Mathematicians Bridge a Gap in Number Theory

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Two mathematicians have proven an important result connecting elliptic curves to modular forms, extending a breakthrough achieved in the 1990s that led to the solution of Fermat’s Last Theorem.

Ana Caraiani of Imperial College London and the University of Bonn, and James Newton of the University of Oxford announced in January that they had shown that elliptic curves are modular for certain imaginary quadratic fields. Their work builds on Andrew Wiles’ 1994 proof that elliptic curves are modular for rational numbers, meaning that for each elliptic curve, there exists a corresponding modular form — an object from analysis, a branch of advanced calculus.

Mathematicians care about the solutions to polynomial equations, like 3×5 + x4 – 9×3 – 4×2 + x – 7 = 0. Elliptic curves, which involve two variables and a cubic (third-degree) polynomial, are complex enough to inspire new mathematics but simple enough to yield to determined investigation. A basic question about an elliptic curve is whether it has finitely or infinitely many solutions. The answer is revealed by pairing the curve with a modular form.

In the 1950s, mathematicians proposed that each elliptic curve corresponds to exactly one modular form. Proving this became a central goal. Wiles proved one direction — you can find an elliptic curve’s modular form. The reverse, proving curves are modular, was harder. Wiles did so for some curves with rational coefficients. Mathematicians then extended this to all rational curves and real quadratic fields. Imaginary quadratic fields, involving √-5 or √-3, remained elusive.

Elliptic curve cryptography is used in applications like Bitcoin to enable secure communication over the Internet. Bitcoin uses the secp256k1 elliptic curve, which is defined over the field of real numbers. Public keys on the Bitcoin network correspond to points on this curve, and private keys correspond to integers that are multiplied by the curve’s base point to yield public keys. The security of the system relies on the elliptic curve discrete logarithm problem, which is computationally infeasible to solve for properly chosen curves.

Caraiani and Newton built on recent work providing a “toehold” on the problem. They spent two years developing a strategy and testing whether it would work. “We started with this optimistic idea that things would work out,” Newton said. By 2021, they had shown infinitely many curves modular and proved modularity for about half of imaginary quadratic fields, including those containing √-1 or √-2.

Their work provides foundations for addressing questions like imaginary Fermat’s Last Theorem. Progress on the modularity of elliptic curves over imaginary quadratic fields could lead to new cryptography protocols secure against quantum computers, which can solve the discrete logarithm problem for real elliptic curves more easily.

But after years on modularity, Caraiani seeks “something with a bit more of a geometric flavor.” Mathematics progresses through the urge to understand simple questions about numbers. This work continues that tradition, revealing secrets encoded in wildly abstract structures.

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