A construction of curves over finite fields
Arnaldo Garcia, Luciane Quoos (2001)
Acta Arithmetica
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Arnaldo Garcia, Luciane Quoos (2001)
Acta Arithmetica
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Xavier Xarles (2013)
Journal de Théorie des Nombres de Bordeaux
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In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.
Bruce W. Jordan (1986)
Journal für die reine und angewandte Mathematik
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Keisuke Arai (2014)
Acta Arithmetica
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In previous articles, we showed that for number fields in a certain large class, there are at most elliptic points on a Shimura curve of Γ₀(p)-type for every sufficiently large prime number p. In this article, we obtain an effective bound for such p.
Keisuke Arai, Fumiyuki Momose (2012)
Acta Arithmetica
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Fernando Torres, Rainer Fuhrmann (1996)
Manuscripta mathematica
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D. Eisenbud, J. Harris (1983)
Inventiones mathematicae
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Fenske, Torsten (1999)
Beiträge zur Algebra und Geometrie
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Stein Arild Stromme (1984)
Mathematica Scandinavica
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Wade Hindes (2015)
Acta Arithmetica
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We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality...
F. Rodríguez Villegas, J. F. Voloch, D. Zagier (2001)
Acta Arithmetica
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S. Kamienny (1990)
Mathematische Annalen
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Nguyen Van Chau (2011)
Annales Polonici Mathematici
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In certain cases the invertibility of a polynomial map F = (P,Q): ℂ²→ ℂ² can be characterized by the irreducibility and the rationality of the curves aP+bQ = 0, (a:b) ∈ ℙ¹.
P. Bayer, A. Travesa (2007)
Acta Arithmetica
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Fumio Sakai, Takashi Matsuoka (1989)
Mathematische Annalen
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Bruce W. Jordan, Livné Ron A. (1987)
Journal für die reine und angewandte Mathematik
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Alexandre Eremenko (2012)
Fundamenta Mathematicae
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Jordan analytic curves which are invariant under rational functions are studied.
Motoko Qiu Kawakita, Shinji Miura (2002)
Acta Arithmetica
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