Local–Global Defects B, Tamagawa Numbers, and Norm-One Tori: Local–Global Identities for Norm-One Tori of Rank One
Year: 2026
Type: preprint
Abstract: This paper establishes, at the elementary level of algebraic tori over numberfields, an identity relating Tamagawa numbers to local–global defects, and completes the construction of its self-contained components using norm-one tori ofquadratic extensions as the worked instance. The instantiation theorem reads asfollows: let K/k be a quadratic extension and T = R1K/kGm the norm-one torus;then Sha(T) is the trivial group and #Sha(T) = 1, whose proof is carried outvia an elementary path through the Hasse–Minkowski theorem for ternary forms;H1(k,X∗(T)) ∼ = Z/2Z, given by an exact computation of cohomology of the character lattice; τ(T) = 2 is output by the Ono–Sansuc identity at the point whereSha is trivial and the cohomological count equals 2. The first two items are selfcontained; the third is an output of an external theorem, and this dependencystratification is explicitly registered in the main text. The arithmetic invariants ofthe three quadratic fields Q(i), Q(√−5), Q(√5) are closed by deterministic exactinteger arithmetic: class numbers, numbers of roots of unity, and discriminants areobtained by reduced-form enumeration and reduced cycle walking; the closed formof L(1,χ) and the partial-sum readings are cross-checked within the strict truncation bound M/(N + 1); the local–global equivalence of norm forms over finiteprime samples holds digit by digit, and the modulo-residue condition at the ramified prime is explicitly witnessed as a guard mechanism. The numerical archivefunctions as an analytic consistency check and does not constitute an independentcomputation of τ(T). The Ono–Sansuc theorem for general tori is stated accordingto the literature, with a six-step proof skeleton registering its dependency positions; the corresponding statement for cyclic extensions of degree n is registered asproposition-level content, in which H1 ∼ = Z/nZ is self-contained, and the weakenable limit of this self-contained computation is the commutativity hypothesis: anyfinite Galois extension yields #H1(G,X∗(T)) = |Gab|, with the noncommutativecase bounded by the order of the abelianization; triviality of Sha is borne by theHasse norm theorem, and τ = n is an identity output. The whole paper is organizedby dependency stratification: self-contained proofs, citation-level inputs, identityoutputs, and cross-check-level readings each occupy their own place and may notsubstitute for one another.
Authors: zhou, changzheng, zhou, ziqing
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225824772
Cites:
Cited by:
Related to: