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Sub-Riemannian Selberg Trace Formulae for Compact Quotients of SL(2,R) and Determinants of Sub-Laplacians
Abstract
We prove sub-Riemannian Selberg trace formulae for compact quotients of SL(2,R). Using the Fourier decomposition along the SO(2)-fibers, we reduce the heat trace computation to the Selberg trace formula for Maass Laplacians on the hyperbolic plane. The resulting formula has an identity contribution and a hyperbolic contribution, the latter involving a theta factor that records both the twisting character and the central sign of the chosen lift to SL(2,\mathbb R)). We then use this trace formula to compute the zeta-regularized determinant of the sub-Laplacian. The determinant formula is expressed as a universal factor depending only on the base hyperbolic surface times an explicit lift- and character-dependent Selberg-type product.
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BibTeX
@misc{baudoin2026subriemannian,
title = {{Sub-Riemannian Selberg Trace Formulae for Compact Quotients of SL(2,R) and Determinants of Sub-Laplacians}},
author = {Fabrice Baudoin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.11813},
eprint = {2606.11813},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11813},
}
RIS
TY - UNPB
TI - Sub-Riemannian Selberg Trace Formulae for Compact Quotients of SL(2,R) and Determinants of Sub-Laplacians
AU - Fabrice Baudoin
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2606.11813
UR - https://arxiv.org/abs/2606.11813
AB - We prove sub-Riemannian Selberg trace formulae for compact quotients of SL(2,R). Using the Fourier decomposition along the SO(2)-fibers, we reduce the heat trace computation to the Selberg trace formula for Maass Laplacians on the hyperbolic plane. The resulting formula has an identity contribution and a hyperbolic contribution, the latter involving a theta factor that records both the twisting character and the central sign of the chosen lift to SL(2,\mathbb R)). We then use this trace formula to compute the zeta-regularized determinant of the sub-Laplacian. The determinant formula is expressed as a universal factor depending only on the base hyperbolic surface times an explicit lift- and character-dependent Selberg-type product.
ER -