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

TL;DR

This paper investigates the ranks of elliptic curves derived from Pythagorean triples, establishing lower bounds and exact ranks for generic and specific families, using advanced cohomological methods.

Contribution

It proves the generic Mordell-Weil rank is at least 1, constructs families with rank 2, and confirms these bounds are optimal through Frobenius automorphism analysis.

Findings

Generic rank over Q is at least 1

Existence of infinite families with rank 2

Lower bounds are proven to be optimal

Abstract

We study the family of elliptic curves $y^2=x(x-a^2)(x-b^2)$ 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 $\mathbb{Q}$ 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 $\mathbb{Q}(t)$ has rank 1 or 2, respectively. In order to prove this, we compute the characteristic polynomials of the Frobenius automorphisms acting on the second $\ell$-adic cohomology groups attached to elliptic surfaces of Kodaira dimensions 0 and 1.