PreprintarXiv licencearXiv
On the cuspidal cohomology of Iwahori congruence subgroups of SL(3, ℤ)
Abstract
We investigate automorphic forms for congruence subgroups of SL(3, ℤ) that are Iwahori at p. In particular, we study the cuspidal cohomology of Iwahori congruence subgroups ℐ(3, p), which are comprised of matrices in SL(3, ℤ) that are upper triangular modulo p. In order to work with ℐ(3, p), we generalize key results from Ash, Grayson, and Green [J.\ Number Theory 19 (1984), pp.\ 412-436]. For levels ℐ(3, p) with p ≤ 227, we found three levels with nonzero cuspidal classes and were able to compute the action of Hecke operators at two of these levels. These are the first examples of non-essentially-self-dual automorphic representations of trivial cohomological weight that are Steinberg at p appearing at Iwahori level.
Related
- Residual finiteness and cuspidal cohomology of Picard modular surfaces
- Tate cohomology of unipotent cuspidal representations of finite groups of Lie type
- Decay of weighted cusp counts for congruence subgroups of SL2 over number fields
- Iwahori Gaiotto conjecture for cotangent type
- Completed Iwahori–Hecke algebra for Kac–Moody groups over local fields
BibTeX
@misc{porat2026cuspidal,
title = {{On the cuspidal cohomology of Iwahori congruence subgroups of $\mathrmSL(3, \mathbbZ)$}},
author = {Zachary Porat},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01776},
eprint = {2609.01776},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01776},
}
RIS
TY - UNPB
TI - On the cuspidal cohomology of Iwahori congruence subgroups of SL(3, ℤ)
AU - Zachary Porat
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01776
UR - https://arxiv.org/abs/2609.01776
AB - We investigate automorphic forms for congruence subgroups of SL(3, ℤ) that are Iwahori at p. In particular, we study the cuspidal cohomology of Iwahori congruence subgroups ℐ(3, p), which are comprised of matrices in SL(3, ℤ) that are upper triangular modulo p. In order to work with ℐ(3, p), we generalize key results from Ash, Grayson, and Green [J.\ Number Theory 19 (1984), pp.\ 412-436]. For levels ℐ(3, p) with p ≤ 227, we found three levels with nonzero cuspidal classes and were able to compute the action of Hecke operators at two of these levels. These are the first examples of non-essentially-self-dual automorphic representations of trivial cohomological weight that are Steinberg at p appearing at Iwahori level.
ER -