Multiplicative and Linear Dependence in Finite Fields and on Elliptic Curves Modulo Primes

Abstract

AbstractFor positive integers $K$ and $L$, we introduce and study the notion of $K$-multiplicative dependence over the algebraic closure ${\overline{{\mathbb{F}}}}_p$ of a finite prime field ${\mathbb{F}}_p$, 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 $\varphi _1,\ldots ,\varphi _m, \varrho _1,\ldots ,\varrho _n\in{\mathbb{Q}}(X)$ and an elliptic curve $E$ defined over the rational numbers ${\mathbb{Q}}$, for any sufficiently large prime $p$, for all but finitely many $\alpha \in{\overline{{\mathbb{F}}}}_p$, at most one of the following two can happen: $\varphi _1(\alpha ),\ldots ,\varphi _m(\alpha )$ are $K$-multiplicatively dependent or the points $(\varrho _1(\alpha ),\cdot ), \ldots ,(\varrho _n(\alpha ),\cdot )$ 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 ${\mathbb{G}}_{\textrm{m}}^m \times E^n$ 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.