Seminar on September 24 - Vijay Patankar

Congruences between Modular Forms and p-Adic Families of Modular Forms

(from Ramanujan to Hida)

Speaker:       Dr Vijay Patankar

                         International Institute of Information Technology, Bangalore

Time:               3 pm, Tuesday, September 24, 2013.

Venue:            TBA

Abstract:  In this expository talk, we will give an introduction to congruences between modular forms as first observed by Ramanujan and how that has led to p-adic families of modular forms. 

Among many other things, Ramanujan studied certain natural arithmetic functions such as the Partition function and the Sum of Divisors function.  In 1916, Ramanujan in his paper On certain arithmetical functions, observed certain congruences between distinct arithmetic functions and hence between the generating functions associated to them (which are in fact modular forms). In 1967, Serre interpreted these congruences in terms of Galois representations and conjectured the existence of Galois representations associated to modular forms (proved by Deligne 1968). In 1972, Serre constructed a p-adic family of Eisenstein forms. In 1986, Hida constructed p-adic families of cusp-forms and the associated p-adic families of Galois representations.

All these developments were essential tools for Andrew Wiles' proof of Fermat's Last Theorem. 

About the Speaker: Dr Patankar obtained his PhD from the University of Toronto in 2005. He has held positions at the Cold Spring Harbor Laboratory (New York), Microsoft Research India (Bangalore), Bhaskaracharya Pratishthana (Pune) and Indian Statistical Institute (Chennai). His research interests are in Number Theory, Algebraic Complexity Theory and Cryptography.