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arxiv: 2605.09896 · v1 · submitted 2026-05-11 · 🧮 math.AG · math.NT

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· Lean Theorem

Homological sieve and Manin's conjecture

Sho Tanimoto

Pith reviewed 2026-05-12 04:24 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords Manin's conjecturehomological sieverational pointsasymptotic countingalgebraic varieties
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The pith

The homological sieve method can be applied to Manin's conjecture.

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

The paper explains the homological sieve method and shows its applications to Manin's conjecture. Manin's conjecture predicts the asymptotic number of rational points of bounded height on algebraic varieties. The survey details how the sieve uses homological constructions to focus on the geometric part of the count. A reader would care because this provides a structured way to approach a central open problem in arithmetic geometry.

Core claim

The homological sieve method can be applied to Manin's conjecture by using homological tools to isolate the necessary geometric contributions for the asymptotic counts.

What carries the argument

Homological sieve method, which employs homological constructions to separate geometric terms in point counting asymptotics.

If this is right

  • Manin's conjecture can be approached through homological methods rather than traditional analytic ones.
  • The asymptotic formula for rational points follows from the sieve's isolation of geometric data.
  • New cases of the conjecture may become provable using these constructions.

Where Pith is reading between the lines

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

  • Similar sieving techniques might apply to other Diophantine problems beyond Manin's conjecture.
  • Computational verification on low-dimensional varieties could test the method's predictions.
  • Links to motivic or categorical approaches in geometry could emerge from the homological focus.

Load-bearing premise

The homological constructions correctly isolate the geometric contributions needed for the asymptotic counts in Manin's conjecture.

What would settle it

A specific variety satisfying the hypotheses of Manin's conjecture where the leading asymptotic term computed via the homological sieve differs from the conjectured one would disprove the claim.

read the original abstract

This is a report of the author's talk at RIMS workshop Algebraic Number Theory and Related Topics 2025 which was held at RIMS Kyoto University during December 15th-19th 2025. In this survey paper, we explain the homological sieve method, which is proposed by Das, Lehmann, Tosteson, and the author, and its applications to Manin's 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.

Referee Report

0 major / 0 minor

Summary. This survey paper reports on the author's talk at the RIMS workshop Algebraic Number Theory and Related Topics 2025. It explains the homological sieve method proposed jointly by Das, Lehmann, Tosteson, and the author, and discusses its applicability to Manin's conjecture on the asymptotic distribution of rational points.

Significance. If the homological constructions in the cited joint works correctly isolate the relevant geometric contributions, the method supplies a new homological framework for deriving the leading constants and error terms in Manin's conjecture. The survey itself adds no new derivations but usefully consolidates the approach and points to its prior applications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance. The report accurately captures the scope of this survey on the homological sieve and its relation to Manin's conjecture.

Circularity Check

0 steps flagged

No significant circularity in survey paper

full rationale

This is a survey paper reporting a talk that explains the homological sieve method proposed in prior joint works (Das, Lehmann, Tosteson, and the author) and states its applicability to Manin's conjecture. No new theorems, proofs, derivations, or asymptotic counts are claimed or derived internally. The text simply outlines the existing method and points to prior applications. All load-bearing constructions are external to this document, so no step reduces by construction to a self-defined input, fitted parameter, or self-citation chain within the paper itself. The central claim relies on the correctness of the cited external papers, which are independent.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a survey the paper does not introduce new free parameters, axioms, or invented entities; it relies on the standard setup of Manin's conjecture and the homological constructions from the cited joint papers.

pith-pipeline@v0.9.0 · 5338 in / 1030 out tokens · 36553 ms · 2026-05-12T04:24:46.978498+00:00 · methodology

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Lean theorems connected to this paper

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

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

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