Elliptic curves maximal over extensions of finite base fields

Given an elliptic curve $E$ over a finite field ${\mathbb{F}}_q$ we study the finite extensions ${\mathbb{F}}_{q^n}$ of ${\mathbb{F}}_q$ such that the number of ${\mathbb{F}}_{q^n}$-rational points on $E$ attains the Hasse upper bound. We obtain an upper bound on the degree $n$ for $E$ ordinary using an estimate for linear forms in logarithms, which allows us to compute the pairs of isogeny classes of such curves and degree $n$ for small $q$. Using a consequence of Schmidt’s Subspace Theorem, we improve the upper bound to $n\le 11$ for sufficiently large $q$. We also show that there are infinitely many isogeny classes of ordinary elliptic curves with $n=3$.