Research Projects

Galois representations (i.e. representations of the absolute Galois groups of number fields) are one of the central objects in number theory. Indeed, a part of the Langlands correspondence, arising from the Langlands program, relates certain 'nice' p-adic Galois representations (popularly known as geometric Galois representation), with automorphic forms (generalisations of modular forms). Automorphic forms are analytic functions with arithmetically interesting symmetries. The study Galois representations in characteristic p and their lifts to characteristic 0 has emerged as a central theme in modern number theory. It has played a crucial role in many important advances in number theory such as the proof of Fermat's last theorem. In this project, I propose to study the properties of modular forms lifting an odd, reducible, semi-simple representation of dimension 2 of the absolute Galois group of Q (the field of rational numbers) over a finite field of characteristic p ? 5. Fix such a representation ?. Given a positive integer M not divisible by p, Carayol has given a necessary condition for the existence of a newform of level M lifting ?. However, it is a priori not clear whether his condition is sufficient. I plan to explore this question in my first project. Note that such results are known as level raising results in the literature and they have become one of the important questions in number theory. This question has been extensively studied for absolutely irreducible representations. However, not many results are known in this direction for reducible representations. In my second project, I propose to study the nature of the congruences mentioned above. To be precise, suppose we know that there exists a newform of level M and weight k lifting ?. In this case, I aim to determine the number of newforms of level M and weight k lifting ?. This question was first formulated by Mazur in his landmark paper on Eisenstein ideal for a special case. Mazur's question was then explored by many authors. However, not many results are known in other cases.