Hasse-Witt invariants of Calabi-Yau varieties

Hossein Movasati; Jin Cao; Mohamed Elmi

arxiv: 2603.03055 · v2 · pith:PMITIJMQnew · submitted 2026-03-03 · 🧮 math.AG · math-ph· math.MP· math.NT

Hasse-Witt invariants of Calabi-Yau varieties

Jin Cao , Mohamed Elmi , Hossein Movasati This is my paper

Pith reviewed 2026-05-15 16:43 UTC · model grok-4.3

classification 🧮 math.AG math-phmath.MPmath.NT

keywords Hasse-Witt invariantCalabi-Yau varietiesCartier operatorCalabi-Yau modular formsconjectureequivalencep-adic cohomologyfinite fields

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The pith

Hasse-Witt invariants of Calabi-Yau varieties admit two definitions that the paper conjectures are equivalent.

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

The paper defines the Hasse-Witt invariant for Calabi-Yau varieties once through the Cartier operator acting on de Rham cohomology and once through the q-expansion of Calabi-Yau modular forms. It conjectures that the two constructions always produce the same number and checks this on many explicit examples such as hypersurfaces and complete intersections. A sympathetic reader would care because the equivalence would connect a direct algebraic computation in positive characteristic to data coming from modular forms, potentially simplifying calculations of these invariants over finite fields.

Core claim

The paper introduces two definitions of the Hasse-Witt invariant of Calabi-Yau varieties: one obtained by applying the Cartier operator to the cohomology of the variety, and the other extracted from the theory of Calabi-Yau modular forms. The central claim is that these two definitions coincide, with the conjecture supported by explicit agreement on numerous examples.

What carries the argument

The Hasse-Witt invariant, obtained dually from the Cartier operator on cohomology and from the modular-form expansion associated to the Calabi-Yau variety.

If this is right

The modular-form construction would become a practical tool for computing Hasse-Witt invariants without direct p-adic cohomology calculations.
Verification of the conjecture on further families would strengthen the link between arithmetic invariants of Calabi-Yau varieties and their associated modular forms.
The equivalence would allow transfer of known properties from one construction to the other, such as integrality or congruence relations.
It would open the possibility of defining analogous invariants for other classes of varieties using the same modular-form machinery.

Where Pith is reading between the lines

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

If the conjecture holds, one could use it to predict the Hasse-Witt invariant of a Calabi-Yau variety whose cohomology is hard to compute directly by instead calculating the corresponding modular form.
The link might extend to questions about the distribution of these invariants in families or their relation to other arithmetic invariants such as the zeta function.
Testing the conjecture on Calabi-Yau varieties with non-trivial fundamental group or on those arising from mirror symmetry could reveal whether the equivalence survives additional geometric structure.

Load-bearing premise

The chosen examples of Calabi-Yau varieties are representative enough that agreement on them implies the two definitions coincide for every Calabi-Yau variety.

What would settle it

A single Calabi-Yau variety in which the integer produced by the Cartier-operator definition differs from the integer produced by the modular-form definition.

read the original abstract

We define the Hasse-Witt invariant of Calabi-Yau varieties in two different ways. The first method is through Cartier operator and the second method is through the theory of Calabi-Yau modular forms developed by the third author. We conjecture that these two definitions are equivalent and provide many examples of Calabi-Yau varieties in support of this conjecture.

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.

Desk Editor's Note private letter to a colleague

The paper defines Hasse-Witt invariants for Calabi-Yau varieties via Cartier operator and via modular forms, conjectures they match, and checks this on examples without a general proof.

read the letter

The one thing to know is that the authors give two definitions of the Hasse-Witt invariant for Calabi-Yau varieties and conjecture they are the same. One uses the Cartier operator on de Rham cohomology; the other comes from the Calabi-Yau modular forms developed by Movasati. They test the match on several explicit families and report agreement in each case. This pairing of the two approaches is new. The modular-form side extends earlier work by one author, but the direct comparison to the Cartier definition and the stated conjecture do not appear in the cited literature. The examples are concrete, covering standard hypersurface and complete-intersection cases, and the side-by-side computations are laid out clearly enough to follow. The paper does a straightforward job presenting both definitions and showing where the numbers line up. That kind of explicit check is useful for making the conjecture look plausible. The main limitation is that it stays a conjecture. No general argument is supplied, and the examples, while numerous, are chosen families rather than a complete sweep. The text does not claim the examples prove the general statement, so the scope is stated accurately. There is no circularity or hidden dependence on the conjecture itself. This is for people already working in arithmetic geometry of Calabi-Yau varieties who care about p-adic invariants or modular-form techniques. A reader hunting for a proved theorem will find it thin, but someone wanting a new computational link or a question worth testing further could get something out of it. I would send it to peer review. The question is concrete enough that referees could usefully comment on the examples, suggest proof directions, or flag families where the two sides might diverge.

Referee Report

0 major / 2 minor

Summary. The manuscript defines two notions of the Hasse-Witt invariant for Calabi-Yau varieties: one via the action of the Cartier operator on de Rham cohomology and the other via the Calabi-Yau modular forms construction. It explicitly states their equivalence only as a conjecture and supports the claim through explicit computations on several families of examples.

Significance. If the conjecture holds, the work would link the Cartier-operator approach in p-adic cohomology with the modular-forms framework, potentially enabling new computational or theoretical tools for invariants of Calabi-Yau varieties. The provision of multiple explicit examples is a concrete strength that allows direct verification and may guide future proofs.

minor comments (2)

[Abstract] The abstract refers to 'many examples' without naming the specific families or dimensions of the Calabi-Yau varieties considered; adding this information would clarify the scope of the computational evidence.
[Introduction] Notation for the two Hasse-Witt invariants should be introduced once in a dedicated definitions subsection and used consistently thereafter to avoid ambiguity when comparing the two constructions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report accurately summarizes that we define the Hasse-Witt invariant in two ways and state their equivalence as a conjecture supported by explicit computations.

Circularity Check

0 steps flagged

No circularity; equivalence stated only as conjecture

full rationale

The paper introduces two independent definitions of the Hasse-Witt invariant—one via the Cartier operator acting on de Rham cohomology and the other via the existing theory of Calabi-Yau modular forms—and explicitly labels their equivalence as a conjecture rather than a derived result. Support consists of explicit computations on external families of Calabi-Yau varieties; no equation or step inside the argument assumes the conjecture to hold, no fitted parameter is relabeled as a prediction, and no uniqueness theorem or ansatz is smuggled in via self-citation to force the outcome. The modular-forms framework is treated as prior independent work, not as an internal input that the present definitions must reproduce by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work assumes standard properties of the Cartier operator on Calabi-Yau varieties and the prior theory of Calabi-Yau modular forms; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The Cartier operator is well-defined and acts on the cohomology of Calabi-Yau varieties in positive characteristic.
Standard background in algebraic geometry over finite fields.
domain assumption Calabi-Yau modular forms are defined and carry the necessary data to produce an invariant.
Relies on the third author's earlier development of the theory.

pith-pipeline@v0.9.0 · 5351 in / 1239 out tokens · 36121 ms · 2026-05-15T16:43:46.070577+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We define the Hasse-Witt invariant of Calabi-Yau varieties in two different ways. The first method is through Cartier operator and the second method is through the theory of Calabi-Yau modular forms... We conjecture that these two definitions are equivalent and provide many examples
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Conjecture 2... HW(Xz, αz) up to sign is the truncation of the holomorphic period at degree p-1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.