Vanishing of degree $3$ unramified cohomology over finite fields
Abstract
Let k be a finite field of characteristic p≠ 3, let E/k be the Fermat cubic curve, and let ℓ≠ p be a prime. If p≡1pmod3, assume moreover that ℓ>3. Then H3nr(k(E3)/k,ℚℓ/ℤℓ(2))=0 and the cycle map CH2(E3)ℤℓ⟶ H4(E3,ℤℓ(2)) is surjective. In particular, H3nr(k̅(E3)/k̅,ℚℓ/ℤℓ(2))=0. Assuming the Tate conjecture for surfaces over finite fields, we prove an analogous surjectivity result, for all but finitely many primes ℓ≠ p, for the integral cycle maps for $1$-cycles on every smooth projective variety of dimension d over a finite field of characteristic different from $2$ which admits a smooth projective lift to the ring of Witt vectors. We apply our results to a conjecture of Colliot-Thélène on the local--global principle for zero-cycles over global function fields. To further illustrate these results, we exhibit examples showing that vanishing of degree-$3$ unramified cohomology over the algebraic closure of the ground field does not imply vanishing over any finite subextension, not even for Fano varieties.
Related
- Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields
- Unramified cohomology and Brauer--Manin pairing
- Maximal unramified 3-extensions of imaginary quadratic fields and \textrm SL\sb 2(\mathbb Z\sb 3)
- Polynomial corners in finite fields beyond the distinct-degree case
- Unramified Geometric Iwasawa theory of function fields
BibTeX
@misc{scavia2026vanishing,
title = {{Vanishing of degree $3$ unramified cohomology over finite fields}},
author = {Federico Scavia and Fumiaki Suzuki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36188},
eprint = {2609.36188},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36188},
}
RIS
TY - UNPB
TI - Vanishing of degree $3$ unramified cohomology over finite fields
AU - Federico Scavia
AU - Fumiaki Suzuki
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36188
UR - https://arxiv.org/abs/2609.36188
AB - Let k be a finite field of characteristic p≠ 3, let E/k be the Fermat cubic curve, and let ℓ≠ p be a prime. If p≡1pmod3, assume moreover that ℓ>3. Then H^3_nr(k(E^3)/k,ℚ_ℓ/ℤ_ℓ(2))=0 and the cycle map CH^2(E^3)_ℤ_ℓ⟶ H^4(E^3,ℤ_ℓ(2)) is surjective. In particular, H^3_nr(k̅(E^3)/k̅,ℚ_ℓ/ℤ_ℓ(2))=0. Assuming the Tate conjecture for surfaces over finite fields, we prove an analogous surjectivity result, for all but finitely many primes ℓ≠ p, for the integral cycle maps for $1$-cycles on every smooth projective variety of dimension d over a finite field of characteristic different from $2$ which admits a smooth projective lift to the ring of Witt vectors. We apply our results to a conjecture of Colliot-Thélène on the local--global principle for zero-cycles over global function fields. To further illustrate these results, we exhibit examples showing that vanishing of degree-$3$ unramified cohomology over the algebraic closure of the ground field does not imply vanishing over any finite subextension, not even for Fano varieties.
ER -