On the coefficients of the transformation polynomials for the elliptic modular function. (English) Zbl 0541.10026

Cohen, Paula

Math. Proc. Camb. Philos. Soc. 95, 389-402 (1984).

Let \(\Phi_ m\) denote the mth transformation polynomial for the elliptic modular invariant j(z). K. Mahler [Acta Arith. 21, 89-97 (1972; Zbl 0257.10013), and Bull. Aust. Math. Soc. 10, 197-218 (1974; Zbl 0269.10013)] investigated the precise rate of growth of the coefficients of \(\Phi_ m\) and showed that \(h(\Phi_ m)\leq c m^{3/2}\) for some absolute constant \(c>0\), where h(P) denotes the logarithm of the maximum of the absolute value of the coefficients of P (the height of P).
In this well written paper the author settles the problem of exactly how large \(h(\Phi_ m)\) becomes for large m. The main theorem is that \[ h(\Phi_ m)=6 \psi(m)\{\log m-2 \kappa(m)+O(1)\} \] where \(\kappa(m)=\sum_{p| m}p^{-1} \log p\) and \(\psi(m)=m\prod_{p| m}(1+p^{-1})\). A corollary is that \(h(\Phi_ m)\sim 6 \psi(m)\quad \log m.\) In particular, when \(m=2^ n\) the height is asymptotic to \(9n 2^ n \log 2\) which, apart from the constants, was an upper bound previously given by Mahler.

Reviewer: T.M.Apostol

MSC:

11F03

Modular and automorphic functions

Keywords:

transformation polynomial; elliptic modular invariant; growth of the coefficients; height

Citations:

Zbl 0257.10013; Zbl 0269.10013

References:

[1]	Mahler, Bull. Austral. Math. Soc. 10 pp 197– (1974)
[2]	Mahler, Acta Arith. 21 pp 89– (1972)
[3]	Hardy, An Introduction to the Theory of Numbers (1979) · Zbl 0423.10001
[4]	Lang, Elliptic Functions (1973)
[5]	Herrmann, J. Reine Angew. Math. 274/275 pp 187– (1975)
[6]	Lang, Elliptic Curves, Diophantine Analysis 231 (1978)