A Componentwise Spectral mod-<em>p</em> Satake Morphism on a Strongly Generic GSp<sub>4</sub> Locus

Source abstract

We study a bounded spectral mod-p Satake problem for GSp4(Q_p) on a strongly generic ordinary locus. The main input is the potentially crystalline and patched local-global framework of Enns-Lee, together with Herzig's characteristic-p Satake algebra and the geometry of the GSp4 Emerton-Gee stack. A natural attempt to realize the normalized spectral generator on finite-group principal-series lattices fails uniformly: on each of the four Weyl branches used in the local-global compatibility theorem, the distinguished Serre weight occurs in Jantzen layer two while the Galois-side normalization exponent is one. We then use the actual patched realization to recover the two noncentral spectral coordinates. For the corresponding W-regular Serre weight, the mod-p Satake algebra is generated by two noncentral antidominant rays and the invertible center, and the recovered coordinates define a direct componentwise algebra morphism. Its kernel is exactly the forced fixed-central-character relation; after central specialization the map is injective, with a proper two-dimensional polynomial image inside the fourteen-dimensional framed special fiber. Finally, the two spectral functions descend through gauge and framing to the formal completion of the corresponding Serre-weight component of the GSp4 Emerton-Gee stack, where they retain algebraic independence and cotangent rank two. All statements are confined to the stated strongly generic, fixed-central, suitable patched locus; no global-functions theorem on the entire component and no all-weight spectral Satake isomorphism is claimed.