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 (p−e)/3(x) of degree (p−e)/3 is a simple linear function of the class number of the quadratic field \(\mathbb{Q}(\sqrt{-3p})\).
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Morton, P. The cubic Fermat equation and complex multiplication on the Deuring normal form. Ramanujan J 25, 247–275 (2011). https://doi.org/10.1007/s11139-010-9286-6
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DOI: https://doi.org/10.1007/s11139-010-9286-6