Elements of large order on varieties over prime finite fields
Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 579-593.

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.

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.

DOI: 10.5802/jtnb.880
Mei-Chu Chang 1; Bryce Kerr 2; Igor E. Shparlinski 3; Umberto Zannier 4

1 Department of Mathematics University of California Riverside, CA 92521, USA
2 Department of Pure Mathematics University of New South Wales Sydney, NSW 2052, Australia
3 Department of Computing Macquarie University Sydney, NSW 2109, Australia
4 Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa, Italy
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     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},
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     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.880/}
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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, Volume 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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