Research Projects

Shimura-Taniyama conjecture is one of the fundamental conjectures in arithmetic geometry and number theory. The study of arithmetic Elliptic curves depends heavily on the fundamental theorem proved by Wiles, Taylor--Wiles on Shimura--Taniyama conjecture. Employing the results of Deligne, one can form a 2-dimensional Galois representation associated with an elliptic curve over rationals. The Shimura-Taniyama conjecture can then be seen as a special case of the Langlands Conjecture (in dimension 2) relating 2-dimensional Galois representations and automorphic forms. So, it is imperative to know more about Langland’s correspondence in more generality. Other than GL_2, our knowledge is quite limited. Again, much is known over global function field in characteristic p by results of Drinfeld, and Lafforgue. In characteristic zero, Harris, Taylor & Henniart have proved the local Langlands correspondence for GL_n for l-adic representations. For p-adic representations, a lot of new technical difficulties arise. Accordingly, the p-adic Langlands programme is one of the most active themes in current research. By fundamental results due to Colmez, Breuil, and Paskunas, the case of GL_2(Q_p) is now resolved. Fontaine's notion of (Phi, Gamma)-modules lies at the heart of the p-adic Langlands correspondence. It has been a major tool in the proof of the 2-dimensional case. One of my research interests lies in dealing with the Iwasawa theory of multi-variable (Phi, Gamma)-modules, which are expected to arise naturally in p-adic Langlands correspondence of higher rank reductive groups.