Rational 1- and 2-cuspidal plane curves.
Fenske, Torsten (1999)
Beiträge zur Algebra und Geometrie
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Displaying similar documents to “Nets, -sequences, and algebraic curves over finite fields with many rational points.”
Rational 1- and 2-cuspidal plane curves.
A construction of curves over finite fields
Arnaldo Garcia, Luciane Quoos (2001)
Acta Arithmetica
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Trivial points on towers of curves
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.
Non-existence of points rational over number fields on Shimura curves
Keisuke Arai (2016)
Acta Arithmetica
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Jordan, Rotger and de Vera-Piquero proved that Shimura curves have no points rational over imaginary quadratic fields under a certain assumption. In this article, we extend their results to the case of number fields of higher degree. We also give counterexamples to the Hasse principle on Shimura curves.
Bounds for the smallest integer point of a rational curve
Dimitrios Poulakis (2003)
Acta Arithmetica
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Trace and families of algebraic curves.
Gerard van der Geer, M. van der Vlugt (1992)
Mathematische Zeitschrift
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Rational approximations to algebraic Laurent series with coefficients in a finite field
Alina Firicel (2013)
Acta Arithmetica
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We give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give a few examples of algebraic Laurent series...
Extrema constrained by curves.
Dogaru, Oltin, Dogaru, Vlad (1999)
Balkan Journal of Geometry and its Applications (BJGA)
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Bounds on the number of non-rational subfields of a function field.
E. Kani (1986)
Inventiones mathematicae
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Appendix to "Plücker conditions on plane rational curves": Families of rational plane curves.
Stein Arild Stromme (1984)
Mathematica Scandinavica
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Divisors on general curves and cuspidal rational curves.
D. Eisenbud, J. Harris (1983)
Inventiones mathematicae
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Rational Bézier curves with infinitely many integral points
Petroula Dospra (2023)
Archivum Mathematicum
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In this paper we consider rational Bézier curves with control points having rational coordinates and rational weights, and we give necessary and sufficient conditions for such a curve to have infinitely many points with integer coefficients. Furthermore, we give algorithms for the construction of these curves and the computation of theirs points with integer coefficients.
On a class of rational cuspidal plane curves.
Hubert Flenner, Mikhail Zaidenberg (1996)
Manuscripta mathematica
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The genus of curves over finite fields with many rational points.
Fernando Torres, Rainer Fuhrmann (1996)
Manuscripta mathematica
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A formula for the Selmer group of a rational three-isogeny
Matt DeLong (2002)
Acta Arithmetica
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An effective bound of p for algebraic points on Shimura curves of Γ₀(p)-type
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.