A lower bound for the r-order of a matrix modulo N

Abstract

For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A k − I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational points of an ordinary elliptic curve E over a finite field with q n elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of ${\Bbb F}_{q^n}-$ rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity.