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 $G={\mathrm{GL}}_n$, $\mu =(1,1,0^{n-2})$, with basic Newton slope $2/n$, for every odd $n\ge 3$, 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 $p=2$.

15 pages, comments welcome!