New Elliptic Curve Breaks 18-Year-Old Record
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New Elliptic Curve Breaks 18-Year-Old Record:
In August, a pair of mathematicians discovered an exotic, record-breaking curve. In doing so, they tapped into a major open question about one of the oldest and most fundamental kinds of equations in mathematics.
Elliptic curves, which date back to at least ancient Greece, are central to many areas of study. They have a rich underlying structure that mathematicians have used to develop powerful techniques and theories. They were instrumental in Andrew Wiles' famous proof of Fermat's Last Theorem in 1994, at the time one of the most important unsolved problems in number theory. And they play a key role in modern cryptography.
Yet mathematicians still can't answer some of the most basic questions about them. For example, they often try to characterize elliptic curves by studying the special "rational points" that live on them. On a given curve, these points form clear and meaningful patterns. But it's not yet known whether there's a limit to how varied and complicated these patterns can get.
Answering this question would allow mathematicians to make sense of the vast and diverse world of elliptic curves, much of which remains uncharted. So they've set out to explore the outer fringes of that world, hunting down outlier curves with stranger and stranger patterns. It's a painstaking process, requiring both creativity and sophisticated computer programs.
Now, two mathematicians - Noam Elkies of Harvard University and Zev Klagsbrun of the Center for Communications Research in La Jolla, California - have found an elliptic curve with the most complicated pattern of rational points to date, breaking an 18-year-old record. "It was a big question whether this barrier could be broken," said Andrej Dujella of the University of Zagreb in Croatia. "It's a very exciting result for all of us working and interested in elliptic curves."
The discovery lays bare an ongoing debate over what mathematicians think they know about elliptic curves.
Elliptic curves don't appear particularly exotic. They're just equations of the form y2 = x3 + Ax + B, where A and B are rational numbers (any number that can be written as a fraction). When you graph the solutions to these equations, they look like this:
Mathematicians are particularly interested in a given elliptic curve's rational solutions - points on the curve whose x- and y-values are both rational numbers. "It's literally one of the oldest math problems in the history of humanity," said Jennifer Park of Ohio State University.
While it's relatively straightforward to find rational solutions to simpler types of equations, elliptic curves are "the first class of equations where there are really a lot of open questions," said Joseph Silverman of Brown University. "It's just two variables in a cubic equation, and that's already complicated enough."
To get a handle on the rational solutions of an elliptic curve, mathematicians often turn to the curve's rank, a number that measures how closely packed the rational points are along the curve. A rank 0 elliptic curve has only a finite number of rational points. A rank 1 elliptic curve has infinitely many rational points, but all of them line up in a simple pattern, so that if you know one, you can follow a well-known procedure to find the rest.
Higher-rank elliptic curves also have infinitely many rational points, but these points have more complicated relationships to each other. For example, if you know one rational solution of a rank 2 elliptic curve, you can use the same procedure you used in the rank 1 case to find a whole family of rational points. But the curve also has a second family of rational points.
The rank of an elliptic curve tells mathematicians how many "independent" points - points from different families - they need in order to define its set of rational solutions. The higher the rank, the richer in rational points the curve will be. A rank 2 and a rank 3 curve both have infinitely many rational solutions, but the rank 3 curve packs in rational points from an additional family, meaning that on average, a given stretch of it will contain more of them.
Almost all elliptic curves are known to be either rank 0 or rank 1. But there are still infinitely many oddballs with higher rank - and they're exceedingly difficult to find.
As a result, mathematicians aren't sure if there's a limit to how high the rank can get. For a while, most experts thought it was theoretically possible to construct a curve of any rank. Recent evidence suggests otherwise. Without a proof either way, mathematicians are left to debate the true nature of elliptic curves, illustrating just how much they have yet to understand about these equations.
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