Isospectral congruence manifolds, spectral rigidity, and a converse to Vignéras
Abstract
Let B be a quaternion division algebra over a number field k that gives rise to arithmetic Fuchsian or Kleinian groups, and let 𝒪 be a maximal order of B. We show that every closed orientable Riemannian manifold isospectral to a principal congruence hyperbolic surface XO(a) is isometric to X𝒪'(a) for some maximal order 𝒪' of B, provided that no prime factor of a ramifies in B and that a satisfies a mild condition at the primes over $2$. This is a converse to the construction of Vignéras, which produces isospectral manifolds from maximal orders of different types. In particular the spectral genus of XO(a) has at most t(B) elements, and XO(a) is spectrally rigid when the type number t(B) is one. In dimension three we prove the same statements for manifolds that, in addition, have the same complex lengths of closed geodesics. The proof rests on a local result that characterizes, by traces alone, the subgroups of SL(2) over a p-adic ring that are conjugate into a principal congruence subgroup by traces. As applications, every principal congruence Hurwitz surface is spectrally rigid, among them Klein's quartic, the Fricke-Macbeath surface and the surfaces of the first Hurwitz triplet, which partially answers a question of Reid.
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BibTeX
@misc{katz2026isospectral,
title = {{Isospectral congruence manifolds, spectral rigidity, and a converse to Vignéras}},
author = {Justin Katz and D. B. McReynolds},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39874},
eprint = {2610.39874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39874},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Isospectral congruence manifolds, spectral rigidity, and a converse to Vignéras
AU - Justin Katz
AU - D. B. McReynolds
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39874
UR - https://arxiv.org/abs/2610.39874
AB - Let B be a quaternion division algebra over a number field k that gives rise to arithmetic Fuchsian or Kleinian groups, and let 𝒪 be a maximal order of B. We show that every closed orientable Riemannian manifold isospectral to a principal congruence hyperbolic surface X_O(a) is isometric to X_𝒪'(a) for some maximal order 𝒪' of B, provided that no prime factor of a ramifies in B and that a satisfies a mild condition at the primes over $2$. This is a converse to the construction of Vignéras, which produces isospectral manifolds from maximal orders of different types. In particular the spectral genus of X_O(a) has at most t(B) elements, and X_O(a) is spectrally rigid when the type number t(B) is one. In dimension three we prove the same statements for manifolds that, in addition, have the same complex lengths of closed geodesics. The proof rests on a local result that characterizes, by traces alone, the subgroups of SL(2) over a p-adic ring that are conjugate into a principal congruence subgroup by traces. As applications, every principal congruence Hurwitz surface is spectrally rigid, among them Klein's quartic, the Fricke-Macbeath surface and the surfaces of the first Hurwitz triplet, which partially answers a question of Reid.
ER -