Explicit points on $y^2 + xy - t^d y = x^3$ and related character sums

Let $\mathbb{F}_q$ denote a finite field of characteristic $p \geq 5$ and let $d = q+1$. Let $E_d$ denote the elliptic curve over the function field $\mathbb{F}_{q^2}(t)$ defined by the equation $y^2 + xy - t^d y = x^3$. Its rank is $q$ when $q \equiv 1 \bmod 3$ and its rank is $q-2$ when $q \equiv 2 \bmod 3$. We describe an explicit method for producing points on this elliptic curve. In case $q \not\equiv 11 \bmod 12$, our method produces points which generate a full-rank subgroup. Our strategy for producing rational points on $E_d$ makes use of a dominant map from the degree $d$ Fermat surface over $\mathbb{F}_{q^2}$ to the elliptic surface associated to $E_d$. We in turn study lines on the Fermat surface $\mathcal{F}_d$ using certain multiplicative character sums which are interesting in their own right. In particular, in the $q \equiv 7 \bmod 12$ case, a character sum argument shows that we can generate a full-rank subgroup using $\mu_d$-translates of a single rational point.