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arxiv: 2606.06008 · v1 · pith:2ZQCJ7AZnew · submitted 2026-06-04 · 🧮 math.NT · math.AG· math.AT

Arithmetic Wu Formulas and the Generalized Hecke Theorem

Shachar Carmeli , Mark Shusterman , Sa'ar Zehavi This is my paper

Pith reviewed 2026-06-27 23:47 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.AT
keywords étale cohomologyWu classesSteenrod squaresHecke theoremChern classesarithmetic schemesmod 2 congruencesBockstein class
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The pith

The absolute étale Wu class of a regular projective flat scheme over an S-integer ring factors as the product of a relative Wu class from its tangent bundle and a base contribution of 1 plus the Bockstein of -1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds Steenrod square operations on modified compactly supported étale cohomology for separated finite-type schemes over rings of S-integers where 2 is invertible. This construction extends the absolute étale Wu class beyond finite fields to these arithmetic bases. Using a modified compactly supported relative Wu formula, the authors prove that for a regular projective flat morphism f from X to B the absolute Wu class of X equals the relative Wu class Sq inverse of the étale Wu class of the relative tangent bundle times the pullback of the absolute Wu class of B. In the S-integer case the base term is exactly 1 plus the Bockstein class, which is also the Kummer class of -1. The resulting arithmetic deformation of Hirzebruch's 2-Todd series then produces an infinite family of universal mod-2 congruences among the Chern classes of such schemes.

Core claim

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 Sq inverse of w_et of tau_f and the pullback of the absolute Wu class of the base; in the S-integer case the base contribution is 1 plus beta_B where beta_B is the Bockstein, equivalently the Kummer class of -1.

What carries the argument

The modified compactly supported relative Wu formula, which extends Benoist's relative Wu formula to the arithmetic compact-support setting and supplies the technical step needed to reach the absolute formula.

If this is right

  • The formula produces an infinite family of universal mod-2 congruences among Chern classes of regular projective flat schemes over these bases, governed by an arithmetic deformation of Hirzebruch's 2-Todd series.
  • In low dimensions the 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.
  • New higher-dimensional congruences appear over both finite and arithmetic bases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same factorization might hold after base change to more general arithmetic rings if the modified relative formula can be established there.
  • The resulting congruences could be used to constrain the possible values of Chern numbers in arithmetic intersection theory.
  • Explicit low-dimensional calculations for elliptic curves or K3 surfaces over S-integers would give concrete numerical tests of the base contribution term.

Load-bearing premise

A modified compactly supported relative Wu formula exists that extends Benoist's relative Wu formula to the arithmetic compact-support setting.

What would settle it

Compute the absolute and relative Wu classes directly for a specific regular projective flat scheme over a ring of S-integers away from 2 and check whether their ratio equals 1 plus the Bockstein class of the base.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper constructs canonical Steenrod square operations on the Geisser–Schmidt/Milne modified compactly supported étale cohomology of separated finite-type schemes over rings of S-integers (2 invertible). It extends Feng's absolute étale Wu class to arithmetic bases and proves an absolute Wu formula for regular projective flat schemes over finite fields of odd characteristic or such S-integer rings: if f: X→B is such a scheme, then the absolute Wu class of X is the product of the relative Wu class Sq^{-1}(w_et(τ_f)) and the pullback of the absolute Wu class of the base (with base contribution 1+β_B in the S-integer case, where β_B is the Bockstein/Kummer class of -1). This yields a generalized Hecke theorem: an infinite family of universal mod-2 congruences among Chern classes governed by an arithmetic deformation of Hirzebruch's 2-Todd series, recovering Hecke, Serre, Atiyah, and Shusterman–Sawin results in low dimensions while producing new higher-dimensional congruences.

Significance. If the constructions hold, the work supplies new Steenrod operations and Wu classes in an arithmetic étale setting, together with a generalized Hecke theorem that unifies several classical results and produces explicit, falsifiable congruences among Chern classes over both finite and arithmetic bases. The extension of Benoist's relative formula to the compact-support arithmetic case is the central technical step enabling these applications.

minor comments (3)
  1. [§2] The definition of the modified compactly supported cohomology (likely in §2) should include an explicit comparison table or diagram relating it to the standard Geisser–Schmidt and Milne versions to clarify the precise modification used for the Steenrod operations.
  2. [Theorem 4.1] In the statement of the absolute Wu formula (likely Theorem 4.1 or 5.2), the notation w_et(τ_f) is introduced without an immediate reminder of its definition from Benoist; adding a one-line cross-reference would improve readability.
  3. [§6] The applications section lists recovered theorems but does not indicate which low-dimensional cases are new verifications versus direct specializations; a short table or sentence clarifying this would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in extending Steenrod operations and Wu classes to the arithmetic étale setting, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external inputs

full rationale

The paper's central derivation extends Benoist's relative Wu formula (external) to a modified compactly supported version in the Geisser–Schmidt/Milne setting, then uses this to obtain the absolute Wu formula as a product involving the relative class and base pullback. No equation or claim reduces by construction to a fitted parameter, self-definition, or self-citation chain. The Shusterman–Sawin reference appears only in an application recovering a known low-dimensional case and is not load-bearing for the main theorems. The construction of Steenrod operations and the arithmetic Wu class are presented as new but independent of the target formulas.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the work rests on standard background results in algebraic geometry and topology with no evident free parameters or newly postulated entities.

axioms (2)
  • domain assumption Standard axioms and properties of étale cohomology, Steenrod operations, and Wu classes continue to hold after the Geisser-Schmidt/Milne modification to compact support.
    Required for the construction of the operations and the subsequent formulas.
  • domain assumption The schemes under consideration are regular, projective, and flat over the stated bases with 2 invertible.
    Explicitly required for the absolute Wu formula to apply as stated.

pith-pipeline@v0.9.1-grok · 5873 in / 1557 out tokens · 36492 ms · 2026-06-27T23:47:17.846463+00:00 · methodology

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