Average Frobenius distribution of elliptic curves
Kevin James, Gang Yu (2006)
Acta Arithmetica
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Kevin James, Gang Yu (2006)
Acta Arithmetica
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Rose, Harvey E. (2000)
Experimental Mathematics
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Jörn Steuding, Annegret Weng (2005)
Acta Arithmetica
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Jörn Steuding, Annegret Weng (2005)
Acta Arithmetica
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Rubin, Karl, Silverberg, Alice (2000)
Experimental Mathematics
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Gang Yu (2005)
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.
Cremona, John E., Mazur, Barry (2000)
Experimental Mathematics
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Lisa Berger (2012)
Acta Arithmetica
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Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)
Acta Arithmetica
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Amílcar Pacheco (2010)
Acta Arithmetica
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Amílcar Pacheco (2003)
Acta Arithmetica
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Andrej Dujella, Kálmán Győry, Ákos Pintér (2012)
Acta Arithmetica
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Joseph H. Silverman, Armand Brumer (1996)
Manuscripta mathematica
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Alf Van Der Poorten (1980)
Mémoires de la Société Mathématique de France
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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.
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...
Delaunay, Christophe (2001)
Experimental Mathematics
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Tom Fisher (2015)
Acta Arithmetica
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We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose 9-torsion subgroups are isomorphic as Galois modules.