arXiv 1806.022442018-06-06math.NT

椭圆$\mathfrak{p}$ -单位的构造

Construction of elliptic $\mathfrak{p}$-units

Werner Bley, Martin Hofer

备注：: 26 pages

摘要：

设$L/k$是虚二次数域$k$的有限阿贝尔扩张.设$\mathfrak{p}$表示$\mathcal{O}_k$的一个素理想落在有理素数$p$上.我们假设$\mathfrak{p}$在$L/k$中完全分裂，并且$p$不划分$k$的类数。如果$p$在$k/\mathbb{Q}$中分裂，则第一命名的作者已经修改了Solomon的构造以在$L$中获得椭圆$\mathfrak{p}$ -单位。在本文中，我们推广这种结构的非分裂的情况下，并获得了一对椭圆$\mathfrak{p}$ -单位依赖于一定的岩泽代数（这里是秩2）的生成元的选择。在我们的主要结果中，我们表示这些$\mathfrak{p}$ -单位的$\mathfrak{p}$ -adic赋值的$p$ -adic对数的显式椭圆单位。证明我们的主要结果的关键输入是一个合适的科尔曼幂级数的常数项的计算，我们依赖于T。清力

英文摘要：

Let $L/k$ be a finite abelian extension of an imaginary quadratic number field $k$. Let $\mathfrak{p}$ denote a prime ideal of $\mathcal{O}_k$ lying over the rational prime $p$. We assume that $\mathfrak{p}$ splits completely in $L/k$ and that $p$ does not divide the class number of $k$. If $p$ is split in $k/\mathbb{Q}$ the first named author has adapted a construction of Solomon to obtain elliptic $\mathfrak{p}$-units in $L$. In this paper we generalize this construction to the non-split case and obtain in this way a pair of elliptic $\mathfrak{p}$-units depending on a choice of generators of a certain Iwasawa algebra (which here is of rank 2). In our main result we express the $\mathfrak{p}$-adic valuations of these $\mathfrak{p}$-units in terms of the $p$-adic logarithm of an explicit elliptic unit. The crucial input for the proof of our main result is the computation of the constant term of a suitable Coleman power series, where we rely on recent work of T. Seiriki.

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