Class invariants by Shimura's reciprocity law

GALOIS ACTIONS OF A CLASS INVARIANT OVER QUADRATIC NUMBER FIELDS WITH DISCRIMINANT D≡64(mod72)

“…For that we need to find the action of M m ∈ GL 2 (Z/mZ) with m = 8, 9. Combining Lemma 6 of [2] and the transformation rule (2.3), we obtain the following: …”

“…The reciprocity law provides not only a method of systematically determining whether f (θ) is a class invariant but also a description of the Galois conjugates of f (θ) under Gal(H O /K). This tool is well illustrated in several works by Alice Gee and Peter Stevenhagen in [2,3,4,9]. The author [5,6,7] compute the Galois actions of certain class invariants over some cases of quadratic number fields.…”

Constructing class invariants

“…The interested reader should consider the more detailed explanations found in [5], [7], [6], [19]. In section 3 we explain our main observation.…”

“…There are many modular functions that can be used for the generation of the ring class field. In a series of articles [5], [7], [6], [19] A. Gee and P. Stevenhagen developed a method based on Shimura reciprocity law, in order to check whether a modular function gives rise to a class invariant. A necessary condition for this is the invariance of the modular function under the action of the group…”

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.…”