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

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


11F03 Modular and automorphic functions
Full Text: DOI


[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.