2006/10/09 by Carlo Magagna, Magagna, Carlo
Computer Science · Mathematics · #11G20 #11Jxx #14G05 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT #msc:11G20 #msc:11Jxx #msc:14G05
paper · pdf · doi:10.48550/arxiv.math/0610277
26 pages
openalex publication_date 2006/10/09 · arxiv created 2006/12/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 minumum integer k, such that Ak-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 which states that the group of rational points of an ordinary elliptic curve E over a finite field with qn 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 \mathbbFqn-rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity.