Cancellations in Short Sums related to Hecke-Cusp Forms

Abstract

In number theory, a problem which arises in a variety of contexts is getting non- trivial cancellation for the general correlation problem, specially when we assume that they are short sums related to Hecke-cusp forms. In my thesis, I have studied the cancellation range for those short sums where they have non-trivial bounds. For these problems, we have used the delta method which was developed by Prof. Ritabrata Munshi in his famous circle method papers. I have studied the delta method in the first chapter of the thesis where the reader will get a notion about the structure of the delta method. In the second and third chapter, I have improved the well-known cancellation range for the short sums related to GL(1) twists of GL(2) Hecke-cusp forms and got significant ranges, without going through the theory of L-functions. In the last chapter, I have studied a subconvexity problem, which, after applying the approximate functional equation, boils down to short sums.