Arithmetic of the modular function \(j_{1,8}\)
The modular curve \(X_1(N)= \Gamma_1(N) \setminus \mathbb{H}^*\) is of genus zero, if and only if \(1\leq N\leq 10\) and \(N=12\) and so, for these values of \(N\) the corresponding function field \(\mathbb{C}(X_1(N))\) is a rational function field. The authors consider the case for \(N=8\), find a generator, \(j_{1,8}\), of this field by means of Jacobi theta series, show that the normalized generator \(N(j_{1,8})\), is an algebraic integer when evaluated at a point of \(\mathbb{H}\cap \mathbb{Q} (\sqrt{-d})\), apply this function to construct some class fields over an imaginary quadratic field and calculate the minimal polynomial of \(N(j_{1,8} (\alpha))\). The article is a continuation of their previous work [Acta Arith. 84, 129-143 (1998; Zbl 0907.11014)].
scientific article; zbMATH DE number 1563056