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 $d\times d$ 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 minumum 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 $\lambda$ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with $\lambda$ 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 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 $\mathbb{F}_{q^n}-$rational points of two non isogenous elliptic curves are almost coprime when $n$ approaches infinity.