Published paper
Role in dependence graphs
Proof-critical source
Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt
This paper is included only for the following marked statements:
- Conjecture 1.1 · journal p. 29The original density-measure conjecture that the focal Theorem 1.5 reformulates.
- Lemma 5.1 · journal pp. 40–41Relates a coefficient box of radius to canonical height of order .
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact Project Euclid journal scan hosted on the author's institutional page
Michel Waldschmidt. Density measure of rational points on Abelian varieties. Nagoya Mathematical Journal 155 (1999), 27–53.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The selected conjecture and coefficient/height comparison are stated consistently; the latter uses the expected quadratic canonical-height scale.
Conjecture 1.1Correctly stated as a conjecture
Uniform density-measure conjecture on a simple abelian variety
journal p. 29 · Conjecture 1.1
The conjecture asks for uniform approximation of every real point by rational points with exponent determined by Mordell–Weil rank and real dimension. Fishman–Lambert–Merrill–Simmons accurately identify this as the assertion reformulated by their Theorem 1.5.
Lemma 5.1No issue found
Coefficient boxes and canonical height have quadratic scale
journal pp. 40–41 · Lemma 5.1
The lemma works with coefficients bounded by a parameter and canonical height bounded on the order of . This is consistent because canonical height is a positive-definite quadratic form on the Mordell–Weil lattice. Renaming the height threshold and coefficient threshold does not create a contradiction; it changes the approximation exponent by the required factor of two.
02Proofs1 reported findingNo proof issues found
The selected Lemma 5.1 argument is coherent and its material simultaneous-approximation input is represented recursively at Waldschmidt 1997.
Proof of Lemma 5.1Correct and complete modulo represented source
The analytic and height comparisons preserve the claimed exponent
journal pp. 40–41
The proof compares the chosen Mordell–Weil coordinates, abelian logarithms, and canonical height, then invokes the earlier simultaneous approximation lemma. Constants and finite-index changes do not affect the exponent. The reused 1997 input and its Cassels transference ancestor are represented separately in the dependency graph.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.