Explicit bounds on the coefficients of modular polynomials for the elliptic \(j\)-invariant (Q6572985)

For any \(N\in \mathbb{N}-\{0\}\), let \(\Phi_N(X,Y)\in\mathbb{Z}[X,Y]\) be the modular polynomial which vanishes at pairs of \(j\)-invariants of curves admitting a cyclic \(N\)-isogeny between them, and define the height \(h(\Phi_N)\) as the \(\log\) of the maximal absolute value of the coefficients of \(\Phi_N(X,Y)\). The authors improve substantially the bound\N\[\Nh(\Phi_N)=6N\prod_{p|N}\left(1+\frac{1}{p}\right)\left[ \log(N)- 2\sum_{p|N} \frac{\log(p)}{p} +O(1)\right] \qquad \text{(as }N\rightarrow +\infty\text{)}\N\]\Nin [\textit{P. Cohen}, Math. Proc. Camb. Philos. Soc. 95, 389--402 (1984; Zbl 0541.10026)] by providing an explicit constant and showing that for all \(N\)\N\[\Nh(\Phi_N)\leqslant 6N\prod_{p|N}\left(1+\frac{1}{p}\right)\left[ \log(N)- 2\sum_{p^n||N} \frac{(p^n-1)\log(p)}{p^{n-1}(p^2-1)}+\log\log(N) +4.436 \right] .\N\]\NThe values for \(N\leqslant 400\) are computed explicitly (via an algorithm in [\textit{R. Bröker} et al., Math. Comput. 81, No. 278, 1201--1231 (2012; Zbl 1267.11125)]), then a general proof for \(N>400\) is obtained by estimating the Mahler measure\N\[\N\sum_{\gamma\in C_N} \log\max\{1,|j(\tau_N)|\}\N\]\Nof the \(j\)-invariants of \(N\)-isogenous curves (where \(C_N\) is the set of \(N\prod_{p|N}\left(1+\frac{1}{p}\right)\) matrices \(\begin{pmatrix} a & b \\ 0&d \end{pmatrix}\) encoding cyclic \(N\)-isogenies).\N\NMost computation use estimates on the modular functions \(j\) and \(\Delta\) and a search for optimal constants by looking at different representatives of \(\tau\) in \(\mathrm{SL}_2(\mathbb{Z})\backslash \mathbb{H}\) (where \(\mathbb{H}\) is the complex upper half plane), to combine the results of [\textit{P. Autissier}, Bull. Soc. Math. Fr. 131, No. 3, 421--433 (2003; Zbl 1067.11022)] and [\textit{F. Pazuki}, Int. J. Number Theory 15, No. 3, 569--584 (2019; Zbl 1446.11126)].