PreprintCC BY 4.0arXiv
Invariant Measures on Smooth Algebraic Groups over Higher Local Fields
Abstract
We construct a positive, finitely additive, left-invariant measure on the congruence cylinder ring of a smooth d-dimensional algebraic group over an n-dimensional local field. The measure takes values in ℝ((X2))⋯((Xn)). A normalized cylinder of depth (i1,…,in) has volume q−di1∏ℓ=2nXℓdiℓ, where q is the final residue-field cardinality. We also obtain coordinate-change and quotient integration formulas for this measure.
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BibTeX
@misc{ma2026invariant,
title = {{Invariant Measures on Smooth Algebraic Groups over Higher Local Fields}},
author = {Xuecai Ma and Jingwen Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11468},
eprint = {2610.11468},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11468},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Invariant Measures on Smooth Algebraic Groups over Higher Local Fields
AU - Xuecai Ma
AU - Jingwen Zhu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11468
UR - https://arxiv.org/abs/2610.11468
AB - We construct a positive, finitely additive, left-invariant measure on the congruence cylinder ring of a smooth d-dimensional algebraic group over an n-dimensional local field. The measure takes values in ℝ((X_2))⋯((X_n)). A normalized cylinder of depth (i_1,…,i_n) has volume q^−di_1∏_ℓ=2^nX_ℓ^di_ℓ, where q is the final residue-field cardinality. We also obtain coordinate-change and quotient integration formulas for this measure.
ER -