Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes

For 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 $arphi_1,ldots,arphi_m, arrho_1,ldots,arrho_ninmathbb{Q}(X)$ and an elliptic curve $E$ defined over the integers $mathbb{Z}$, for any sufficiently large prime $p$, for all but finitely many $alphainoverline{mathbb{F}}_p$, at most one of the following two can happen: $arphi_1(alpha),ldots,arphi_m(alpha)$ are $K$-multiplicatively dependent or the points $(arrho_1(alpha),cdot), ldots,(arrho_n(alpha),cdot)$ 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 $mathbb{G}_{mathrm{m}}^m imes E^n$ 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.