Supersingular parameters of the Deuring normal form
Let \(E_3(\alpha)\) be the elliptic curve in Deuring normal form \(Y^2+\alpha XY+Y=X^3\) with \(\alpha\in\overline{\mathbb{F}}_p\) (the algebraic closure of \(\mathbb{F}_p\)). The paper deals with the values of \(\alpha\) for which \(E_3(\alpha)\) is supersingular (i.e., \(E_3(\alpha)[p]=0\)) and shows that they are all of the type \(\alpha=3+\gamma^3\) for some \(\gamma\in\mathbb{F}_{p^2}\) (assuming \(p>3\)). Considering \(\alpha\) as an indeterminate and using the \(j\)-invariant \(j(\alpha):=\frac{\alpha^3(\alpha^3-24)^3}{\alpha^3-27}\), the author computes the normal closure \(N\) of \(\mathbb{F}_{p^2}(\alpha)/\mathbb{F}_{p^2}(j(\alpha))\) and uses a result of \textit{P. Morton} [Ramanujan J. 25, No. 2, 247--275 (2011; Zbl 1277.11029)], connecting supersingular parameters to solutions of the Fermat curve \(27X^3+27Y^3=X^3Y^3\), to prove the main theorem (for \(j(\alpha)\neq 0,1728\), these remaining cases are treated separately by direct computations). \noindent The computation of \(N\) also provides some connection with the theory of modular functions and information on the structure of the supersingular parameters \(\alpha\). Indeed \(\mathrm{Gal}(N/\mathbb{F}_{p^2}(j(\alpha)))\simeq \widetilde{\Gamma}(9):=\mathrm{SL}_2(\mathbb{Z}/9)/\pm 1\) and the author writes down an isomorphism which (together with results of \textit{V. Fleckinger} [Ann. Inst. Fourier 38, No. 1, 17--57 (1988; Zbl 0634.12012)]) provides an explicit action of \(\widetilde{\Gamma}(9)\) on the set of \(\alpha\)'s. As a consequence of the computations the author also proves (or re-proves) some interesting identities involving Fleckinger's functions, Dedekind's \(\eta\)-function and Borwein's cubic theta functions (introduced in [\textit{J. M. Borwein} and \textit{P. B. Borwein}, Trans. Am. Math. Soc. 323, No. 2, 691--701 (1991; Zbl 0725.33014)]).
scientific article; zbMATH DE number 6348217
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