Published paper
Abstract
The paper proves the Lang–Weil point-count estimate and its consequence that a nonempty open subset acquires rational points over every sufficiently large finite extension.
Role in dependence graphs
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statements:
- Theorem 1 · exact volume scan physical PDF p. 823 / printed p. 819Bounds the rational-point count of a projective variety over a finite field.
- Open-subset extension-field consequence · exact volume scan physical PDF p. 827 / printed p. 823Gives a rational point on a nonempty open subset over every sufficiently large finite extension, as required in Weil 1955.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact digitized volume scan · American Journal of Mathematics 76(4) (1954), 819–827
Serge Lang and André Weil. Number of Points of Varieties in Finite Fields. American Journal of Mathematics 76(4) (1954), 819–827.
Open audited source ↗01Statements4 reported findingsNo statement issues found
The exact publication states a correct uniform point-count estimate and derives the open-subset extension-field consequence used by Weil in 1955. The consequence holds for every sufficiently large extension degree, so it supports Weil's additional choice of a prime degree greater than a prescribed integer.
Theorem 1No issue found
The finite-field point count has the asserted leading term and error
Exact volume scan · physical PDF p. 823 / printed p. 819 · Theorem 1
For a projective variety of dimension and degree over a field of elements, the theorem bounds by a term of order plus a constant depending only on times . This is the standard Lang–Weil estimate in the form proved in the article.
Extension-field asymptoticNo issue found
The same estimate applies uniformly over every finite extension
Exact volume scan · physical PDF p. 827 / printed p. 823 · opening of Section 2
Over the degree- extension, the field has elements, so Theorem 1 gives with constants independent of .
Open-subset consequenceNo issue found
Removing a proper frontier still leaves points over all large extensions
Exact volume scan · physical PDF pp. 827–828 / printed pp. 823–824 · frontier and abstract-variety discussion
A proper algebraic frontier has dimension at most and therefore contributes only points. Subtracting that count leaves a positive leading term for every sufficiently large , hence a rational point on the open complement.
Weil 1955 dependency matchDependency match verified
The cited consequence permits a large prime extension degree
Weil 1955 exact issue scan · physical PDF p. 182 / printed p. 374
Weil needs a point on over an extension of degree , with prime and . Since Lang–Weil gives a point for every sufficiently large degree, one may choose any sufficiently large prime .
02Proofs6 reported findingsNo proof issues found
The selected estimate and corollary are proved correctly. The induction controls good hyperplane sections with the lower-dimensional theorem and controls bad sections uniformly through the Chow-coordinate locus. The proof-critical Weil Section 1.6 article branch is represented separately rather than treated as a book terminal.
Base caseNo issue found in the reviewed proof
The curve estimate follows from the function-field Riemann hypothesis
Exact volume scan · physical PDF p. 823 / printed p. 819 · first paragraph of the proof
Passing to a nonsingular projective model changes the count by a bounded amount depending on the degree, while the curve Riemann-hypothesis bound supplies the error. The genus is bounded in terms of , so the constant is uniform in the variety.
Cycle boundNo issue found in the reviewed proof
Lemma 1 uniformly bounds points on lower-dimensional cycles
Exact volume scan · physical PDF pp. 824–825 / printed pp. 820–821 · Lemma 1
Induction cuts a prime rational cycle by members of a pencil that cover all rational points. Bézout preserves the degree of each proper section, and the number of prime rational components is at most the total degree.
Bad hyperplanesNo issue found in the reviewed proof
Lemma 2 gives a uniform codimension-one bound for bad sections
Exact volume scan · physical PDF pp. 824–825 / printed pp. 820–821 · Lemma 2
The Chow coordinates of are homogeneous polynomials of fixed degree in . Pulling back a nonzero equation for the closed non-variety locus places every bad hyperplane in a hypersurface of uniformly bounded degree, and Lemma 1 bounds its rational points.
Hyperplane averagingNo issue found in the reviewed proof
The two-way incidence count yields the claimed exponent
Exact volume scan · physical PDF pp. 826–827 / printed pp. 822–823 · proof of Theorem 1
Counting pairs consisting of a rational point of and a rational hyperplane through it expresses as an average of the section counts. Good sections use the induction hypothesis; the number and size of bad sections are both controlled by Lemmas 1–2, and their total contribution is of lower order.
Required Section 1.6 branchRequired historical article branch represented
Weil's Chow-locus result is represented by an exact separate node
Printed pp. 820–821, citation [4] · ordinary-audits/weil-equivalence-criteria-algebraic-geometry.json
The algebraicity, field of definition, properness, and fixed-degree Chow equations used in Lemma 2 come from Weil's 1954 Section I, no. 6. That original article dependency is audited directly in its own ordinary node; it is not replaced by a modern textbook citation.
Open-subset deductionNo issue found in the reviewed proof
The consumer corollary is an immediate asymptotic comparison
Exact volume scan · physical PDF p. 827 / printed p. 823 · Section 2
The main term dominates both the theorem's error and the frontier's bound as tends to infinity. No unmentioned congruence condition on is introduced.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.