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arxiv: 2606.27129 · v1 · pith:I6V6VGE5new · submitted 2026-06-25 · 🧮 math.NT · math.AG

Quantitativity in the Mordell Conjecture

Xinyi Yuan This is my paper

Pith reviewed 2026-06-26 02:51 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords Mordell conjecturerational pointsuniform boundsquantitative estimatesalgebraic curvesnumber fieldsFaltings theorem
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The pith

Yu-Yuan-Zhou establish explicit quantitative bounds for the number of rational points on curves of genus at least two.

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

This survey presents the quantitative version of the uniform Mordell problem. The Mordell conjecture, proved by Faltings, asserts only finitely many rational points exist on a smooth projective curve of genus at least two over a number field. The uniform version supplies bounds on this number that depend only on the genus and the degree of the number field, and was settled by combining Vojta's work with results of Dimitrov-Habegger-Gao and Kühne. Yu-Yuan-Zhou add explicit size estimates to these uniform bounds. A sympathetic reader cares because the result turns a statement of finiteness into one with concrete, usable upper limits.

Core claim

The recent work of Yu-Yuan-Zhou proves a quantitative refinement of the uniform Mordell problem by supplying explicit upper bounds on the number of rational points, obtained by combining the uniformity theorems of Vojta, Dimitrov-Habegger-Gao and Kühne with additional estimates.

What carries the argument

The quantitative uniformity problem for rational points on curves of genus at least two, which produces explicit bounds by merging prior uniformity results with new estimates.

If this is right

  • The number of rational points on such curves is bounded by an explicit function of the genus and the degree of the number field.
  • These bounds are effective and in principle allow computation of all rational points once the bound is known.
  • Finiteness statements in the Mordell conjecture become effective rather than purely existential.
  • The same combination of uniformity and estimates applies to related Diophantine finiteness problems.

Where Pith is reading between the lines

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

  • The explicit bounds may permit practical enumeration algorithms for rational points on individual curves once the constants are computed.
  • The approach could extend to give quantitative statements for other uniform finiteness results in arithmetic geometry.
  • Height functions and their distribution on moduli spaces become more directly usable for bounding point counts.

Load-bearing premise

The uniformity theorems of Vojta, Dimitrov-Habegger-Gao and Kühne can be combined with the estimates supplied by Yu-Yuan-Zhou to yield explicit bounds.

What would settle it

An explicit curve of genus at least two over a number field whose number of rational points exceeds the quantitative upper bound stated in the Yu-Yuan-Zhou theorem.

read the original abstract

The Mordell conjecture asserts that there are only finitely many rational points on a smooth projective curve of genus at least two over a number field. The uniform Mordell problem asks for suitable upper bounds on the number of rational points in the Mordell conjecture, and has been solved by combining works of Vojta, Dimitrov--Habegger--Gao and Kuhne. In this survey, we will introduce a quantitative version of the uniformity problem proved by the recent work of Yu--Yuan--Zhou.

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 / 1 minor

Summary. The manuscript is a survey on the Mordell conjecture, which asserts finiteness of rational points on smooth projective curves of genus at least 2 over number fields. It recalls that the uniform Mordell problem (uniform upper bounds on the number of such points) has been solved by combining Vojta's work with results of Dimitrov--Habegger--Gao and Kuhne. The paper then introduces a quantitative version of this uniformity problem, attributing its proof to the recent work of Yu--Yuan--Zhou.

Significance. If the summary of the cited external results is accurate, the survey offers a concise overview of progress toward quantitative bounds in Diophantine geometry. It explicitly credits the combination of uniformity theorems with the additional estimates supplied by Yu--Yuan--Zhou, providing a clear pointer to the literature for readers interested in effective versions of the Mordell conjecture.

minor comments (1)
  1. The abstract refers to 'the recent work of Yu--Yuan--Zhou' without a citation; adding the arXiv or journal reference in the introduction would improve traceability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report correctly captures the scope of the survey as an overview of the uniform Mordell problem and its quantitative strengthening due to Yu--Yuan--Zhou.

Circularity Check

0 steps flagged

No circularity; survey reports external result

full rationale

The manuscript is explicitly a survey whose sole purpose is to introduce a quantitative uniformity result already proved in the cited external work of Yu--Yuan--Zhou. No derivations, equations, or load-bearing steps are advanced inside the paper itself. The central claim therefore rests entirely on the correctness of that prior proof and does not reduce to any self-referential construction, fitted input, or self-citation chain within the present text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an expository survey; no free parameters, axioms, or invented entities are introduced by the paper itself.

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