An effective Shafarevich theorem for elliptic curves
Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)
Acta Arithmetica
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Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)
Acta Arithmetica
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Amílcar Pacheco (2003)
Acta Arithmetica
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Amílcar Pacheco (2010)
Acta Arithmetica
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M. Skałba (2005)
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.
A. Schinzel, M. Skałba (2004)
Bulletin of the Polish Academy of Sciences. Mathematics
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Solutions of the equations y² = xⁿ+k (n = 3,4) in a finite field are given almost explicitly in terms of k.
Tomasz Jędrzejak (2012)
Colloquium Mathematicae
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It is classical that a natural number n is congruent iff the rank of ℚ -points on Eₙ: y² = x³-n²x is positive. In this paper, following Tada (2001), we consider generalised congruent numbers. We extend the above classical criterion to several infinite families of real number fields.
Takaaki Kagawa (2001)
Acta Arithmetica
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Armand Brumer, Oisín McGuinness (1992)
Inventiones mathematicae
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Lisa Berger (2012)
Acta Arithmetica
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Cremona, John E., Mazur, Barry (2000)
Experimental Mathematics
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Rose, Harvey E. (2000)
Experimental Mathematics
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Benedict H. Gross, Joe P. Buhler (1985)
Inventiones mathematicae
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Hizuru Yamagishi (1998)
Manuscripta mathematica
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MacLeod, Allan J. (2006)
Journal of Integer Sequences [electronic only]
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Campbell, Garikai (2003)
Journal of Integer Sequences [electronic only]
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Min Sha, Igor E. Shparlinski (2015)
Acta Arithmetica
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We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterised families of elliptic curves when the parameter runs through a set of rational numbers of...
Patrick Ingram (2008)
Acta Arithmetica
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