Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-12T09:04:52.542Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Artin'S Conjecture for Primitive Roots and Cyclicity Mod of Elliptic Curves Over Function Fields

Published online by Cambridge University Press:  20 November 2018

David A. Clark  and
Masato Kuwata
David A. Clark
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal, Québec H3A 2K6
Masato Kuwata
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal, Québec H3A 2K6
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

Keywords

11R5811G05elliptic curvesfunction fields
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 2 , 01 June 1995 , pp. 167 - 173
Copyright
Copyright © Canadian Mathematical Society 1995

References

[B] Bilharz, H., Primdivisoren mit vorgegebener Primitivwurzel, Math. Ann. 114(1937), 476–492.Google Scholar
[CP1 Cox, D. A. and Parry, W. P., Representations associated with elliptic surfaces, Pacific J Math. 114(1984), 309–323.Google Scholar
[FJ] Fried, M. D. and Jarden, M., Field Arithmetic, Springer-Verlag, Berlin, Heidelberg, 1986.Google Scholar
[GM1] Gupta, R. and Murty, M. R., Primitive points on elliptic curves, Compositio Math. 58(1986), 13–44.Google Scholar
[GM2] Gupta, R. and Murty, M. R., Cyclicity and generation of points mod/? on elliptic curves, Invent. Math. 101 ( 1990), 225–235.Google Scholar
[Ho] Hooley, C., On Artin s conjecture, J. Reine Angew. Math. 225( 1967), 197–218.Google Scholar
[I] Igusa, J., Fiber system of Jacobian varieties III, Amer. J. Math. 81(1959), 453–476.Google Scholar
[LT] Lang, S. and H. Trotter, Primitive points on elliptic curves, Bull. Amer. Math. Soc. 83(1977), 289–292.Google Scholar
[M] Murty, M. R., On Artin s conjecture, J. Number Theory 16(1983), 147–168.Google Scholar
[S1]J.-P. SerrQ.Abelian l-adic Representation and Elliptic Curves, Benjamin, New York, 1968.Google Scholar
[S2], Resume des cours de l'année scolaire 1976-1977. Google Scholar
[Si] Silverman, J. H., The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986.Google Scholar
[W] Weil, A., Sur les Courbes Algébriques et Variétés qui s'en Déduisent, Hermann, Paris, 1948 Google Scholar