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

2018/01/09 by Upton, James
#11R29 (secondary) #11S15 (primary) 11S20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1801.03056

Abstract

In this paper we consider a tower of number fields ⋯ ⊇ K(1) ⊇ K(0) ⊇ K arising naturally from a continuous p-adic representation of Gal(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 \mathbbF_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,pⁱ for i sufficiently large. This generalizes a previously known result on discriminant-growth in ℤ_p-towers of local fields of characteristic zero.

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