Steinness of the Basic $\mathrm{GL}_n$ Local Shimura Tower in Odd Rank

Abstract

We prove Steinness over $\breve{\mathbf{Q}}_p$ for the basic local Shimura tower attached to $G=\mathrm{GL}_n$, $μ=(1,1,0^{n-2})$, with basic Newton slope $2/n$, for every odd $n\ge3$, at every finite level and for every prime $p$. The proof proceeds by constructing global analytic functions from the crystalline period map, proving compactness of their sublevel sets via a perfectoid normalization of Tate lattices, and showing that the resulting function map to affine space is finite. The key geometric inputs are the Fargues--Fontaine realization of the universal cover, a normalized determinant on the rank-two locus, and an integral PEL realization over $W(\overline{\mathbf{F}}_p)$ used to produce bounded global Hodge generators. The integral constructions include the prime $p=2$.