arXiv math/06127252006-12-28math.NT

秩1可解p-adic微分方程与Lubin-Tate群的有限交换特征标

Rank One Solvable p-adic Differential Equation and Finite Abelian Characters via Lubin-Tate groups

Andrea Pulita

备注：: 53 Pages, this is not the published version. To appear in Math.Annalen. On line version avaiable at the following addres: http://springerlink.metapress.com/content/1432-1807/?sortorder=asc&sw=rank+one
DOI：: 10.1007/s00208-006-0040-8

摘要：

本文引入了一类新的Artin-Hasse型指数，称为$\boldsymbolπ$ -指数.这些指数依赖于Lubin-Tate群$\mathfrak{G}$对$\mathbb{Z}_p$的Tate模块的生成器$\boldsymbolπ$的选择。它们自然地产生为Robba环上的可解微分模的解。如果$\mathfrak{G}$在$\mathbb{Z}_p$上同构于$\hat{\mathbb{G}}_m$，我们给出了检验其过收敛性的方法，从而得到了微分方程的Frobenius结构定理的一个更强的形式.我们定义了一个自然变换的阿廷-施赖埃复杂的库默尔复杂。这提供了$\mathcal{E}^†_{K_{\infty}}$的Kummer非分歧扩张的显式生成元，其剩余域是k（（t））的给定Artin-Schreier扩张，其中k是K的剩余域。在张量积下，我们计算了秩为1的可解微分方程的同构类的群。此外，我们还得到了一个计算$\mathcal{E}^†_{K_{\infty}}$上的秩一$ϕ$ -模的规范方法，该模附加在$Gal(k((t))^{sep}/k((t)))$的秩一表示上，由Artin-Schreier特征标定义.

英文摘要：

We introduce a new class of exponentials of Artin-Hasse type, called $\boldsymbolπ$-exponentials. These exponentials depends on the choice of a generator $\boldsymbolπ$ of the Tate module of a Lubin-Tate group $\mathfrak{G}$ over $\mathbb{Z}_p$. They arise naturally as solutions of solvable differential modules over the Robba ring. If $\mathfrak{G}$ is isomorphic to $\hat{\mathbb{G}}_m$ over $\mathbb{Z}_p$, we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation of the Artin-Schreier complex into the Kummer complex. This provides an explicit generator of the Kummer unramified extension of $\mathcal{E}^†_{K_{\infty}}$, whose residue field is a given Artin-Schreier extension of k((t)), where k is the residue field of K. We then compute explicitely the group, under tensor product, of isomorphism classes of rank one solvable differential equations. Moreover, we get a canonical way to compute the rank one $ϕ$-module over $\mathcal{E}^†_{K_{\infty}}$ attached to a rank one representation of $Gal(k((t))^{sep}/k((t)))$, defined by an Artin-Schreier character.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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