In this paper we consider a tower of number fields $\cdots \supseteq K(1)\supseteq K(0)\supseteq K$ arising naturally from a continuous p-adic representation of $\mathrm{Gal}(\overline{\mathbb{Q}}/K)$, referred to as a p-adic Lie tower over K. A recent conjecture of Daqing Wan hypothesizes, for certain p-adic Lie towers of curves over ${\mathbb{F}}_p$, a stable (polynomial) growth formula for the genus. Here we prove the analogous result in characteristic zero, namely: the p-adic valuation of the discriminant of the extension $K(i)/K$ is given by a polynomial in $i,p^i$ for i sufficiently large. This generalizes a previously known result on discriminant-growth in ${\mathbb{Z}}_p$-towers of local fields of characteristic zero.

在本文中，我们考虑一个数字域的塔$\cdots \supseteq ķ(1)\supseteq ķ(0)\supseteq ķ$自然产生于一个连续的p -adic 表示$\mathrm{加尔}(\overline{\mathbb{问}}/ķ)$，称为K上的p进李塔。大庆万最近的一个猜想假设，对于某些p -adic Lie towers of curve over${\mathbb{F}}_p$，属的稳定（多项式）增长公式。这里我们证明了特征零的类似结果，即：扩展判别式的p进估值$ķ(\mathrm{一世})/ķ$由多项式给出$\mathrm{一世},p^{\mathrm{一世}}$因为i足够大。这概括了先前已知的关于判别增长的结果${\mathbb{Z}}_p$- 特征零的局部场的塔。