ALGEBRAIC INTEGERS AS SPECIAL VALUES OF MODULAR UNITS

Abstract

Let phi(tau) = eta(1/2(tau + 1))(2) / root 2 pi exp{1/4 pi i}eta(tau + 1), where eta(tau) is the Dedekind eta function. We show that if tau(0) is an imaginary quadratic argument and m is an odd integer, then root m phi(m tau(0))/phi(tau(0)) is an algebraic integer dividing root m. This is a generalization of a result of Berndt, Chan and Zhang. On the other hand, when K is an imaginary quadratic field and theta(K) is an element of K with lm(theta(K)) > 0 which generates the ring of integers of K over Z, we find a sufficient condition on in which ensures that root m phi(m theta(K))/phi(theta(K)) is a unit.