Sequences of consecutive squares on quartic elliptic curves

Abstract

Let $C:y^2=a{x}^4+b{x}^2+c$, be an elliptic curve defined over $\mathbb{Q}$. A set of rational points $(x_i,y_i)\in C(\mathbb{Q})$, $i=1,2,\cdots ,$ is said to be a sequence of consecutive squares if $x_i=(u+i{)}^2$, $i=1,2,\cdots$, for some $u\in \mathbb{Q}$. Using ideas of Mestre, we construct infinitely many elliptic curves $C$ with sequences of consecutive squares of length at least $6$. It turns out that these 6 rational points are independent. We then strengthen this result by proving that for a fixed $6$-term sequence of consecutive squares, there are infinitely many elliptic curves $C$ with the latter sequence forming the $x$-coordinates of six rational points in $C(\mathbb{Q})$.