Elements of large order on varieties over prime finite fields
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(𝒱) telle que pour presque tout nombre premier p, tous les points de la réduction de 𝒱 modulo p, sauf peut-être C(𝒱) 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(𝒱) such that for almost all primes p for all but at most C(𝒱) 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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DOI : https://doi.org/10.5802/jtnb.880
@article{JTNB_2014__26_3_579_0,
     author = {Mei-Chu Chang and Bryce Kerr and Igor E. Shparlinski and Umberto Zannier},
     title = {Elements of large order on varieties over prime finite fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {579--593},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {3},
     year = {2014},
     doi = {10.5802/jtnb.880},
     mrnumber = {3320493},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.880/}
}
Mei-Chu Chang; Bryce Kerr; Igor E. Shparlinski; Umberto Zannier. Elements of large order on varieties over prime finite fields. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 579-593. doi : 10.5802/jtnb.880. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.880/

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