For positive integers K and L, we introduce and study the notion of K-multiplicative dependence over the algebraic closure of a finite prime field Fp, 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 φ1,…,φm,ϱ1,…,ϱn ∈ Q(X) and an elliptic curve E defined over the integers Z, for any sufficiently large prime p, for all but finitely many α in the algebraic closure of F_p, at most one of the following two can happen: φ1(α),…,φm(α) are K-multiplicatively dependent or the points (ϱ1(α),⋅),…,(ϱn(α),⋅) 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 G^k_m×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.

Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes / Barroero, Fabrizio; Capuano, Laura; Mérai, Laszlo; Ostafe, Alina; Sha, Min. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - ELETTRONICO. - 2022:20(2022), pp. 16094-16137. [10.1093/imrn/rnab171]

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

Laura Capuano;
2022

Abstract

For positive integers K and L, we introduce and study the notion of K-multiplicative dependence over the algebraic closure of a finite prime field Fp, 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 φ1,…,φm,ϱ1,…,ϱn ∈ Q(X) and an elliptic curve E defined over the integers Z, for any sufficiently large prime p, for all but finitely many α in the algebraic closure of F_p, at most one of the following two can happen: φ1(α),…,φm(α) are K-multiplicatively dependent or the points (ϱ1(α),⋅),…,(ϱn(α),⋅) 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 G^k_m×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.
File in questo prodotto:
File Dimensione Formato  
12.Multiplicative and Linear dependence.pdf

accesso aperto

Tipologia: 1. Preprint / submitted version [pre- review]
Licenza: PUBBLICO - Tutti i diritti riservati
Dimensione 392.57 kB
Formato Adobe PDF
392.57 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2849211