Supersingular parameters of the Deuring normal form

It is proved that the supersingular parameters α of the elliptic curve E3(α) : Y 2 + αXY + Y = X3 in Deuring normal form satisfy α = 3 + γ3, where γ lies in the finite field Fp2 . This is accomplished by finding explicit generators for the normal closure N of the finite extension k(α)/k(j(α)), where α is an indeterminate over k = Fp2 and j(α) is the j-invariant of E3(α). The function field N is constructed over any field k containing a primitive cube root of unity whose characteristic is different from 2 and 3, and contains the function field of the cubic Fermat curve. This is used to study solutions of the cubic Fermat equation in Hilbert class fields of imaginary quadratic fields in which the prime 3 splits, as well as solutions in modular functions given in terms of the Dedekind η-function. It has been known since Hasse’s 1934 paper [h] that there are only finitely many isomorphism classes of elliptic curves E defined over the algebraic closure of the finite field Fp, for which E has no points of order p. Such a curve is said to be supersingular, and Deuring [d] showed that its j-invariant j(E) lies in Fp2 . Supersingular j-invariants are somewhat sparse: in characteristic p there are roughly (p− 1)/12 of them. They can be characterized as the roots of a certain polynomial (see [d], [brm], [m1]), and it is of interest to find other arithmetic relations that they satisfy. For example, it is proved in [m2] that the j-invariant of any supersingular curve E in characteristic p is a perfect cube in Fp2 . For certain families of elliptic curves, the values of the parameters for which these curves are supersingular also satisfy interesting arithmetic relationships in finite ∗MSC2010: 14H52, 14H05, 11D41, 11F20 †