On the Evaluation of Singular Invariants for Canonical Generators of Certain Genus One Arithmetic Groups. Issue 1 (2nd March 2020) - Jorgenson, Jay; Smajlović, Lejla; Then, Holger
ABSTRACT: Let N be a positive square-free integer such that the discrete group Γ0 ( N ) + has genus one. In a previous article, we constructed canonical generators xN and yN of the holomorphic function field associated with Γ0 ( N ) + as well as an algebraic equation PN ( xN, yN ) = 0 with integer coefficients satisfied by these generators. In the present paper, we study the singular moduli problem corresponding to xN and yN, by which we mean the arithmetic nature of the numbers xN (τ) and yN (τ) for any CM point τ in the upper half plane H . If τ is any CM point which is not equivalent to an elliptic point of Γ0 ( N ) +, we prove that the complex numbers xN (τ) and yN (τ) are algebraic integers. Going further, we characterize the algebraic nature of xN (τ) as the generator of a certain ring class field of Q ( τ ) of prescribed order and discriminant depending on properties of τ and level N . The theoretical considerations are supplemented by computational examples. As a result, several explicit evaluations are given for various N and τ, and further arithmetic consequences of our analysis are presented. In one example, we explicitly construct a set of minimal polynomials for the Hilbert class field of Q ( - 74 ) whose coefficients are less than 2.2 × 10 4, whereas the minimal polynomial obtained from the Hauptmodul of PSL ( 2, Z ) have coefficients as large as 6.6 × 10 73 .