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arXiv 2201.082032022-01math.NT

关于 Leopoldt 和 Gross 的调节映射的秩

On the rank of Leopoldt's and Gross's regulator maps

Universität Paderborn（帕德博恩大学）

Alexandre Maksoud

备注：: 20 pages, comments welcome!
发表信息：: Documenta Mathematica 28 (2023), 1441-1471
DOI：: 10.4171/dm/935

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AI总结：

本文推广了 Waldschmidt 对 Leopoldt 缺陷的界，并证明了 Gross 缺陷的界，同时证明了 Gross 有限性猜想的新情况，并展示了 Gross p 进调节的秩。

摘要：

我们推广了 Waldschmidt 对 Leopoldt 缺陷的界，并为数字段扩展的 Gross 缺陷证明了类似的界。作为应用，我们证明了 Gross 的有限性猜想（也称为 Gross-Kuz'min 猜想）在经典阿贝尔情况之外的新情况，并表明 Gross 的 p 进调节具有至少猜想中所预测的一半的秩。我们还描述并计算了 Gross 缺陷的非 cyclotomic 类比。

英文摘要：

We generalize Waldschmidt's bound for Leopoldt's defect and prove a similar bound for Gross's defect for an arbitrary extension of number fields. As an application, we prove new cases of Gross's finiteness conjecture (also known as the Gross-Kuz'min conjecture) beyond the classical abelian case, and we show that Gross's $p$-adic regulator has at least half of the conjectured rank. We also describe and compute non-cyclotomic analogues of Gross's defect.

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