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

Abstract

Let ${\mathbb{F}}_q$ denote a finite field of characteristic $p\ge 5$ 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 $q\equiv 1\mathrm{mod}3$ and its rank is $q-2$ when $q\equiv 2\mathrm{mod}3$. We describe an explicit method for producing points on this elliptic curve. In case $q\not\equiv 11\mathrm{mod}12$, 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 $q\equiv 7\mathrm{mod}12$ case, a character sum argument shows that we can generate a full-rank subgroup using $\hat{I}{¼}_d$-translates of a single rational point.