Petersson-rigid lattices of signature (2,1) and their harmonic weak Maass forms
Abstract
An even lattice L of signature (2,1) is Petersson-rigid if the orthogonal group of its discriminant form acts absolutely irreducibly on the space of cusp forms of weight 3/2 for its Weil representation, a space of dimension at least two. We prove, with computer assistance, that there are exactly twelve such lattices up to isometry, and we determine their root bases. Four of them are theta lattices, whose cusp forms are lifts of unary theta series. On each, the space is ℂ F0⊗σ0 for one form F0 and an irreducible σ0, so the Petersson norm is |F|2=tμ|cF(μ)|2 for all cusp forms F, with one tμ>0, μ the least exponent of F0 and cF(μ) the coefficient of qμ in F. We prove that the harmonic weak Maass forms of weight 1/2 in the σ0-isotypic part with a pole of least possible order are exactly the forms ℋF with principal part q−μcF(μ)̅, and ξ1/2ℋF=F/tμ. For the four theta lattices it is a theorem that tμ=(s−1)/12√s, with s the square-free part of |det L|/2, and the ℋF come from umbral moonshine. For the eight others we compute F0, tμ and ℋF0: the Hecke eigenvalues are integers and, at the primes computed, the ap of an elliptic curve, and for three of the eight F0 is explicit as an eta-theta product. We prove, for every prime p>3, a closed formula in p and one class number for the difference g(X0(4p)/wp)−g(X0(2p)/wp) of the genera of two Atkin-Lehner quotients of modular curves. In a family 𝒟p of discriminant forms, p an odd prime, containing three of the eight, comparison with that classification shows that 𝒟p is rigid exactly when X0(4p)/wp has genus $1$ and X0(2p)/wp has genus $0$.
Related
- Petersson-Rigid Lattices in a Census of 100 Rank-Three Root Bases
- Singularity of harmonic measures for hyperbolic lattices
- Ramanujan-Type Series of Signature 2: Analytical Evaluation via Degree-2 Transformations and Associated Harmonic Expansions
- Local Weak Maass Forms and Traces of Cycle Integrals
- Quantitative nodal domain count for Maass forms
BibTeX
@misc{cho2026peterssonrigid,
title = {{Petersson-rigid lattices of signature (2,1) and their harmonic weak Maass forms}},
author = {Eungang Cho},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40519},
eprint = {2610.40519},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40519},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Petersson-rigid lattices of signature (2,1) and their harmonic weak Maass forms
AU - Eungang Cho
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40519
UR - https://arxiv.org/abs/2610.40519
AB - An even lattice L of signature (2,1) is Petersson-rigid if the orthogonal group of its discriminant form acts absolutely irreducibly on the space of cusp forms of weight 3/2 for its Weil representation, a space of dimension at least two. We prove, with computer assistance, that there are exactly twelve such lattices up to isometry, and we determine their root bases. Four of them are theta lattices, whose cusp forms are lifts of unary theta series. On each, the space is ℂ F_0⊗σ_0 for one form F_0 and an irreducible σ_0, so the Petersson norm is |F|^2=t_μ|c_F(μ)|^2 for all cusp forms F, with one t_μ>0, μ the least exponent of F_0 and c_F(μ) the coefficient of q^μ in F. We prove that the harmonic weak Maass forms of weight 1/2 in the σ_0-isotypic part with a pole of least possible order are exactly the forms ℋ_F with principal part q^−μc_F(μ)̅, and ξ_1/2ℋ_F=F/t_μ. For the four theta lattices it is a theorem that t_μ=(s−1)/12√s, with s the square-free part of |det L|/2, and the ℋ_F come from umbral moonshine. For the eight others we compute F_0, t_μ and ℋ_F_0: the Hecke eigenvalues are integers and, at the primes computed, the a_p of an elliptic curve, and for three of the eight F_0 is explicit as an eta-theta product. We prove, for every prime p>3, a closed formula in p and one class number for the difference g(X_0(4p)/w_p)−g(X_0(2p)/w_p) of the genera of two Atkin-Lehner quotients of modular curves. In a family 𝒟_p of discriminant forms, p an odd prime, containing three of the eight, comparison with that classification shows that 𝒟_p is rigid exactly when X_0(4p)/w_p has genus $1$ and X_0(2p)/w_p has genus $0$.
ER -