Manjul Bhargava - Contributions

Contributions

His Ph.D. thesis generalized the classical Gauss composition law for quadratic forms to many other situations. One major use of his results is the parametrization of quartic and quintic orders in number fields, thus allowing the study of asymptotic behavior of arithmetic properties of these orders and fields.

His research also includes fundamental contributions to the representation theory of quadratic forms, to interpolation problems and p-adic analysis, to the study of ideal class groups of algebraic number fields, and to the arithmetic theory of elliptic curves. A short list of his specific mathematical contributions are:

• 14 new Gauss-style composition laws.
• Determination of the asymptotic density of discriminants of quartic and quintic number fields.
• Proofs of the first known cases of the Cohen-Lenstra-Martinet heuristics for class groups.
• Proof of the 15 theorem, including an extension of the theorem to other number sets such as the odd numbers and the prime numbers.
• Proof (with Jonathan Hanke) of the 290 theorem.
• A novel generalization of the factorial function, resolving a decades-old conjecture by George Pólya.
• Proof (with Arul Shankar) that the average rank of all elliptic curves over Q (when ordered by height) is bounded.

In July 2010 Manjul Bhargava and Arul Shankar proved the Birch and Swinnerton-Dyer conjecture for a positive proportion of elliptic curves.