Maximal curves and Tate-Shafarevich results for quartic and sextic twists

We study elliptic surfaces corresponding to an equation of the specific type $y^2=x^3+f(t)x$, defined over the finite field ${\mathbb{F}}_q$ for a prime power $q\equiv 3\mathrm{mod}4$. It is shown that if $s^4=f(t)$ defines a curve that is maximal over ${\mathbb{F}}_{q^2}$ then the rank of the group of sections defined over ${\mathbb{F}}_q$ on the elliptic surface is determined in terms of elementary properties of the rational function $f(t)$. Similar results are shown for elliptic surfaces given by $y^2=x^3+g(t)$ using prime powers $q\equiv 5\mathrm{mod}6$ and curves $s^6=g(t)$. Finally, for each of the forms used here, existence of curves with the property that they are maximal over ${\mathbb{F}}_{q^2}$ is discussed, as well as various examples.