Pell's Equation over Polynomial Rings

It is classical that for any positive integer $d$ not a perfect square there are integers $x$ and $y\neq 0$ with $x^{2}-dy^{2}=1$. The analogous assertion for $D,\ X,\ Y\neq 0$ in $\mathrm{C}[\mathrm{t}]$ with $X^{2}-DY^{2}=1$ clearly requires that the degree of $D$ be even, and it is easy to see that it holds for all quadratic $D$. However for quartic $D$ the problem changes character; this can be seen from that fact that the set of complex $\lambda$ such that $X,\ Y$ exist for $D=t^{4}+t +\lambda$ is infinite but ``scarce'' ; for example it contains at most finitely many rationals. For sextic $D$ things change again, and we will sketch a proof that there are at most finitely many complex $\lambda$ such that $X,\ Y$ exist for $D=t^{6}+t +\lambda$. This follows from recent work with Umberto Zannier about unlikely intersections for simple abelian schemes.