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The Ramanujan Journal

, Volume 25, Issue 2, pp 247–275 | Cite as

The cubic Fermat equation and complex multiplication on the Deuring normal form

  • Patrick Morton
Article

Abstract

The arithmetic on elliptic curves in Deuring normal form is shown to be related to solutions of the Fermat equation 27X 3+27Y 3=X 3 Y 3. This arithmetic is used to give conditions for the existence of multipliers μ on supersingular elliptic curves in characteristic p for which μ 2=−3p. Together with an explicit factorization of a certain class equation, these conditions imply that the number of irreducible binomial quadratic factors (mod p) of the Legendre polynomial P (pe)/3(x) of degree (pe)/3 is a simple linear function of the class number of the quadratic field \(\mathbb{Q}(\sqrt{-3p})\).

Keywords

Legendre polynomial Class number Elliptic curve Deuring normal form Fermat equation Complex multipliers 

Mathematics Subject Classification (2000)

11C08, 11G07, 14K22 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Dept. of Mathematical SciencesIndiana University—Purdue University at Indianapolis (IUPUI)IndianapolisUSA

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