Elements of large order on varieties over prime finite fields

Chang, Mei-Chu; Kerr, Bryce; Shparlinski, Igor E.; Zannier, Umberto

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 579-593.

Soit $𝒱$ une variété algébrique fixée définie par $m$ polynomes en $n$ variables à coefficients entiers. Nous montrons qu’il existe une constante $C\left(𝒱\right)$ telle que pour presque tout nombre premier $p$, tous les points de la réduction de $𝒱$ modulo $p$, sauf peut-être $C\left(𝒱\right)$ d’entre eux, possède une composante d’ordre multiplicatif grand. Ceci généralise plusieurs résultats précédents et constitue un pas en direction d’une conjecture de B. Poonen.

Let $𝒱$ be a fixed algebraic variety defined by $m$ polynomials in $n$ variables with integer coefficients. We show that there exists a constant $C\left(𝒱\right)$ such that for almost all primes $p$ for all but at most $C\left(𝒱\right)$ points on the reduction of $𝒱$ modulo $p$ at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

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