A minimal bound and divergence case on the elliptic curves over finite fields

Abstract

The main goal of this thesis is the study of elliptic curves over finite fields and the number of points of them. A study of interest is to analyze and find the cases when the difference #E(F_q^n )-#E(F_q) vanishes, where #E(F_q) denotes the number of points of an elliptic curve E over the finite field F_q. Moreover we can show that the sequence a_n=#E(F_q^n ) - #E(F_q)=q^n-q for n odd, q prime of the form q=4k+3 where k is a nonnegative integer and any elliptic curve of the form E : y^2=x^3+tx over F_q. Also a_n=#E(F_q^n ) - #E(F_q)=q^n-q for q prime of the form 3k+2 where k is a positive integer, n odd and any elliptic curve of the form E: y^2=x^3+b over F_q.