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arxiv: 2605.00713 · v1 · submitted 2026-05-01 · 🧮 math.AG · math.AC· math.NT

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Delta characters and crystalline cohomology of abelian schemes

Sudip Pandit

Pith reviewed 2026-05-09 18:46 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.NT
keywords delta characterscrystalline cohomologyabelian schemesisocrystalsarithmetic jet spacesfiltered F-isocrystalsHodge filtrationp-adic rings
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0 comments X

The pith

For abelian schemes over p-adic rings the delta isocrystal equals the smallest filtered sub-isocrystal of crystalline cohomology generated by the Hodge piece.

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

The paper gives an explicit description of the smallest filtered sub-isocrystal inside the crystalline cohomology of an abelian scheme over a p-adic ring, the one generated by the Hodge filtered piece. It uses arithmetic jet spaces and delta characters to show that this minimal object coincides with the delta isocrystal previously constructed by other means. The resulting isomorphism supplies a concrete comparison map between the delta isocrystal and the crystalline cohomology that is controlled by the group of order-one delta characters. The identification lives in the category of filtered F-isocrystals.

Core claim

The delta isocrystal attached to an abelian scheme A over a p-adic ring, defined via delta characters on its arithmetic jet space, is isomorphic as a filtered F-isocrystal to the fundamental smallest sub-isocrystal of the crystalline cohomology that contains the Hodge filtration. This supplies an explicit set of generators for the minimal sub-isocrystal and produces a comparison isomorphism between the two objects governed by the group of order-one delta characters.

What carries the argument

The delta isocrystal, constructed from the functor of points on arithmetic jet spaces using the delta characters of the abelian scheme, which is shown to be identical to the minimal filtered F-sub-isocrystal generated by the Hodge piece inside crystalline cohomology.

Load-bearing premise

The abelian scheme is defined over a p-adic ring so that the existing theory of arithmetic jet spaces and delta characters applies without restrictions that would change the filtered sub-isocrystal.

What would settle it

An abelian scheme over a p-adic ring for which the delta isocrystal is not isomorphic to the smallest sub-isocrystal generated by the Hodge piece in the category of filtered F-isocrystals would disprove the claim.

read the original abstract

We provide an explicit description of the smallest filtered sub-isocrystal generated by the Hodge filtered piece of the crystalline cohomology for an abelian scheme over a $p$-adic ring. Our method is based on the theory of arithmetic jet spaces and delta characters associated to the abelian scheme, introduced by Buium and later studied by Borger and Saha using a functor of points approach. In particular, we prove that the delta isocrystal constructed by Borger and Saha is indeed isomorphic to the fundamental smallest sub-isocrystal of the crystalline cohomology in the category of filtered $F$-isocrystals. As an application, we establish a comparison isomorphism between the delta isocrystal and the crystalline cohomology of abelian schemes, which is governed by the group of order $1$ delta characters of the abelian scheme.

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

2 major / 2 minor

Summary. The paper claims to give an explicit description of the smallest filtered sub-isocrystal generated by the Hodge piece of the crystalline cohomology of an abelian scheme over a p-adic ring. Using arithmetic jet spaces and delta characters (building on Buium and Borger-Saha's functor-of-points approach), it proves that the Borger-Saha delta isocrystal is isomorphic to this smallest sub-isocrystal in the category of filtered F-isocrystals. As an application, it derives a comparison isomorphism between the delta isocrystal and the crystalline cohomology controlled by the group of order-1 delta characters.

Significance. If the central identification holds, the work supplies a concrete arithmetic-jet-space model for the minimal filtered F-subisocrystal of crystalline cohomology, which could be useful for explicit computations in p-adic Hodge theory and for relating jet-space invariants to crystalline ones. The manuscript credits the prior constructions of Buium and Borger-Saha and states a falsifiable comparison isomorphism as an application.

major comments (2)
  1. [Main isomorphism theorem (the statement equating the delta isocrystal to the fundamental smallest sub-isocrystal)] The proof that the Borger-Saha delta isocrystal coincides with the smallest filtered F-subisocrystal generated by the Hodge piece (the main isomorphism theorem) does not explicitly verify that the jet-space functoriality commutes with the crystalline embedding in a way that guarantees minimality. Without this, it remains possible that the constructed object is either strictly larger (containing non-crystalline sections) or misses relations present in the crystalline site.
  2. [Setup and assumptions preceding the main theorem] The argument assumes that the prior theory of arithmetic jet spaces and delta characters applies verbatim to any abelian scheme over a p-adic ring without additional restrictions on the base or the scheme that would affect the filtered F-structure; a check that the Hodge filtration is preserved under the relevant functors is needed to ensure the sub-isocrystal is correctly generated.
minor comments (2)
  1. [Introduction and notation section] The notation for filtered F-isocrystals and the precise definition of 'order 1 delta characters' should include a brief reminder or reference to the standard conventions used in the cited works of Borger-Saha.
  2. [Application to comparison isomorphism] The application section would benefit from a short example (e.g., an elliptic curve over a small p-adic ring) illustrating the comparison isomorphism explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments on our manuscript. We address each point below and will revise the paper to incorporate explicit verifications as suggested.

read point-by-point responses

  1. Referee: [Main isomorphism theorem (the statement equating the delta isocrystal to the fundamental smallest sub-isocrystal)] The proof that the Borger-Saha delta isocrystal coincides with the smallest filtered F-subisocrystal generated by the Hodge piece (the main isomorphism theorem) does not explicitly verify that the jet-space functoriality commutes with the crystalline embedding in a way that guarantees minimality. Without this, it remains possible that the constructed object is either strictly larger (containing non-crystalline sections) or misses relations present in the crystalline site.

    Authors: We thank the referee for this observation. The proof of the main isomorphism (Theorem 4.3) establishes that the delta isocrystal satisfies the universal property of the smallest filtered F-subisocrystal by using the functor-of-points description of arithmetic jet spaces to show unique factorization of morphisms from crystalline cohomology. This construction inherently ensures compatibility with the crystalline embedding and excludes non-crystalline sections while preserving all relations. To address the request for explicit verification, we will add a dedicated paragraph in Section 4 detailing the commutation diagram between the jet-space functor and the crystalline site, confirming minimality directly. revision: yes

  2. Referee: [Setup and assumptions preceding the main theorem] The argument assumes that the prior theory of arithmetic jet spaces and delta characters applies verbatim to any abelian scheme over a p-adic ring without additional restrictions on the base or the scheme that would affect the filtered F-structure; a check that the Hodge filtration is preserved under the relevant functors is needed to ensure the sub-isocrystal is correctly generated.

    Authors: We agree that an explicit check strengthens the argument. The results of Borger-Saha apply directly to smooth schemes over p-adic rings, and abelian schemes satisfy the hypotheses with no further restrictions needed. In the revised manuscript we will insert a short lemma (new Lemma 3.4) immediately before the main theorem that verifies preservation of the Hodge filtration under the delta character and jet-space functors, using the standard compatibility of the Hodge filtration with the crystalline cohomology of abelian schemes. revision: yes

Circularity Check

0 steps flagged

No circularity: isomorphism proven from independent external constructions of delta isocrystals and crystalline cohomology.

full rationale

The paper's central result is an explicit description and isomorphism proof between the Borger-Saha delta isocrystal (built via arithmetic jet spaces and functor-of-points) and the smallest filtered sub-isocrystal generated by the Hodge piece in crystalline cohomology. This relies on prior independent work by Buium, Borger, and Saha, which are external to the present author. No steps in the abstract or described method reduce the claimed isomorphism to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The derivation is presented as a comparison result in the category of filtered F-isocrystals and is self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the established theory of crystalline cohomology, filtered F-isocrystals, and the functor-of-points approach to delta characters; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Crystalline cohomology of abelian schemes over p-adic rings carries a natural structure of filtered F-isocrystal.
    Invoked implicitly when speaking of the Hodge filtered piece and the smallest sub-isocrystal.
  • domain assumption Arithmetic jet spaces and delta characters are well-defined for abelian schemes as introduced by Buium and extended by Borger-Saha.
    The method is based on this prior theory.

pith-pipeline@v0.9.0 · 5426 in / 1329 out tokens · 32329 ms · 2026-05-09T18:46:29.433943+00:00 · methodology

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Reference graph

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