Discriminant-Stability in $p$-adic Lie Towers of Number Fields

TL;DR

This paper proves that in certain $p$-adic Lie towers of number fields, the discriminant's $p$-adic valuation grows polynomially with the tower level, extending known results from local fields to global number fields.

Contribution

It establishes polynomial growth of discriminants in $p$-adic Lie towers over number fields, generalizing prior local field results to a broader global context.

Findings

Discriminant valuation is polynomial in tower level for large $i$

Generalizes local field discriminant growth to number fields

Supports conjecture on stable genus growth in characteristic zero

Abstract

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 $\mathrm{Gal}(\bar{\mathbb{Q}}/K)$, 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.