On the ranks of the Jacobians of curves defined by Jacobi polynomials

Abstract

We show that the degree-3 Jacobi Polynomials define two one-parameter families of elliptic curves $\E_{r}/\q(r)$ and $\F_{s}/\q(s)$ with positive rank outside of an explicit finite set of $(r,s) \in \q \times \q$. In addition we initiate a program to study the ranks of the Jacobians of these curves in higher degree.