Points on elliptic curves over finite fields
M. Skałba (2005)
Acta Arithmetica
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M. Skałba (2005)
Acta Arithmetica
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Rubin, Karl, Silverberg, Alice (2000)
Experimental Mathematics
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Sungkon Chang (2010)
Acta Arithmetica
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Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)
Acta Arithmetica
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Yuji Hasegawa (1997)
Manuscripta mathematica
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Sheldon Kamienny, Filip Najman (2012)
Acta Arithmetica
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P. G. Walsh (2009)
Acta Arithmetica
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Masanari Kida (2001)
Journal de théorie des nombres de Bordeaux
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We prove that the -invariant of an elliptic curve defined over an imaginary quadratic number field having good reduction everywhere satisfies certain Diophantine equations under some hypothesis on the arithmetic of the quadratic field. By solving the Diophantine equations explicitly in the rings of quadratic integers, we show the non-existence of such elliptic curve for certain imaginary quadratic fields. This extends the results due to Setzer and Stroeker.
Amílcar Pacheco (2003)
Acta Arithmetica
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Keisuke Arai, Fumiyuki Momose (2012)
Acta Arithmetica
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Amílcar Pacheco (2010)
Acta Arithmetica
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Tim Dokchitser, Vladimir Dokchitser (2009)
Acta Arithmetica
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Rose, Harvey E. (2000)
Experimental Mathematics
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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...
Lisa Berger (2012)
Acta Arithmetica
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Cremona, John E., Mazur, Barry (2000)
Experimental Mathematics
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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.
Fabien Pazuki (2014)
Publications mathématiques de Besançon
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We compare general inequalities between invariants of number fields and invariants of elliptic curves over number fields. On the number field side, we remark that there is only a finite number of non-CM number fields with bounded regulator. On the elliptic curve side, assuming the height conjecture of Lang and Silverman, we obtain a Northcott property for the regulator on the set of elliptic curves with dense rational points over a number field. This amounts to say that the arithmetic...
Dongho Byeon (2004)
Acta Arithmetica
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