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.

Barroero, F., Capuano, L., Mérai, L., Ostafe, A., Sha, M. (2022). Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2022(20), 16094-16137 [10.1093/imrn/rnab171].

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

Fabrizio Barroero;Laura Capuano;
2022-01-01

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 $arphi_1,ldots,arphi_m, arrho_1,ldots,arrho_ninmathbb{Q}(X)$ 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.
Barroero, F., Capuano, L., Mérai, L., Ostafe, A., Sha, M. (2022). Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2022(20), 16094-16137 [10.1093/imrn/rnab171].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/391730
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