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arxiv: 2507.10111 · v3 · submitted 2025-07-14 · 🧮 math.NT · math.AG

Generalised height pairings and the Albanese kernel

Netan Dogra This is my paper

Pith reviewed 2026-05-19 05:07 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords generalised height pairingsAlbanese kernelChabauty-Coleman-Kim methodrational points on curvesJacobianBeilinson-Bloch conjecturesp-adic methodsmotivic refinements
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The pith

If the Albanese kernel of X×X is torsion for a smooth projective curve X, then an algorithm computes the generalised height pairing on rational points of the Jacobian.

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

The paper shows that for a smooth projective curve X, the assumption that the Albanese kernel of X×X is torsion makes it possible to compute the generalised height pairing on pairs of rational points in the Jacobian. This pairing extends Nekovář's p-adic height pairing by using a higher Chow group in place of the multiplicative group, as needed in the depth-two version of the Chabauty-Coleman-Kim method. A reader would care because the result supplies both a clear domain for the pairing and a concrete computational procedure, while tying the problem to the Beilinson-Bloch conjectures and opening the way to motivic refinements of nonabelian cohomology varieties.

Core claim

If X is a smooth projective curve and the Albanese kernel of X×X is torsion, then there is an algorithm to compute the generalised height pairing on a pair of rational points on the Jacobian. The argument proceeds by examining the relevance of the Beilinson-Bloch conjectures to the definition and evaluation of the generalised p-adic height pairing that appears in the Chabauty-Coleman-Kim method in depth two.

What carries the argument

The generalised height pairing obtained by replacing G_m with a higher Chow group in Nekovář's p-adic height pairing; the torsion hypothesis on the Albanese kernel of X×X renders this pairing computable on rational points of the Jacobian.

If this is right

  • The domain of definition of the generalised height pairing is clarified under the torsion hypothesis.
  • Motivic refinements of the nonabelian cohomology varieties that appear in nonabelian Chabauty become available for study.
  • The depth-two Chabauty-Coleman-Kim method gains an explicit computational step for the height pairing.
  • The Beilinson-Bloch conjectures acquire a direct computational link to the description of rational points on curves.

Where Pith is reading between the lines

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

  • If the torsion condition holds for a useful range of curves, the algorithm could be run explicitly to locate rational points in concrete examples.
  • The same conditional computability might apply to height pairings attached to higher-dimensional varieties.
  • Verification on curves already known to have torsion Albanese kernels would test whether the resulting height values match independent calculations.

Load-bearing premise

The Albanese kernel of X times X must be torsion.

What would settle it

Exhibit a smooth projective curve for which the Albanese kernel of X×X is torsion yet no algorithm computes the generalised height pairing on Jacobian points, or find a curve without the torsion condition where the pairing is nevertheless algorithmically computable.

read the original abstract

The Chabauty--Coleman--Kim method in depth two describes the rational points on a curve in terms of a generalisation of Nekov\'a\v{r}'s $p$-adic height pairing which replaces $\mathbb{G}_m$ with a higher Chow group. It is unclear both what the domain of definition of this pairing is, and how to compute it. This paper explores the relevance of the Beilinson--Bloch conjectures to this problem. In particular, it is shown that if $X$ is a smooth projective curve and the Albanese kernel of $X\times X$ is torsion, then there is an algorithm to compute the generalised height pairing on a pair of rational points on the Jacobian. This leads to the consideration of certain `motivic refinements' of the nonabelian cohomology varieties which arise in nonabelian Chabauty.

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 / 2 minor

Summary. The manuscript explores the domain and computability of a generalised p-adic height pairing in the Chabauty-Coleman-Kim method at depth two, where G_m is replaced by a higher Chow group. It proves that if X is a smooth projective curve and the Albanese kernel of X×X is torsion, then there exists an algorithm to compute this pairing for rational points on the Jacobian of X. The result is conditional on this torsion hypothesis and contextualised by the Beilinson-Bloch conjectures, and it introduces motivic refinements of the nonabelian cohomology varieties arising in nonabelian Chabauty.

Significance. If the result holds, it would provide a valuable algorithmic tool for computing generalised height pairings under a verifiable hypothesis, advancing computational aspects of nonabelian Chabauty methods for finding rational points on curves. The explicit reduction to finite motivic cohomology data once the kernel is torsion is a positive aspect, as is the clear statement of the conditional nature of the theorem. This could have implications for effective Diophantine geometry if the torsion condition holds for curves of interest.

minor comments (2)
  1. The discussion of the domain of definition of the pairing could benefit from a more explicit statement of the conditions under which it is defined, beyond the abstract.
  2. Ensure that all references to external conjectures like Beilinson-Bloch are accompanied by precise citations to the relevant statements used in the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, for highlighting its potential value as an algorithmic tool in nonabelian Chabauty methods, and for recommending minor revision. We are pleased that the conditional nature of the result, the explicit reduction to finite motivic cohomology data, and the relevance of the Beilinson-Bloch conjectures were noted as strengths.

Circularity Check

0 steps flagged

No significant circularity; result is explicitly conditional on external torsion hypothesis

full rationale

The central claim is a conditional implication: given the independent hypothesis that the Albanese kernel of X×X is torsion, the generalised height pairing (defined via higher Chow groups) admits an algorithm for rational points on the Jacobian. This reduces the pairing computation to finite motivic cohomology data once the kernel is torsion. Beilinson-Bloch is invoked only to contextualise the domain, not as an internal assumption or self-citation load-bearing step. No equations or derivations reduce by construction to fitted inputs or prior self-citations; the torsion condition is stated as an external premise rather than derived. The manuscript is self-contained against external benchmarks for the conditional algorithm.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Beilinson-Bloch conjectures (to guarantee existence or properties of the generalised pairing) and on the torsion assumption for the Albanese kernel; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Beilinson-Bloch conjectures
    The paper explores the relevance of these conjectures to defining and computing the generalised height pairing.

pith-pipeline@v0.9.0 · 5667 in / 1372 out tokens · 32434 ms · 2026-05-19T05:07:17.993299+00:00 · methodology

discussion (0)

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Reference graph

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