Kronecker's congruence

Short description: Theorem on a polynomial involving the elliptic modular function

In mathematics, Kronecker's congruence, introduced by Kronecker, states that

[math]\displaystyle{ \Phi_p(x,y)\equiv (x-y^p)(x^p-y)\bmod p, }[/math]

where p is a prime and Φ_p(x,y) is the modular polynomial of order p, given by

[math]\displaystyle{ \Phi_n(x,j) = \prod_\tau (x-j(\tau)) }[/math]

for j the elliptic modular function and τ running through classes of imaginary quadratic integers of discriminant n.

References

• Lang, Serge (1987), Elliptic functions, Graduate Texts in Mathematics, 112 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96508-6