分圆Iwasawa伪测度的布洛赫-正则元主部

中文摘要 AI 辅助

设$p$为奇素数，$K/\boldsymbol{\text{Q}}_p$为有限非分歧扩张，我们由与$p$互素阶的单位根的有限表示，构造$\boldsymbol{Z}_p^\times$上的局部化Iwasawa伪测度。尽管该伪测度依赖所选表示，但其模有界测度的像仅依赖关联布洛赫类：它是该类的弗罗贝尼乌斯亏缺Coleman正则元乘以通用半平移ζ主部。因此，每个非例外权分量是有界的，而例外分量至多有一个单极点，其留数可明确确定。当$p>3$且系数取$\boldsymbol{Z}_p$值时，主部消失等价于$K_3(K;\boldsymbol{Z}_p)$中对应类消失。分圆细化保持布洛赫类，并通过明确的Iwasawa乘子作用于关联伪测度。细化的归一化有限线性组合可插值有界权空间数据的任意有限喷流，仅受例外点处归一化的约束。我们还建立半平移复Mellin分解，将该构造与GSWZ芽族比较，并在$p>3$的素数上方的简单一次位处，推导纽结$5_2$的局部$K_3$-类的有限多重对数准则。

英文摘要

Let $p$ be an odd prime and let $K/\mathbb{Q}_p$ be a finite unramified extension. From a finite presentation by roots of unity of order prime to $p$, we construct a localized Iwasawa pseudomeasure on $\mathbb{Z}_p^\times$. Although the pseudomeasure depends on the chosen presentation, its image modulo bounded measures depends only on the associated Bloch class: it is the Frobenius-depleted Coleman regulator of that class multiplied by a universal half-shifted zeta principal part. Consequently, every nonexceptional weight component is bounded, while the exceptional component has at most a simple pole with explicitly determined residue. For $p>3$ and $\mathbb{Z}_p$-valued coefficients, vanishing of the principal part is equivalent to vanishing of the corresponding class in $K_3(K;\mathbb{Z}_p)$. Cyclotomic refinements preserve the Bloch class and act on the associated pseudomeasures by explicit Iwasawa multipliers. Normalized finite linear combinations of refinements interpolate arbitrary finite jets of the bounded weight-space data, subject only to the normalization at the exceptional point. We also establish a half-shifted complex Mellin factorization, compare the construction with the GSWZ germ family, and derive, at simple degree-one places above primes $p>3$, a finite-polylogarithm criterion for the local $K_3$-class of the knot $5_2$.

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64 pages, no figures