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 \(k\le 446\)). We show that \(\varphi _k\) is irreducible for infinitely many k.
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References
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Acknowledgements
The author thanks Scott Ahlgren for his insightful suggestions. The author was partially supported by the Alfred P. Sloan Foundation’s MPHD Program, awarded in 2017.
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González, O.E. Irreducibility of the zero polynomials of Eisenstein series. Arch. Math. 119, 351–358 (2022). https://doi.org/10.1007/s00013-022-01766-6
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DOI: https://doi.org/10.1007/s00013-022-01766-6