The limiting zero counting measure for Eisenstein series
Abstract
For an arbitrary congruence group Γ, we study the zeros of a generating set of the space of Eisenstein series of varying even weight for all Γ(1)-conjugates of Γ. These zeros form a Γ(1)-invariant set that descends to the modular orbifold Y(1). It is known that, as the (even) weight tends to infinity, the set of zeros converges in the Hausdorff metric to finitely many geodesic segments. In this paper, we find the weak limit of the normalised counting measure of the zeros in the (natural) orbifold sense: if a geodesic occurring in the Hausdorff limit is parametrised by the unit circle S1 via a Möbius transformation then the limit measure on this segment is the pullback of the Haar measure on S1 by that transformation. For the proof, we first show a version of the Poincaré--Lelong theorem for orbifold curves, and then apply it on Y(1) to reduce the problem to computing the scaled limit of the subharmonic functions log |E| associated to Eisenstein series E. This turns out to be the maximum of the negative logarithm of the modulus of finitely many linear functions supported on the relevant geodesic segments, and we finish the proof by showing that the associated measure has no atoms and making explicit, on each open segment, the distributional Laplacian of this function in terms of the argument measure. The non-atomicity allows us to conclude a sublinear growth rate in the weight for the total order of all Eisenstein series at a given point in Y(1), and weak convergence also of the non-orbifold measure to the same limit.
Related
- Zero-one laws for uniform approximation via Gaussian and Eisenstein integers
- Second moment of incomplete Eisenstein series revisited
- Fourier Coefficients of Siegel-Eisenstein Series of Degree 2m and Weight m+1
- Modularity criteria for multiple Eisenstein series
- Zagier's 2-3-2 formula for multiple Eisenstein series
BibTeX
@misc{cornelissen2026limiting,
title = {{The limiting zero counting measure for Eisenstein series}},
author = {Gunther Cornelissen and Mahya Mehrabdollahei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32590},
eprint = {2610.32590},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32590},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The limiting zero counting measure for Eisenstein series
AU - Gunther Cornelissen
AU - Mahya Mehrabdollahei
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32590
UR - https://arxiv.org/abs/2610.32590
AB - For an arbitrary congruence group Γ, we study the zeros of a generating set of the space of Eisenstein series of varying even weight for all Γ(1)-conjugates of Γ. These zeros form a Γ(1)-invariant set that descends to the modular orbifold Y(1). It is known that, as the (even) weight tends to infinity, the set of zeros converges in the Hausdorff metric to finitely many geodesic segments. In this paper, we find the weak limit of the normalised counting measure of the zeros in the (natural) orbifold sense: if a geodesic occurring in the Hausdorff limit is parametrised by the unit circle S^1 via a Möbius transformation then the limit measure on this segment is the pullback of the Haar measure on S^1 by that transformation. For the proof, we first show a version of the Poincaré--Lelong theorem for orbifold curves, and then apply it on Y(1) to reduce the problem to computing the scaled limit of the subharmonic functions log |E| associated to Eisenstein series E. This turns out to be the maximum of the negative logarithm of the modulus of finitely many linear functions supported on the relevant geodesic segments, and we finish the proof by showing that the associated measure has no atoms and making explicit, on each open segment, the distributional Laplacian of this function in terms of the argument measure. The non-atomicity allows us to conclude a sublinear growth rate in the weight for the total order of all Eisenstein series at a given point in Y(1), and weak convergence also of the non-orbifold measure to the same limit.
ER -