Elliptic curves with bounded ranks in function field towers
Lisa Berger (2012)
Acta Arithmetica
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Lisa Berger (2012)
Acta Arithmetica
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Touafek, Nouressadat (2008)
Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică
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Andreas Schweizer (1998)
Journal de théorie des nombres de Bordeaux
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For let be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve . We determine all and for which the quotient curve is rational or elliptic.
Yamada, Yasuhiko (2009)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
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R. M. Avanzi, U. M. Zannier (2001)
Acta Arithmetica
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Everett W. Howe (1993)
Compositio Mathematica
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Andrew Bremner (2003)
Acta Arithmetica
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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.
Graham Everest, Patrick Ingram, Valéry Mahé, Shaun Stevens (2008)
Acta Arithmetica
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J. F. Voloch (1990)
Compositio Mathematica
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P. G. Walsh (2009)
Acta Arithmetica
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Dorian Goldfeld, Lucien Szpiro (1995)
Compositio Mathematica
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Franz Lemmermeyer (2003)
Acta Arithmetica
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Bartosz Naskręcki (2016)
Banach Center Publications
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We discuss the distribution of Mordell-Weil ranks of the family of elliptic curves y² = (x + αf²)(x + βbg²)(x + γh²) where f,g,h are coprime polynomials that parametrize the projective smooth conic a² + b² = c² and α,β,γ are elements from ℚ̅. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.