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 $\varphi _1,\ldots ,\varphi _m, \varrho _1,\ldots ,\varrho _n\in{\mathbb{Q}}(X)$ 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.

中文翻译：

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有限域和椭圆曲线模素数中的乘法和线性相关

摘要

对于正整数 $K$ 和 $L$，我们引入并研究了 $K$-对有限素域 $ 的代数闭包 ${\overline{{\mathbb{F}}}}_p$ 的乘法依赖的概念{\mathbb{F}}_p$，以及$L$-点在归约模素数椭圆曲线上的线性相关性。我们的主要结果之一表明，给定非零有理函数 $\varphi _1,\ldots ,\varphi _m, \varrho _1,\ldots ,\varrho _n\in{\mathbb{Q}}(X)$ 和在有理数 ${\mathbb{Q}}$ 上定义的椭圆曲线 $E$，对于任何足够大的素数 $p$，对于除有限多个 $\alpha \in{\overline{{\mathbb{F} }}}_p$，最多可能发生以下两种情况之一： $\varphi _1(\alpha ),\ldots ,\varphi _m(\alpha )$ 是 $K$-乘法依赖或点 $(\varrho _1(\alpha ),\cdot ), \ldots ,(\varrho _n(\alpha ), \cdot )$ 是 $L$-线性依赖于 $E$ 模 $p$ 的减少。作为我们的主要工具之一，我们证明了关于分裂半阿贝尔变体 ${\mathbb{G}}_{\textrm{m}}^m \times E^n$ 中的不可约曲线与余维的代数子群至少 $2$。作为我们结果的应用，我们改进了 MC Chang 的结果并扩展了 JF Voloch 关于有限域中大阶元素在某些特殊情况下的结果。