Zeros of E2, quasi-periods, and Schwarz maps
Abstract
We study the zeros of the Eisenstein series E2, their distribution near rational cusps, and the values of modular functions at these zeros. Using Watson's inversion formula, we prove that the number of zeros in 0≤Reτ<1 with imaginary part at least Y satisfies N(Y)=1/2πY +O (Y−1/2log1/Y) (Y→0+). We also determine the limiting distribution of the scaled displacements from the cusps and prove that two zeros whose difference is rational differ by an integer. At every zero ρ, classical transcendence theorems imply that both j(ρ) and λ(ρ) are transcendental. We use this fact to obtain an exact multiplicity formula for arithmetic quasimodular forms at arbitrary congruence level and to show that all ordinary derivatives of arithmetic modular functions on congruence subgroups, evaluated at ρ, are algebraic over ℚ̅(j(ρ),j'(ρ)). We then relate Watson's formula to the Schwarz map of a rational transform of the Legendre hypergeometric equation. Finally, we give conditional results on the transcendence of ρ under modular Schanuel and period conjectures, and determine why the associated analytic subgroup fails the rank condition in Waldschmidt's theorem.
Related
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- Generic reconstruction of rational maps from multipliers of periods one and two
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BibTeX
@misc{sebbar2026zeros,
title = {{Zeros of $E_2$, quasi-periods, and Schwarz maps}},
author = {Abdellah Sebbar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30025},
eprint = {2610.30025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30025},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Zeros of E_2, quasi-periods, and Schwarz maps
AU - Abdellah Sebbar
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30025
UR - https://arxiv.org/abs/2610.30025
AB - We study the zeros of the Eisenstein series E_2, their distribution near rational cusps, and the values of modular functions at these zeros. Using Watson's inversion formula, we prove that the number of zeros in 0≤Reτ<1 with imaginary part at least Y satisfies N(Y)=1/2πY +O (Y^−1/2log1/Y) (Y→0^+). We also determine the limiting distribution of the scaled displacements from the cusps and prove that two zeros whose difference is rational differ by an integer. At every zero ρ, classical transcendence theorems imply that both j(ρ) and λ(ρ) are transcendental. We use this fact to obtain an exact multiplicity formula for arithmetic quasimodular forms at arbitrary congruence level and to show that all ordinary derivatives of arithmetic modular functions on congruence subgroups, evaluated at ρ, are algebraic over ℚ̅(j(ρ),j'(ρ)). We then relate Watson's formula to the Schwarz map of a rational transform of the Legendre hypergeometric equation. Finally, we give conditional results on the transcendence of ρ under modular Schanuel and period conjectures, and determine why the associated analytic subgroup fails the rank condition in Waldschmidt's theorem.
ER -