Steinness of the Basic GL_n Local Shimura Tower in Odd Rank

Abstract

We prove Steinness over Q̆ₚ for the basic local Shimura tower attached to G = GLₙ, μ = (1, 1, 0ⁿ⁻²), with basic Newton slope 2/n, for every odd n ≥ 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(F̄ₚ) used to produce bounded global Hodge generators. The integral constructions include the prime p = 2.

Topics

• p-adic geometry
• Local Shimura varieties
• Local Shimura towers
• Perfectoid geometry
• Fargues–Fontaine curve
• GL_n
• Stein spaces