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2008•Pure and Applied Mathematics QuarterlyOpen access

Minimal CM Liftings of Supersingular Elliptic Curves

Tonghai Yang

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Abstract

In this paper, we prove that if every supersingular elliptic curve over F p can be lifted to a CM elliptic curve by an imaginary order O D for some D p θ , then θ ≥ 1 2 .We also prove that if every supersingular elliptic curve over Fp can be lifted to a CM elliptic curve by an imaginary order O D for some D p θ , then θ ≥ 2 3 as suggested by Elkies.

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What this paper is about

In this paper, we prove that if every supersingular elliptic curve over F p can be lifted to a CM elliptic curve by an imaginary order O D for some D p θ , then θ ≥ 1 2 .We also prove that if every supersingular elliptic curve over Fp can be lifted to a CM elliptic curve by an imaginary order O D for some D p θ , then θ ≥ 2 3 as suggested by Elkies.

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Available abstract

In this paper, we prove that if every supersingular elliptic curve over F p can be lifted to a CM elliptic curve by an imaginary order O D for some D p θ , then θ ≥ 1 2 .We also prove that if every supersingular elliptic curve over Fp can be lifted to a CM elliptic curve by an imaginary order O D for some D p θ , then θ ≥ 2 3 as suggested by Elkies.

Key concepts: Supersingular elliptic curve, Mathematics, Elliptic curve, Order (exchange), Elliptic curve point multiplication, Schoof's algorithm, Half-period ratio, Hessian form of an elliptic curve

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