Arithmetic of the modular function j(1,8)

Abstract

Since the genus of the modular curve X-1(8) = Gamma (1)(8)\H* is zero, we find a field generator j(1),(8)(z) = theta (3)(2z)/theta (3)(4z) (theta (3)(z) := Sigma (n is an element ofe pi Ze pi in2z)) such that the function field over X-1(8) is C(j(1),(8)). We apply this modular function j(1,8) to the construction of some class fields over an imaginary quadratic field K, and compute the minimal polynomial of the singular value of the Hauptmodul N(j(1,8)) of C(j(1,8)).