Functiones et Approximatio Commentarii Mathematici

Explicit congruences for class equations

Patrick Morton

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Explicit congruences (mod $p$) are proved for the class equations corresponding to discriminants $D=-8p, -3p, -12p$ in the theory of complex multiplication, where $p$ is an odd prime. They are explicit in that they can be computed directly from a formula for the supersingular polynomial without first having to know the coefficients of the class equation in characteristic zero. These congruences have previously appeared in print without proof, and have been used to study factorizations of certain Legendre polynomials (mod $p$).

Article information

Funct. Approx. Comment. Math., Volume 51, Number 1 (2014), 77-110.

First available in Project Euclid: 24 September 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]
Secondary: 11H52 11R11: Quadratic extensions

class equation supersingular polynomial modular equation class number complex multiplication


Morton, Patrick. Explicit congruences for class equations. Funct. Approx. Comment. Math. 51 (2014), no. 1, 77--110. doi:10.7169/facm/2014.51.1.4.

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