Displaying similar documents to “Corrigendum to 'Distribution of the traces of Frobenius on elliptic curves over function fields' (Acta Arith. 106 (2003), 255-263)”

Distribution of Mordell-Weil ranks of families of elliptic curves

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.

Counting elliptic curves of bounded Faltings height

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.

Heights and regulators of number fields and elliptic curves

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...

On Equations y² = xⁿ+k in a Finite Field

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.

Congruent numbers over real number fields

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.