On the coefficients of transformation polynomials for the modular function

Abstract

In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial F_m (u, v) of the Weber modular function j(ω) of level 1 is not greater than ₂(36n+57)2ⁿ when m = 2ⁿ is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form e^O(m10gm) should hold, but I have not so far succeeded in proving such a result.