arXiv:1111.0163v5
S-adic version of Minkowski's geometry of numbers and Mahler's compactness criterion
math.DS
Abstract
In this note we give a detailed proof of certain results on geometry of numbers in the -adic case. These results are well-known to experts, so the aim here is to provide a convenient reference for the people who need to use them.
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Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The S-adic Mahler compactness criterion and Minkowski-type bounds for successive minima are correct with the stated module and covolume hypotheses.
Theorem 1.1Correct
The S-adic compactness criterion is correct
Pages 2 and 10–12 · Theorem 1.1 · arXiv:1111.0163v5
Uniform exclusion of nonzero vectors from a fixed product neighborhood is necessary and sufficient for relative compactness in the quotient. The module normalization and the contribution from every place are retained in both directions.
Full paper, version 5 ↗Theorems 1.2 and 4.4Correct
The successive-minima product bounds are correct
Pages 2–3 and 8–10 · Minkowski-type theorems · arXiv:1111.0163v5
The product of successive minima is bounded above and below by constants depending only on the ambient S-adic data times the covolume. The statement correctly accommodates nonprincipal modules through the ideal-class reduction.
02Proofs2 reported findingsCorrect
The ideal-class reduction, volume estimates, and compactness argument are correct and complete.
Proof of the Minkowski theoremCorrect and complete
The local-volume and module-index factors balance correctly
Pages 5–10 · proof culminating in Theorem 4.4 · arXiv:1111.0163v5
The adelic convex body is compared with a fundamental region, successive independent vectors are selected in order, and the finite ideal-class decomposition gives uniform constants. No place or rank case is omitted.
Proof of Theorem 1.1Correct and complete
Mahler compactness follows in both directions
Pages 10–12 · proof of Theorem 1.1 · arXiv:1111.0163v5
A convergent family cannot acquire a nonzero vector tending to zero, while the lower bound on the first minimum and the covolume normalization bound all successive minima. A bounded choice of bases then yields a convergent subsequence in the quotient.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.