ELLIPTIC CURVES ARISING FROM THE TRIANGULAR NUMBERS

摘要

We study the Legendre family of elliptic curves Et : y2 = x(x 1)(x t), parametrized by triangular numbers t = t(t + 1)/2. We prove that the rank of Et over the function field Q(t) is 1, while the rank is 0 over Q(t). We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.