Q-curves over quadratic fields.
Yuji Hasegawa (1997)
Manuscripta mathematica
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Yuji Hasegawa (1997)
Manuscripta mathematica
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Leprévost, Franck (1993)
Experimental Mathematics
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Takaaki Kagawa (2001)
Acta Arithmetica
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Glenn Stevens (1989)
Inventiones mathematicae
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Rubin, Karl, Silverberg, Alice (2000)
Experimental Mathematics
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M. Skałba (2005)
Acta Arithmetica
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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.
Daeyeol Jeon, Chang Heon Kim (2004)
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.
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.
François Brunault (2008)
Acta Arithmetica
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Andreas Enge, Reinhard Schertz (2005)
Acta Arithmetica
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Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)
Acta Arithmetica
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Galbraith, Steven D. (1999)
Experimental Mathematics
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Joseph H. Silverman (1987)
Journal für die reine und angewandte Mathematik
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Ruthi Hortsch (2016)
Acta Arithmetica
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We give an asymptotic formula for the number of elliptic curves over ℚ with bounded Faltings height. Silverman (1986) showed that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in ℝ².