Improved bounds on the number of low-degree points on certain curves
Pavlos Tzermias (2005)
Acta Arithmetica
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Pavlos Tzermias (2005)
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.
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.
Alf Van Der Poorten (1980)
Mémoires de la Société Mathématique de France
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J. E. Cremona (1993)
Journal de théorie des nombres de Bordeaux
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In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class ; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be distinguished from other curves in their isogeny class.
Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)
Acta Arithmetica
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Joseph H. Silverman (1987)
Journal für die reine und angewandte Mathematik
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Dimitrios Poulakis (2003)
Acta Arithmetica
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F. Rodríguez Villegas, J. F. Voloch, D. Zagier (2001)
Acta Arithmetica
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Carlos Munuera Gómez (1991)
Extracta Mathematicae
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Let p be a prime number, p ≠ 2,3 and Fp the finite field with p elements. An elliptic curve E over Fp is a projective nonsingular curve of genus 1 defined over Fp. Each one of these curves has an isomorphic model given by an (Weierstrass) equation E: y2 = x3 + Ax + B, A,B ∈ Fp with D = 4A3 + 27B2 ≠ 0. The j-invariant of E is defined...
M. Anderson, David W. Masser (1980)
Mathematische Zeitschrift
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B. B. Epps (1973)
Colloquium Mathematicae
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