Arithmetic Wu Formulas and the Generalized Hecke Theorem

We construct canonical Steenrod square operations on the Geisser--Schmidt/Milne modified compactly supported \'etale cohomology of separated finite-type schemes over rings of $S$-integers in which $2$ is invertible. This lets us extend Feng's notion of the absolute \'etale Wu class from the finite-field setting to arithmetic bases away from $2$. A key technical input is a modified compactly supported relative Wu formula, extending Benoist's relative Wu formula to the arithmetic compact-support setting. Using this, we prove an absolute Wu formula for regular projective flat schemes over either finite fields of odd characteristic or rings of $S$-integers away from $2$: if $f\colon X\to B$ is such a scheme, then the absolute Wu class of $X$ is the product of the relative Wu class $\operatorname{Sq}^{-1}(w_{\mathrm{et}}(\tau_f))$ and the pullback of the absolute Wu class of the base. In the $S$-integer case, the base contribution is $1+\beta_B$, where $\beta_B$ is the Bockstein, equivalently the Kummer class of $-1$. As an application, we obtain an infinite family of universal mod-$2$ congruences among the Chern classes of regular projective flat schemes over such bases, governed by an arithmetic deformation of Hirzebruch's $2$-Todd series; this is the generalized Hecke theorem. In low dimensions these congruences recover Hecke's theorem on the different away from $2$, Serre's Riemann--Hurwitz theorem for spin bundles, Atiyah's theorem on theta characteristics over finite fields, and the smooth $3$-manifold branched-cover analogue of the Shusterman--Sawin theorem, while yielding new higher-dimensional congruences over both finite and arithmetic bases.