Research Interest
Eisenstein cycles, also known as Eisenstein elements, are elements in the Eisenstein cohomology of SL_2. These cycles are, by definition, those over which every integral of holomorphic differential forms vanish. Our Project however is very much concerned with the study of Binachi modular forms which are basically modular forms over imaginary quadratic fields. One of the tools we would like to use for the research purpose is Hecke operator. In the theory of modular forms Hecke operator is a certain kind of averaging operators that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.
The main work of the research project is to extend the study of Eisenstein elements to Bianchi groups. The study of modular symbols over K which is a imaginary quadratic field, sometimes called Bianchi modular symbols were initiated in the works of Kurcanov and Cremona several decades ago. Also the Eisenstein elements are closely connected to modular symbols and value of L-functions. We expect that our research will have numerous applications in values of L functions.