arXiv 1802.093092018-02-26math.NT

分歧素数的稠密性

The density of ramified primes

Jyoti Prakash Saha

摘要：

设$F$是数域，$\mathcal{O}$是具有特征为零的分数域$\mathcal{K}$的域，$ρ: \mathrm{Gal}(\overline F/F) \to \mathrm{GL}_n(\mathcal{O})$是使得$ρ\otimes\overline{\mathcal{K}}$是半单的表示。如果$\mathcal{O}$允许系数在$p$ -adic整数环中的幂级数环的有限单态（分别为. $\mathcal{O}$是$p$ -adic数域上的仿射代数）和$ρ$关于最大理想adic拓扑是连续的（分别是Banach代数拓扑），然后证明了$ρ$的分歧素数集是密度为零的.若$\mathcal{O}$是$\mathbb{Z}_p$上的完备局部Noether环，其特征为$p$的有限剩余域，$ρ$关于极大理想adic拓扑是连续的，且$ρ$的纯特化核构成$\mathrm{Spec} \mathcal{O}$的一个Zeriki稠密子集，则证明了$ρ$的分歧素数集的密度为零.这些结果在大伽罗瓦表示的背景下是Khare和Rajan的结果的类似物，并且依赖于他们的结果被证明。

英文摘要：

Let $F$ be a number field, $\mathcal{O}$ be a domain with fraction field $\mathcal{K}$ of characteristic zero and $ρ: \mathrm{Gal}(\overline F/F) \to \mathrm{GL}_n(\mathcal{O})$ be a representation such that $ρ\otimes\overline{\mathcal{K}}$ is semisimple. If $\mathcal{O}$ admits a finite monomorphism from a power series ring with coefficients in a $p$-adic integer ring (resp. $\mathcal{O}$ is an affinoid algebra over a $p$-adic number field) and $ρ$ is continuous with respect to the maximal ideal adic topology (resp. the Banach algebra topology), then we prove that the set of ramified primes of $ρ$ is of density zero. If $\mathcal{O}$ is a complete local Noetherian ring over $\mathbb{Z}_p$ with finite residue field of characteristic $p$, $ρ$ is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of $ρ$ form a Zariski-dense subset of $\mathrm{Spec} \mathcal{O}$, then we show that the set of ramified primes of $ρ$ is of density zero. These results are analogues, in the context of big Galois representations, of a result of Khare and Rajan, and are proved relying on their result.

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