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 $f:X\to B$ of pure relative dimension, with $X$ regular and $B$ such a base, we prove $v_X={\mathrm{Sq}}^{-1}(w_{\mathrm{et}}({\tau }_f+{\mathcal{O}}_X^{\oplus 3}))$ in completed mod-$2$ étale cohomology. Here ${\tau }_f$ is the virtual relative tangent bundle, $w_{\mathrm{et}}$ the total étale Stiefel--Whitney class, and ${\mathrm{Sq}}^{-1}$ 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.

93 pages, comments are welcome