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Distribution of Mordell-Weil ranks of families of elliptic curves

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Naskręcki B.

Banach Center Publications

2016

tom 108

nr 1

201-229

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.

Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples

100%

Naskręcki B.

Acta Arithmetica

2013

tom 160

nr 2

159-183

We study the family of elliptic curves y² = x(x-a²)(x-b²) parametrized by Pythagorean triples (a,b,c). We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over ℚ is 1, and for some explicitly given infinite family the rank is 2. To each family we attach an elliptic surface fibered over the projective line. We show that the lower bounds for the rank are optimal, in the sense that for each generic fiber of such an elliptic surface its corresponding Mordell-Weil group over the function field ℚ(t) has rank 1 or 2, respectively. In order to prove this, we compute the characteristic polynomials of the Frobenius automorphisms acting on the second ℓ -adic cohomology groups attached to elliptic surfaces of Kodaira dimensions 0 and 1.

Ograniczanie wyników

1 Acta Arithmetica

1 Banach Center Publications

2 Naskręcki B.

1 2016

1 2013

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Distribution of Mordell-Weil ranks of families of elliptic curves

Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples