An effective bound of p for algebraic points on Shimura curves of Γ₀(p)-type

Keisuke Arai

Displaying similar documents to “An effective bound of p for algebraic points on Shimura curves of Γ₀(p)-type”

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.

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.

A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.

Joseph H. Silverman (1987)

Journal für die reine und angewandte Mathematik

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A Family of Abelian Surfaces and Curves of Genus Four.

H. Lange, Ch. Birkenhake (1994)

Manuscripta mathematica

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Linear systems over ℙ¹×ℙ¹ with base points of multiplicity bounded by three

Tomasz Lenarcik (2011)

Annales Polonici Mathematici

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We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general position. As an application, we obtain a classification of special linear systems on ℙ¹×ℙ¹ with multiplicities not exceeding 3.

The analytic order of III for modular elliptic curves

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.

An effective Shafarevich theorem for elliptic curves

Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)

Acta Arithmetica

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Elliptic curves on Abelian surfaces.

Ernst Kani (1994)

Manuscripta mathematica

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Linear forms in logarithms, effectivity and elliptic curves

Alf Van Der Poorten (1980)

Mémoires de la Société Mathématique de France

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Refined Kodaira classes and conductors of twisted elliptic curves

Jerzy Browkin, Daniel Davies

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We consider elliptic curves defined over ℚ. It is known that for a prime p > 3 quadratic twists permute the Kodaira classes, and curves belonging to a given class have the same conductor exponent. It is not the case for p = 2 and 3. We establish a refinement of the Kodaira classification, ensuring that the permutation property is recovered by {refined} classes in the cases p = 2 and 3. We also investigate the nonquadratic twists. In the last part of the paper we discuss the number...