Divisibility of the coefficients of modular polynomials

Abstract

Let $N>1$ and let $\Phi_N(X,Y)\in\mathbb{Z}[X,Y]$ be the modular polynomial which vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of degree $N$. In this note we study the divisibility of the coefficients of $\Phi_N(X+J, Y+J)$ for certain algebraic numbers $J$, in particular $J=0$ and other singular moduli. It turns out that these coefficients are highly divisible by small primes at which $J$ is supersingular.