On explicit formulas for the modular equation (Q5940641)

For every \(n\in {\mathbb N}\) there exists the so-called modular polynomial \(\Phi_n(X,J)=\sum C_{i,j}X^i J^j\in {\mathbb Z}[J][X]\). It encodes a lot of information: modular curves, \(n\)-isogenies between elliptic curves, Hecke operators, class equations, \(\ldots\) . Correspondingly, even for very small values of \(n\) the coefficients \(C_{i,j}\) are astronomically large, and only afew of the \(\Phi_n(X,J)\) are explicitly known. NEWLINENEWLINENEWLINEFor primes \(p\), \textit{N. Yui} [J. Reine Angew. Math. 299-300, 185--200 (1978; Zbl 0368.14012)] described how to determine \(\Phi_p(X,J)\) using the \(q\)-expansion (i.e., the Fourier expansion around the cusp \(i\infty\)) of the elliptic modular function \(J(z)\). For composite \(n\) one can theoretically obtain \(\Phi_n(X,J)\) from the \(\Phi_m(X,J)\) for divisors \(m\) of \(n\). But this is not suitable for explicit computation, since it involves calculations with the zeroes of \(\Phi_m(X,J)\). NEWLINENEWLINENEWLINEThe authors describe a more practical algorithm to determine \(\Phi_{p^2}(X,J)\) by using the Fourier expansions of the function \(J(p^2 z)\) around the cusps \(i\infty\) and \(\frac{-1}{p}\) of the group \(\Gamma_0(p^2)\). NEWLINENEWLINENEWLINEAs an example they calculate \(\Phi_9(X,J)\). Obviously, the coefficient \(C_{10,10}=-1/2\) must be a typo, since the correct value has to be an integer. We also point out that some of the rational numbers in Lemma 2 represent the same cusps; but this does not affect the algorithm.