ELLIPTIC CURVES MAXIMAL OVER EXTENSIONS OF FINITE BASE FIELDS

Given an elliptic curve E over a finite field F-q we study the finite extensions F(q)n of F-q such that the number of 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 <= 11 for sufficiently large q. We also show that there are infinitely many isogeny classes of ordinary elliptic curves with n = 3.