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

Abstract

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 ${\phi }_1,\dots ,{\phi }_m,{\varrho }_1,\dots ,{\varrho }_n\in \mathbb{Q}(X)$ and an elliptic curve $E$ defined over the integers $\mathbb{Z}$, 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: ${\phi }_1(\alpha ),\dots ,{\phi }_m(\alpha )$ are $K$-multiplicatively dependent or the points $({\varrho }_1(\alpha ),\cdot ),\dots ,({\varrho }_n(\alpha ),\cdot )$ 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 ${\mathbb{G}}_{\mathrm{m}}^m\times E^n$ 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.

32 pages. To appear in International Mathematics Research Notices