Irreducibility of the zero polynomials of Eisenstein series

Abstract

Let $E_k$ be the normalized Eisenstein series of weight $k$ on $\mathrm{SL}_{2}(\mathbb{Z})$. Let $\varphi_k$ be the polynomial that encodes the $j$-invariants of non-elliptic zeros of $E_k$. In 2001, Gekeler observed that the polynomials $\varphi_k$ seem to be irreducible (and verified this claim for $k\leq 446$). We show that $\varphi_k$ is irreducible for infinitely many $k$.