Irreducibility of the zero polynomials of Eisenstein series

TL;DR

This paper proves that the polynomials encoding non-elliptic zeros of Eisenstein series are irreducible for infinitely many weights, extending previous computational evidence and contributing to understanding their algebraic properties.

Contribution

The paper establishes the irreducibility of the polynomials _k for infinitely many weights k, advancing the knowledge on their algebraic structure beyond computational verification.

Findings

Proves _k is irreducible for infinitely many k

Extends Gekeler's computational results from k46 to an infinite set

Provides new insights into the algebraic nature of Eisenstein series zeros

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 $k$.