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

• Autorzy: Bartosz Naskręcki
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11D25: Cubic and quartic equations
• 11D45: Counting solutions of Diophantine equations
• 11G05: Elliptic curves over global fields
• 14H52: Elliptic curves

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 160
• Numer: 2
• Strony: 159-183

Daty

wydano

2013

Twórcy

autor

Bartosz Naskręcki

• Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 61-614 Poznań, Poland

Identyfikatory

DOI

10.4064/aa160-2-5