Chennai Mathematical Institute


Dr. F.C. Kohli Centre of Excellence

Perspectives in Mathematical Sciences

January 10–February 4, 2022

Wednesday, 2 February 2022, 21:00 IST

Chandrashekhar Khare, University of California at Los Angeles (UCLA)

Title Modular forms, Galois representations and the Ramanujan prime 691 (Video Recording)


The number 1729 is part of the Ramanujan lore, being famously the number of the taxi which Hardy took to visit Ramanujan. In this talk I would like to argue that the prime number 691 is a number of greater significance than 1729 to the mathematics that Ramanujan discovered. It occurs in his paper "On certain arithmetical functions" published in 1916 in which he observed a congruence between modular forms modulo 691. The observations Ramanujan made in that paper had a great influence on the developments in number theory in the 20th century which led to a proof of Fermat's Last Theorem by Andrew Wiles. The 1916 observations of Ramanujan led to Serre formulating his influential modularity conjecture which was proved in 2009 by Jean-Pierre Wintenberger and myself. I will explain some of these developments and current work on the connection between modular forms and Galois representations.

About the speaker

Chandrashekhar Khare photo Chandrashekhar Khare was born in Mumbai obtained his Ph.D. in 1995 at UCLA and Caltech. He returned to India to work for almost a decade at the Tata Institute of Fundamental Research in Mumbai. He moved thereafter to the University of Utah, and has been at the University of California at Los Angeles since 2007.

He is a number theorist and works on the connection between modular forms and Galois representations (such a connection plays a key role in Wiles's solution of Fermat's Last Theorem). His work with Jean-Pierre Wintenberger gave a proof of a celebrated conjecture of J.-P. Serre in the subject. The conjecture had motivated much work in this central area of number theory and had remained unresolved for more than three decades after it was first formulated. His later work has centered around developing the connection between Galois representations and modular forms in situations that go beyond the context of Serre's modularity conjecture.

Prof. Khare has received a number of honors and awards in recognition of his work. He received the Humboldt Research Award in 2011, Cole Prize in 2011, Infosys Prize in 2010, Guggenheim fellowship in 2008, Fermat prize in 2007, and the INSA Young Scientist Award in 1999. In 2012, he was elected as a Fellow of the Royal Society.