From Common-Slot Chains to Heisenberg Central Products over Global Fields
Abstract
Let F be a global field and let p be an odd prime with p≠charF. Assuming first that μp⊂ F, we record in a uniform global-field form the length-four common-slot chain for equal degree-p symbol classes and emphasize the signed normalization adapted to explicit norm constructions. Thus, from (a,b)p=(c,d)p∈Br(F)[p] one obtains x,y∈ F× such that (a,b)p=(x−1,b)p=(x,y)p=(c−1,y)p=(c,d)p. For number fields, the underlying chain is the length-four chain lemma of Gille--Szamuely, based on Tate's simultaneous local--global theorem. The point developed here is that its signed form yields four compatible norm equations that can be used constructively. For the extraspecial central product Hp3*Hp3 over a Cp4-Kummer extension, the central-embedding obstruction is (a,b)p−(c,d)p, while the four norm equations supplied by the chain assemble, under a natural independence hypothesis on the auxiliary Kummer classes, into an explicit factorized radical realization of the central product. Finally, when the ground field does not contain μp, we show by a restriction--corestriction argument that the central-embedding obstruction is detected after passage to the cyclotomic extension F(μp). Over that field the problem is Kummer, and the obstruction is again the difference of the two symbol classes.
Related
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- Global Central Limit Theorems for Markov chains with normal transition operator
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BibTeX
@misc{palaisti2026from,
title = {{From Common-Slot Chains to Heisenberg Central Products over Global Fields}},
author = {Marina Palaisti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00244},
eprint = {2609.00244},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00244},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Common-Slot Chains to Heisenberg Central Products over Global Fields
AU - Marina Palaisti
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00244
UR - https://arxiv.org/abs/2609.00244
AB - Let F be a global field and let p be an odd prime with p≠charF. Assuming first that μ_p⊂ F, we record in a uniform global-field form the length-four common-slot chain for equal degree-p symbol classes and emphasize the signed normalization adapted to explicit norm constructions. Thus, from (a,b)_p=(c,d)_p∈Br(F)[p] one obtains x,y∈ F^× such that (a,b)_p=(x^−1,b)_p=(x,y)_p=(c^−1,y)_p=(c,d)_p. For number fields, the underlying chain is the length-four chain lemma of Gille--Szamuely, based on Tate's simultaneous local--global theorem. The point developed here is that its signed form yields four compatible norm equations that can be used constructively. For the extraspecial central product H_p^3*H_p^3 over a C_p^4-Kummer extension, the central-embedding obstruction is (a,b)_p−(c,d)_p, while the four norm equations supplied by the chain assemble, under a natural independence hypothesis on the auxiliary Kummer classes, into an explicit factorized radical realization of the central product. Finally, when the ground field does not contain μ_p, we show by a restriction--corestriction argument that the central-embedding obstruction is detected after passage to the cyclotomic extension F(μ_p). Over that field the problem is Kummer, and the obstruction is again the difference of the two symbol classes.
ER -