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Supersingular parameters of the Deuring normal form

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Abstract

It is proved that the supersingular parameters α of the elliptic curve E 3(α): Y 2+αXY+Y=X 3 in Deuring normal form satisfy α=3+γ 3, where γ lies in the finite field \(\mathbb{F}_{p^{2}}\). 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 \(k=\mathbb{F}_{p^{2}}\), and j(α) is the j-invariant of E 3(α). Computing an explicit algebraic form for the elements of the Galois group of the extension N/k(j) leads to some new relationships between supersingular parameters for the Deuring normal form. The function field N, which contains the function field of the cubic Fermat curve, is then used to show how the results of Fleckinger for the Deuring normal form are related to cubic theta functions.

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  1. Department of Mathematical Sciences, Indiana University—Purdue University at Indianapolis (IUPUI), 402 N. Blackford St., LD 270, Indianapolis, IN, 46202, USA

    Patrick Morton

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Morton, P. Supersingular parameters of the Deuring normal form. Ramanujan J 33, 339–366 (2014). https://doi.org/10.1007/s11139-012-9459-6

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  • DOI: https://doi.org/10.1007/s11139-012-9459-6

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