![]() ![]() Mathematicians have cared about the solutions to polynomial equations - combinations of variables raised to constant powers - since at least the ancient Greeks. The work is a technical achievement in its own right, and it opens the door to making progress on some of the most important questions in math in the imaginary setting. They improved the second step of Wiles,” said Chandrashekhar Khare of the University of California, Los Angeles. “That’s where the beautiful work of Caraiani and Newton came in. ![]() Imaginary numbers play a crucial role in mathematics because for many problems, they’re easier to work with than real numbers.īut proving that elliptic curves are modular over imaginary quadratic fields has long remained out of reach, because the techniques for proving modularity over real quadratic fields don’t work.Ĭaraiani and Newton achieved modularity - for all elliptic curves over about half of all imaginary quadratic fields - by figuring out how to adapt a process for proving modularity pioneered by Wiles and others to elliptic curves over imaginary quadratic fields. (This includes all the irrational numbers, like $latex \sqrt$. Quadratic fields are a mathematical steppingstone between the rational numbers and the real numbers, which include every possible decimal number, even those with infinite patterns to the right of the decimal point that never repeat. In 2013, three mathematicians including Samir Siksek of the University of Warwick proved that elliptic curves are also modular over real quadratic fields (meaning that the variables and coefficients are taken from a number system called a real quadratic field).Īs the advances mounted, one particular goal remained out of reach: proving that elliptic curves are modular over imaginary quadratic fields. In 2001 four mathematicians proved that all elliptic curves are modular over the rational numbers (whereas Wiles had only proved this for some curves). After his work, mathematicians endeavored to establish modularity in a broader variety of contexts. Wiles proved that certain kinds of elliptic curves are modular - meaning that there is a particular modular form that corresponds to each curve - when the two variables and two coefficients involved in defining the curve are all rational numbers, values that can be written as fractions. This past January, Ana Caraiani of Imperial College London and the University of Bonn and James Newton of the University of Oxford opened a new vein of research in this area when they proved that a relationship Wiles had established between elliptic curves and modular forms also holds for some mathematical objects called imaginary quadratic fields. It has involved understanding the relationship between certain kinds of polynomial equations called elliptic curves and more esoteric objects called modular forms, which burst to prominence in mathematics in 1994 when Andrew Wiles used them to prove Fermat’s Last Theorem, among the most celebrated results of 20th century mathematics. Over the past several decades, one of the most exciting lines of research in mathematics has followed this form. ![]() Instead, mathematicians find ways to connect the solutions to wildly abstract structures whose complexity encodes their secrets. But it is often impossible to do so by tinkering with the equation itself. How are prime numbers distributed in the integers? Are there perfect cubes (like 8 = 2 3 or 27 = 3 3) that can be written as the sum of two other cubes? More generally, mathematicians might want to solve an equation. Many complicated advances in research mathematics are spurred by a desire to understand some of the simplest questions about numbers. ![]()
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