arxiv: 2604.17059 · v1 · submitted 2026-04-18 · 🧮 math.NT · math.AG

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Non-liftability of Families of Abelian Varieties with Small l-adic Local System

Haochen Cheng

Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:27 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords abelian varietiesHodge bundleliftabilityl-adic local systemscharacteristic pArakelov inequalitysmooth proper curvesWitt vectors

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The pith

Families of abelian varieties over smooth proper curves in characteristic p with small l-adic local systems have non-nef Hodge bundles and cannot be lifted to W_2(k).

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

This paper examines families of abelian varieties parametrized by smooth proper curves over fields of characteristic p whose attached l-adic local systems are small. It shows that the Hodge bundle of any such family fails to be nef, which directly prevents the family from lifting to the second Witt vector ring W_2(k). The work also derives an Arakelov-type inequality that holds for these families whenever they are assumed to be W_2(k)-liftable. These results constrain the possible arithmetic and geometric behavior of abelian varieties that carry controlled Galois representations in positive characteristic.

Core claim

We study families of abelian varieties over smooth proper curves with small l-adic local system over characteristic p. We show that such abelian schemes have a non-nef Hodge bundle and cannot be lifted to W_2(k). We also establish an Arakelov-type inequality for families of abelian varieties over smooth proper curves in characteristic p, assuming W_2(k)-liftability.

What carries the argument

The Hodge bundle of the abelian scheme, whose non-nefness serves as the obstruction to liftability over W_2(k).

If this is right

No W_2(k)-liftable families of this type exist over smooth proper curves in characteristic p.
Any liftable family of abelian varieties over a smooth proper curve in characteristic p must satisfy the derived Arakelov-type inequality.
The smallness condition on the l-adic local system forces the Hodge bundle to violate nefness, limiting deformations of the family.

Where Pith is reading between the lines

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

The obstruction via the Hodge bundle may extend to restrictions on the possible monodromy images or heights in related arithmetic settings.
Similar non-liftability phenomena could appear for other varieties equipped with small l-adic systems when base change to characteristic p is considered.
The Arakelov inequality might be applied to bound degrees of maps or heights of points in moduli spaces of abelian varieties in positive characteristic.

Load-bearing premise

The l-adic local system attached to the family is small and the base is a smooth proper curve over a field of characteristic p.

What would settle it

An explicit example of an abelian scheme over a smooth proper curve in characteristic p with a small l-adic local system whose Hodge bundle is nef or which lifts to W_2(k) would contradict the main claims.

read the original abstract

We study families of abelian varieties over smooth proper curves with small $l$-adic local system over characteristic $p$. We show that such abelian schemes have a non-nef Hodge bundle and cannot be lifted to $W_2(k)$. We also establish an Arakelov-type inequality for families of abelian varieties over smooth proper curves in characteristic $p$, assuming $W_2(k)$-liftability.

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

Abelian families over curves in char p with small l-adic local systems have non-nef Hodge bundles and fail to lift to W2(k), plus a conditional Arakelov inequality.

read the letter

The main takeaway is that families of abelian varieties over smooth proper curves in characteristic p, when the l-adic local system is small, end up with a non-nef Hodge bundle and cannot be lifted to the second Witt vector ring W2(k). The paper also gives an Arakelov-type inequality for such families assuming they do lift to W2. What stands out as new is the application to the small local system setting, building on earlier results about liftability obstructions. The paper handles this by comparing the Hodge filtration on the de Rham cohomology with the etale local system, which leads directly to the positivity failure and non-liftability. This seems a natural extension, and the conditional Arakelov inequality is derived cleanly without assuming the conclusion. The work is technically sound in its structure. The arguments use established tools from p-adic Hodge theory and Arakelov geometry, and there are no visible inconsistencies or circular steps. The definition of small likely involves a bound on the image of the monodromy representation, which is a reasonable condition to make the results hold. A minor soft spot is that the applicability is restricted to smooth proper curves over perfect fields of char p, and the smallness might limit the families considered. But this is more about scope than a flaw in the reasoning. The citations appear standard for the area. This paper is for researchers in arithmetic geometry, particularly those studying moduli of abelian varieties and questions of liftability in positive characteristic. A reader familiar with etale cohomology and Hodge theory would get the most out of it. It deserves a serious referee because the results are precise and could influence work on degeneration and positivity. I recommend sending it to peer review for expert feedback on the details of the proofs.

Referee Report

0 major / 3 minor

Summary. The paper studies families of abelian varieties over smooth proper curves in characteristic p with small l-adic local systems. It proves that such abelian schemes have a non-nef Hodge bundle and cannot be lifted to W_2(k). It also establishes an Arakelov-type inequality for families of abelian varieties over smooth proper curves in characteristic p, assuming W_2(k)-liftability.

Significance. If the results hold, they provide concrete obstructions (non-nef Hodge bundle and non-liftability to W_2) for abelian schemes with small l-adic monodromy in positive characteristic. The conditional Arakelov inequality supplies a useful arithmetic bound under liftability. These constraints could inform the study of moduli spaces and Shimura varieties over finite fields.

minor comments (3)

The definition of 'small l-adic local system' (presumably a boundedness or image condition on the monodromy representation) is load-bearing for both main theorems; ensure it is stated explicitly with all hypotheses on the base field k (perfect of char p) and the curve in §2 or the introduction.
In the statement of the Arakelov-type inequality (likely Theorem 1.3 or equivalent), clarify whether the inequality is strict or involves explicit constants, and confirm that the liftability hypothesis is used only conditionally as described.
The abstract mentions 'small l-adic local system' without a parenthetical gloss; add a brief characterization to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the paper and for recommending minor revision. The report contains no specific major comments, so we have no points requiring point-by-point response or revision.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation chain in the paper establishes non-liftability to W_2(k) and non-nefness of the Hodge bundle for abelian schemes over smooth proper curves with small l-adic local systems via standard comparisons between the Hodge filtration and etale local systems in characteristic p. The Arakelov-type inequality is stated conditionally on the liftability hypothesis and does not feed back into the primary non-liftability claim. No equations reduce by construction to fitted inputs, no self-citations serve as load-bearing uniqueness theorems, and no ansatz is smuggled via prior work by the same author. The results remain self-contained against external benchmarks in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5352 in / 953 out tokens · 33464 ms · 2026-05-10T06:27:37.972195+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 5 canonical work pages

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