This note contains an application of the algebraic study by Schütt and Shioda of the elliptic modular surface attached to the commutator subgroup of the modular group. This is used here to provide algebraic descriptions of certain coverings of a j-invariant 0 elliptic curve, unramified except over precisely one point.

No abstract

This paper intends to focus on the universal property of this Hesse pencil and of its twists. The main goal is to do this as explicit and elementary as possible, and moreover to do it in such a way that it works in every characteristic different from three.

Given an elliptic curve E over a finite field Fq we study the finite extensions Fqn of Fq such that the number of Fqn-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 = 3.2. If E is ordinary, that is gcd (a 1 , q) = 1, then there are at most finitely many extensions of F q over which E is maximal. Furthermore if q is a square, then such extensions do not exist.