Generating class fields using Shimura reciprocity

Schläfli modular equations for generalized Weber functions

“…We should note immediately that various functions of this type already appear in the literature (see for example [12] or even [17] §72), however they have different normalizations to ours (or none at all).…”

“…The author's interest in these modular equations arose from a study of the use of modular equations in evaluating singular values of quotients of the Dedekind eta function. These turn out to provide explicit generators for ring class fields of certain imaginary quadratic number fields (see [12]). The final section of this paper details a simple evaluation along these lines by making use of the modular equations derived earlier.…”

“…We explicitly evaluate a specific quotient of the Dedekind eta function. Evaluation of such quantities is important in obtaining explicit generators of ring class fields in explicit class field theory (see [12] for further details). Our example will come from level three functions.…”

Galois action on special theta values

“…In this section we follow the interpretation of Shimura's reciprocity law (see [11]) by Gee and Stevenhagen (see [5], [4], [12]). Let K be an imaginary quadratic field and O an order in K with basis [α, 1].…”

Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

“…For this to be doable, new invariants had to be used, minimizing the height of their minimal polynomials. This task was done using Schertz's formulation of Shimura's reciprocity law [39], with the invariants of [18,16] (alternatively see [25,24]). Note that replacing j by other functions does not change the complexity of the algorithm, though it is crucial in practice.…”