Major progress on a problem of Ramanujan

Krishnaswami Alladi

KUMBAKONAM: There has been sensational recent progress in the study of universal quadratic forms, a subject which originated in 1910 when Ramanujan wrote down 54 examples of such quadratic forms. These recent developments were related by Professor Manjul Bhargava of Princeton University while delivering the Third Ramanujan Commemoration Lecture at SASTRA University, Kumbakonam, on December 22.

One of the most important results in number theory is a 1770 theorem of Lagrange which states that every positive integer is a sum of four squares. This motivated Ramanujan to investigate those quadratic forms which would represent all positive integers. In his notebooks Ramanujan wrote 54 examples of such quadratic forms. Ramanujan's discovery resulted in a flood of activity in the ensuing decades in the study of universal quadratic forms, namely, quadratic forms representing all positive integers.

A startling result

In 1993 Conway and Schneebeger announced a startling result that in order to decide whether certain special quadratic forms defined via matrices are universal, one need only check whether these represent the integers from 1 to 15. Their proof of this result, which was very intricate, was never published. Prof. Bhargava has found a new and much simpler proof of this result using geometric notions, which he presented in his lecture.

Conway had conjectured that in the general case of integer valued quadratic forms, in order to decide which of those are universal, it suffices to check whether a certain special set of 29 integers up to 290 can be represented. This is a very difficult problem and Conway said he did not expect to see a proof in his lifetime! Quite surprisingly, during the summer of 2005, Prof. Bhargava and Jonathan Hanke proved Conway's conjecture, and following this, were able to determine the complete list of all 6,436 integer valued quadratic forms that are universal.

In establishing this result, they used a variety of techniques and results due to Ramanajun such as the circle method that Hardy and Ramanujan introduced in the asymptotic study of partitions, and Ramanujan's bounds for the coefficients of certain modular forms of integral weight that were proved by Fields Medalist Pierre Deligne.

Prof. Bhargava said that Ramanujan would have been quite pleased not only at the complete resolution of the problem of universal quadratic forms, but also at the methods employed in the proofs.

"I am pleased to present these results in Ramanujan's home town on Ramanujan's birthday," he said.

The Ramanujan Commemoration Lecture by Prof. Bhargava was the concluding event for the International Conference on Number Theory and Mathematical Physics held at SASTRA University in Kumbakonam. On the opening day of the conference, he and Kannan Soundararajan (University of Michigan) were each awarded the First SASTRA Ramanujan Prizes of US $10,000 for outstanding contribution in areas of mathematics influenced by Ramanujan.

Prof. Bhargava announced that he would be donating a portion of the prize to SASTRA to support mathematically gifted students.

(Krishnaswami Alladi is with the University of Florida, Gainesville, U.S.)

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