Taras Banakh, Fedor Bogomolov, Andrij Gatalevych, Ihor Guran, Yurij Ishchuk, Mykola Komarnytskyi, Igor Kuz, Ivanna Melnyk, Vasyl Petrychkovych, Yaroslav Prytula, Oleh Romaniv, Oleh Skaskiv, Ludmyla Stakhiv, Georgiy Sullym, Bohdan Zabavskyi, Volodymir Zelisko, Mykhajlo Zarichnyi (2013)
Open Mathematics
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Alberto Conte and Sebastià Xambó (1989)
Collectanea Mathematica
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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.
Arnaldo Garcia, Luciane Quoos (2001)
Acta Arithmetica
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Itenberg, Ilia, Kharlamov, Viatcheslav, Shustin, Eugenii (2005)
Geometry & Topology
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Constantinescu, Adrian (2008)
Acta Universitatis Apulensis. Mathematics - Informatics
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Fabrizio Catanese (2009)
Bollettino dell'Unione Matematica Italiana
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The theory of algebraic surfaces, according to Federigo Enriques, revealed `riposte armonie' (hidden harmonies) who the mathematicians to undertook their investigation. Purpose of this article is to show that this holds still nowadays; and point out, while reviewing recent progress and unexpected new results, the many faceted connections of the theory, among others, with algebra (Galois group of the rational numbers), with real geometry, and with differential and symplectic topology...
Jeremy Gray (1997)
Revue d'histoire des mathématiques
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Mathematicians and historians generally regard the modern period in algebraic geometry as starting with the work of Kronecker and Hilbert. But the relevant papers by Hilbert are often regarded as reformulating invariant theory, a much more algebraic topic, while Kronecker has been presented as the doctrinaire exponent of finite, arithmetical mathematics. Attention is then focused on the Italian tradition, leaving the path to Emmy Noether obscure and forgotten.There was, however, a steady...
Günter Lettl (2005)
Acta Arithmetica
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Moshe Jarden (1988)
Manuscripta mathematica
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Pol Vanhaecke (2005)
Annales de l’institut Fourier
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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...
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.