Arithmetic Wu Formulas and the Generalized Hecke Theorem
Abstract
We construct canonical stable Steenrod squares on modified compactly supported étale cohomology of separated finite-type schemes over rings of S-integers in number fields with $2$ invertible, extending Feng's absolute Wu classes to arithmetic bases. For a flat projective morphism f:X→ B of pure relative dimension, with X regular and B such a base, we prove vX=Sq−1(wet(τf+𝒪X⊕3)) in completed mod-$2$ étale cohomology. Here τf is the virtual relative tangent bundle, wet the total étale Stiefel--Whitney class, and Sq−1 the inverse total Steenrod square. Over finite fields of odd characteristic, the formula holds without the three trivial summands. The proof uses a modified compactly supported relative Wu formula extending Benoist's theorem. Our generalized Hecke theorem gives universal mod-$2$ relations involving Chern classes and the Kummer class of −1, governed by an arithmetic deformation of Hirzebruch's $2$-Todd series. These hold modulo an explicit archimedean ideal and become vanishing identities over finite fields, over totally imaginary arithmetic bases, or when −1 is a square on X. Nonempty real loci force infinitely many nonzero Wu components. Applications include Hecke's theorem on the different away from $2$, a finite-field analog of Atiyah's theorem on theta characteristics, and new higher-dimensional relations. We also revisit Serre's Riemann--Hurwitz theorem for spin bundles, recover the Shusterman--Sawin theorem for smooth branched covers of closed $3$-manifolds and prove a function-field analog of the Lusztig--Milnor--Peterson formula, expressing the difference between mod-$2$ and $2$-adic semicharacteristics using the middle Wu class and its Tate-twisted Bockstein.
Related
BibTeX
@misc{carmeli2026arithmetic,
title = {{Arithmetic Wu Formulas and the Generalized Hecke Theorem}},
author = {Shachar Carmeli and Mark Shusterman and Sa'ar Zehavi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.06008},
eprint = {2606.06008},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.06008},
}
RIS
TY - UNPB
TI - Arithmetic Wu Formulas and the Generalized Hecke Theorem
AU - Shachar Carmeli
AU - Mark Shusterman
AU - Sa'ar Zehavi
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2606.06008
UR - https://arxiv.org/abs/2606.06008
AB - We construct canonical stable Steenrod squares on modified compactly supported étale cohomology of separated finite-type schemes over rings of S-integers in number fields with $2$ invertible, extending Feng's absolute Wu classes to arithmetic bases. For a flat projective morphism f:X→ B of pure relative dimension, with X regular and B such a base, we prove v_X=Sq^−1(w_et(τ_f+𝒪_X^⊕3)) in completed mod-$2$ étale cohomology. Here τ_f is the virtual relative tangent bundle, w_et the total étale Stiefel--Whitney class, and Sq^−1 the inverse total Steenrod square. Over finite fields of odd characteristic, the formula holds without the three trivial summands. The proof uses a modified compactly supported relative Wu formula extending Benoist's theorem. Our generalized Hecke theorem gives universal mod-$2$ relations involving Chern classes and the Kummer class of −1, governed by an arithmetic deformation of Hirzebruch's $2$-Todd series. These hold modulo an explicit archimedean ideal and become vanishing identities over finite fields, over totally imaginary arithmetic bases, or when −1 is a square on X. Nonempty real loci force infinitely many nonzero Wu components. Applications include Hecke's theorem on the different away from $2$, a finite-field analog of Atiyah's theorem on theta characteristics, and new higher-dimensional relations. We also revisit Serre's Riemann--Hurwitz theorem for spin bundles, recover the Shusterman--Sawin theorem for smooth branched covers of closed $3$-manifolds and prove a function-field analog of the Lusztig--Milnor--Peterson formula, expressing the difference between mod-$2$ and $2$-adic semicharacteristics using the middle Wu class and its Tate-twisted Bockstein.
ER -