Irreducibility of the zero polynomials of Eisenstein series

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 verifie...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Author: González, Oscar E
Format: Article
Language:English
Subjects:
Polynomials
Online Access:Get full text
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Summary: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 \(k\).
ISSN:2331-8422