M Bhargava

Biography

Professor Manjul Bhargava is one of the most famous number theorists of this generation. He has done spectacular work with his discovery of composition laws of higher degree, originally discovered by CF Gauss in the nineteenth century for quadratic forms. His work in algebraic number theory is profound and extraordinarily original. It has revolutionized the way in which number fields on elliptic curves are counted. He has had a dramatic impact on three important areas of modern number theory – rings of small rank, the Cohen-Lenstra conjectures for ideal class groups and the average rank of elliptic curves over the rational numbers. His thesis gave a reformulation of Gauss’s law for the composition of binary quadratic forms in terms of the orbits of the group SL(2,Z)3 on the tensor product of the three standard representations. He then turned to the study of cubic, quartic, and quintic rings – parametrizing each by integral orbits in more complicated representations, then counting the integral orbits of bounded discriminant using the calculation of volumes of fundamental domains. He then had the amazing idea that his methods could be generalized to co-regular representations, with a polynomial ring of invariants. Applying this to representation of PGL(2,Z) on binary quartic forms, which has two independent polynomial invariants, he was able to show that the orbits parametrized classes in the 2-Selmer groups of elliptic curves over the rational numbers, and by an application of delicate analytic techniques to obtain the average order of these groups. From there, he could bound the average rank of the group of rational solutions, for all elliptic curves over the rational numbers. This is one of the most interesting results in arithmetic geometry proved in the past 25 years.

Research Interest

Number Theory