Complex Multiplication - Sample Consequence

Sample Consequence

It is no accident that

or equivalently,

is so close to an integer. This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of modular forms, and the fact that

is a unique factorization domain.

Here satisfies α² = α − 41. In general, S denotes the set of all polynomial expressions in α with coefficients in S, which is the smallest ring containing α and S. Because α satisfies this quadratic equation, the required polynomials can be limited to degree one.

Alternatively,

an internal structure due to certain Eisenstein series, and with similar simple expressions for the other Heegner numbers.