, Volume 24, Issue 1, pp 67-83
Date: 24 Nov 2010

Singular values of some modular functions

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Abstract

For an integer N greater than 5 and a triple \({\mathfrak{a}}=[a_{1},a_{2},a_{3}]\) of integers with the properties 0<a i N/2 and a i a j for ij, we consider a modular function \(W_{\mathfrak{a}}(\tau)=\frac{\wp (a_{1}/N;L_{\tau})-\wp (a_{3}/N;L_{\tau})}{\wp (a_{2}/N;L_{\tau})-\wp(a_{3}/N;L_{\tau})}\) for the modular group Γ 1(N), where ℘(z;L τ ) is the Weierstrass ℘-function relative to the lattice L τ generated by 1 and a complex number τ with positive imaginary part. For a pair of such triples \({\mathfrak{A}}=[{\mathfrak{a}},{\mathfrak{b}}]\) and a pair of non-negative integers F=[m,n], we define a modular function \(T_{{\mathfrak{A}},F}\) for the group Γ 0(N) as the trace of the product \(W_{\mathfrak{a}}^{m}W_{\mathfrak{b}}^{n}\) to the modular function field of Γ 0(N). In this article, we study the integrality of singular values of the functions \(W_{\mathfrak{a}}\) and \(T_{{\mathfrak{A}},F}\) by using their modular equations. We prove that the functions \(T_{{\mathfrak{A}},F}\) for suitably chosen \({\mathfrak{A}}\) and F generate the modular function field of Γ 0(N), and from Shimura reciprocity and Gee–Stevenhagen method we obtain that singular values \(T_{{\mathfrak{A}},F}(\tau)\) for suitably chosen \({\mathfrak{A}}\) and F generate ring class fields. Further, we study the class polynomial of \(T_{{\mathfrak{A}},F}\) for Schertz N-system.