Supersingular parameters of the Deuring normal form
type: Item
instance of: scholarly article
title: Supersingular parameters of the Deuring normal form
zbMATH Open document ID: 1307.14048
author: Patrick Morton
published in: The Ramanujan Journal
publication date: 2014-09-26T00:00:00Z
review text: 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)]).
reviewed by: Andrea Bandini
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11F20
zbMATH DE Number: 6348217
zbMATH Keywords: elliptic curves
zbMATH Keywords: Deuring normal form
zbMATH Keywords: supersingular
zbMATH Keywords: algebraic function fields
zbMATH Keywords: cubic Fermat curve
MaRDI profile type: Publication
full work available at URL: s11139-012-9459-6
OpenAlex ID: W1982777350
cites work: A Cubic Counterpart of Jacobi's Identity and the AGM
cites work: Some Cubic Modular Identities of Ramanujan
cites work: Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial
cites work: An identity for the Dedekind eta-function involving two independent complex variables
cites work: Rational analogues of Ramanujan's series for 1/π
cites work: A new class of theta function identities in two variables
cites work: Cubic theta functions.
cites work: A simple proof of an expansion of an eta-quotient as a Lambert series
cites work: Monogénéité de l'anneau des entiers de certains corps de classes de rayon. (Monogeneity of the ring of integers of certain ray class fields)
cites work: Q4758296
cites work: Q4693118
cites work: Q3414861
cites work: Note on an eta identity presented by B. C. Berndt and W. B. Hart
cites work: Eta Products and Theta Series Identities
cites work: Q3801701
cites work: Explicit identities for invariants of elliptic curves
cites work: The cubic Fermat equation and complex multiplication on the Deuring normal form
cites work: Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)
cites work: Realising wreath products of cyclic groups as Galois groups
cites work: Q5666618
cites work: Q3562284
cites work: Q4771024
cites work: Q4317891
cites work: Algebraic function fields and codes
cites work: Q3655906
DOI: 10.1007/S11139-012-9459-6
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description: scientific article; zbMATH DE number 6348217