We give an algorithm to compute the generalized division polynomials for elliptic curves with complex multiplication. These polynomials can be used to generate the ray class fields of imaginary quadratic fields over the Hilbert class field with no restriction on the conductor. | Turk J Math (2015) 39: 547 – 555 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article On the computation of generalized division polynomials ∗ ¨ ¨ ¸ UKSAKALLI ¨ Omer KUC Department of Mathematics, Middle East Technical University, Ankara, Turkey Received: • Accepted/Published Online: • Printed: Abstract: We give an algorithm to compute the generalized division polynomials for elliptic curves with complex multiplication. These polynomials can be used to generate the ray class fields of imaginary quadratic fields over the Hilbert class field with no restriction on the conductor. Key words: Complex multiplication, division polynomial, Hurwitz number 1. Introduction A fundamental problem in algebraic number theory is to construct a polynomial that generates a given number field. Inspired by the Kronecker–Weber theorem, Hilbert’s twelfth problem asks us to generate abelian extensions of number fields using singular values of analytical functions. There are two cases, namely the cyclotomic case and the elliptic case, for which this problem has an affirmative answer. The cyclotomic case has been investigated deeply and there is a vast literature. On the other hand, our information for the elliptic case is limited. Unlike the cyclotomic case there is not even a formula for polynomials generating the simplest extensions, such as the extensions of prime conductor. Let K be an imaginary quadratic field with ring of integers OK and let f be an ideal of OK . The fundamental theorem of complex multiplication states that the ray class field Kf of conductor f can be generated by the singular values of the j -function and Weber functions [9]. The first step of such a construction is to generate the Hilbert class field H using the j -function. Instead of the j -function one can use alternative functions and produce relatively smaller class polynomials, but there is no .
