surprising order can emerge.
“Number theory uses tools from so many different areas of math that it serves as a cross-pollinator of ideas,” Wake said. “Techniques first developed for number theory often end up becoming essential in other areas of mathematics — and sometimes those areas do have real-world applications.”
