A Note on a Problem Proposed by Kim and Lisonek

Abstract

Let $\ell$ a prime number and $\Phi _{\ell }(X,Y)$ the modular polynomial of level $\ell$. Since this polynomial has integer coefficients one may compute it modulo primes. Petr Lisonek and Yung-Jung Kim compute the polynomial $\Phi _{\ell }(X,Y)$ modulo $2$ explicitly for several $\ell$ and they conjecture that the coefficients under the diagonal vanish. In this note we prove their conjecture and that the same property holds modulo the primes $3$ and $5$.

C.-S. Radu—The research was supported by the strategic program “Innovatives OÖ 2010 plus” by the Upper Austrian Government in the frame of project W1214-N15-DK6 of the Austrian Science Fund (FWF).