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.
