Points on Elliptic Curve over finite fields [HBNI Th85]

Abstract

This thesis is divided into two parts. In the first part,the main topic of interest, the title of this thesis is studied. While the second part studies a problem related to additive representation function related to sum-set. In this thesis, the average of K*(N) over (N less than or equal to x ) , is computed. This asymptotic result improves an earlier result significantly and checks the consistency of the conditional result with other unconditional ones. Further, this work investigates the distribution of ME(N), that is the probability of the event { ME(N) = l } for a fixed integer l and N. For that purpose, taking an average of the indicator function of the event { ME(N) = l } over a class C of curves and prove that ME(N) follows a Poisson distribution on average with a mean equals to the average of ME(N) over the same class C. The second part of the thesis, discusses a problem in additive number theory.