Legendre Polynomials and Complex Multiplication, I Legendre Polynomials and Complex Multiplication, I
نویسنده
چکیده
The factorization of the Legendre polynomial of degree (p− e)/4, where p is an odd prime, is studied over the finite field Fp. It is shown that this factorization encodes information about the supersingular elliptic curves in Legendre normal form which admit the endomorphism √ −2p, by proving an analogue of Deuring’s theorem on supersingular curves with multiplier √ −p. This is used to count the number of irreducible binomial quadratic factors of P(p−e)/4(x) over Fp in terms of the class number h(−2p). 1 2
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