Determination of elliptic curves with everywhere good reduction over real quadratic fields ℚ(√(3p))
Takaaki Kagawa (2001)
Acta Arithmetica
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Takaaki Kagawa (2001)
Acta Arithmetica
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Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)
Acta Arithmetica
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J. W. S. Cassels (1970-1971)
Séminaire de théorie des nombres de Bordeaux
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Amílcar Pacheco (2003)
Acta Arithmetica
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Rubin, Karl, Silverberg, Alice (2000)
Experimental Mathematics
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Amílcar Pacheco (2010)
Acta Arithmetica
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Rose, Harvey E. (2000)
Experimental Mathematics
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Andrej Dujella, Kálmán Győry, Ákos Pintér (2012)
Acta Arithmetica
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P. G. Walsh (2009)
Acta Arithmetica
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Keisuke Arai, Fumiyuki Momose (2012)
Acta Arithmetica
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Lisa Berger (2012)
Acta Arithmetica
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Sheldon Kamienny, Filip Najman (2012)
Acta Arithmetica
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Sungkon Chang (2010)
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.
Yuji Hasegawa (1997)
Manuscripta mathematica
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Cremona, John E., Mazur, Barry (2000)
Experimental Mathematics
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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...
Gang Yu (2005)
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.