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

Bartosz Naskręcki. "Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples." Acta Arithmetica 160.2 (2013): 159-183. <http://eudml.org/doc/279803>.

@article{BartoszNaskręcki2013,
abstract = {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.},
author = {Bartosz Naskręcki},
journal = {Acta Arithmetica},
keywords = {Mordell-Weil group; elliptic curves; elliptic surfaces},
language = {eng},
number = {2},
pages = {159-183},
title = {Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples},
url = {http://eudml.org/doc/279803},
volume = {160},
year = {2013},
}

TY - JOUR
AU - Bartosz Naskręcki
TI - Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples
JO - Acta Arithmetica
PY - 2013
VL - 160
IS - 2
SP - 159
EP - 183
AB - 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.
LA - eng
KW - Mordell-Weil group; elliptic curves; elliptic surfaces
UR - http://eudml.org/doc/279803
ER -