整体域上从公共 slot 链到海森堡中心积
From Common-Slot Chains to Heisenberg Central Products over Global Fields
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中文总结 AI 辅助
本文研究整体域上的公共 slot 链与海森堡中心积，基于链引理构造范数方程实现中心积，基域不含 μ_p 时通过分圆扩域检测嵌入障碍。
中文摘要 AI 辅助
设 F 为整体域，p 为满足 p≠char F 的奇素数。首先假设 μ_p⊂F，我们以统一的整体域形式记录等p次符号类的长度为4的公共 slot 链，并强调适配于显式范数构造的带符号正规化。由此，从 (a,b)_p=(c,d)_p∈Br(F)[p] 可得到 x,y∈F^×，使得 (a,b)_p=(x^{-1},b)_p=(x,y)_p=(c^{-1},y)_p=(c,d)_p。对于数域，该链是 Gille–Szamuely 的长度为4的链引理，基于 Tate 的同时局部-整体定理。本文的核心是，其带符号形式给出四个相容的范数方程，可用于构造。对于 C_p^4 -库默尔扩域上的特殊中心积 H_{p^3}*H_{p^3}，中心嵌入障碍为 (a,b)_p-(c,d)_p；在辅助库默尔类满足自然独立性假设下，链提供的四个范数方程可组装成该中心积的显式因式分解根式实现。最后，当基域不含 μ_p 时，我们通过限制- corestriction 论证表明，中心嵌入障碍在经过分圆扩域 F(μ_p) 后可被检测；在该域上问题为库默尔型，障碍仍为两个符号类的差。
英文摘要
Let $F$ be a global field and let $p$ be an odd prime with $p\neq\operatorname{char}F$. Assuming first that $μ_p\subset F$, 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 \[(a,b)_p=(c,d)_p\in\textrm{Br}(F)[p]\] one obtains $x,y\in F^\times$ 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.
