2020/08/02 by Barroero, Fabrizio, Capuano, Laura, Mérai, László +2
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2008.00389
For positive integers K and L, we introduce and study the notion of K-multiplicative dependence over the algebraic closure \mathbbFp of a finite prime field \mathbbFp, as well as L-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions φ1,…,φm, \varrho1,…,\varrhon∈ℚ(X) and an elliptic curve E defined over the integers ℤ, for any sufficiently large prime p, for all but finitely many α∈\mathbbFp, at most one of the following two can happen: φ1(α),…,φm(α) are K-multiplicatively dependent or the points (\varrho1(α),⋅), …,(\varrhon(α),⋅) are L-linearly dependent on the reduction of E modulo p. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety \mathbbGmm × En with the algebraic subgroups of codimension at least 2. As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases.