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Quantitative nodal domain count for Maass forms

  • Peter Zenz
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Abstract

We consider a real even Hecke-Maass cusp form ϕ on the modular surface, with Laplace eigenvalue 1/4+T2. We prove that its restriction to any fixed compact segment β of the vertical geodesic above i has ≫β,ε T1/6−ε sign changes, yielding the same lower bound for the number of distinct inert nodal domains meeting the segment. Building on ideas of Jang-Jung and Jung-Young, we study correlations between values at points on the vertical geodesic separated by distances of order T−1. Using arithmetic quantum unique ergodicity and Dirichlet-polynomial estimates, we show that the correlation is negative for a suitable choice of separation. An elementary counting argument and the Iwaniec-Sarnak supremum bound then give the claimed lower bound.

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Cite

BibTeXRIS
BibTeX
@misc{zenz2026quantitative,
  title = {{Quantitative nodal domain count for Maass forms}},
  author = {Peter Zenz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31857},
  eprint = {2610.31857},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31857},
}
RIS
TY  - UNPB
TI  - Quantitative nodal domain count for Maass forms
AU  - Peter Zenz
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31857
UR  - https://arxiv.org/abs/2610.31857
AB  - We consider a real even Hecke-Maass cusp form ϕ on the modular surface, with Laplace eigenvalue 1/4+T^2. We prove that its restriction to any fixed compact segment β of the vertical geodesic above i has ≫_β,ε T^1/6−ε sign changes, yielding the same lower bound for the number of distinct inert nodal domains meeting the segment. Building on ideas of Jang-Jung and Jung-Young, we study correlations between values at points on the vertical geodesic separated by distances of order T^−1. Using arithmetic quantum unique ergodicity and Dirichlet-polynomial estimates, we show that the correlation is negative for a suitable choice of separation. An elementary counting argument and the Iwaniec-Sarnak supremum bound then give the claimed lower bound.
ER  -