ℓ-Independence, slopes, and weights of ℓ-adic cohomology groups of rigid spaces
Abstract
Let ℓ and p be distinct prime numbers, K a local field with residue characteristic p, C a complete algebraically closed extension of K, and X a quasi-compact quasi-separated rigid space over K. When mathopchar K = 0, we prove that the alternating sum of traces of elements in the positive part of the Weil group acting on RΓ(XC,ℚℓ) is independent of ℓ and is an integer, extending a result of Mieda to the singular case. We also give optimal bounds for the p-adic valuations of the eigenvalues of geometric Frobenius liftings acting on Hi(XC,ℚℓ), proving a conjecture of Lu--Zheng. In the general case, we give optimal bounds for the p-adic valuations of the eigenvalues of geometric Frobenius liftings acting on Hic(XC,ℚℓ), and we show that these eigenvalues are algebraic integers and Weil numbers, thereby extending results of Lu--Zheng and Mieda to the positive characteristic case.
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BibTeX
@misc{wu2026ℓindependence,
title = {{$\ell$-Independence, slopes, and weights of $\ell$-adic cohomology groups of rigid spaces}},
author = {Xiangdong Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17852},
eprint = {2610.17852},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17852},
}
RIS
TY - UNPB
TI - ℓ-Independence, slopes, and weights of ℓ-adic cohomology groups of rigid spaces
AU - Xiangdong Wu
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17852
UR - https://arxiv.org/abs/2610.17852
AB - Let ℓ and p be distinct prime numbers, K a local field with residue characteristic p, C a complete algebraically closed extension of K, and X a quasi-compact quasi-separated rigid space over K. When mathopchar K = 0, we prove that the alternating sum of traces of elements in the positive part of the Weil group acting on RΓ(X_C,ℚ_ℓ) is independent of ℓ and is an integer, extending a result of Mieda to the singular case. We also give optimal bounds for the p-adic valuations of the eigenvalues of geometric Frobenius liftings acting on H^i(X_C,ℚ_ℓ), proving a conjecture of Lu--Zheng. In the general case, we give optimal bounds for the p-adic valuations of the eigenvalues of geometric Frobenius liftings acting on H^i_c(X_C,ℚ_ℓ), and we show that these eigenvalues are algebraic integers and Weil numbers, thereby extending results of Lu--Zheng and Mieda to the positive characteristic case.
ER -