Spectral Fields and Unitary Descent: Local splitting, simultaneous norms, and transcendental obstructions

Abstract

Spectral fields can recover matrix coefficients without standardizing the invariant Hermitian form. Over an algebraic subfield of $\mathbb{C}$ stable under nontrivial complex conjugation, we prove that every Zariski-dense subgroup in a normalized Hermitian model of dimension $m\ge4$ and signature $(m-1,1)$ at the fixed embedding contains a loxodromic element whose spectrum makes all diagonal coefficients norms. The element can be chosen so that every positive power is regular and generates the same splitting field. For an arbitrary initial Hermitian model over such a field, two spectra suffice for standard unitary descent over an extension of relative degree at most $(m!)^2$. Over every quadratic extension $F/k$ of nonarchimedean local fields of characteristic zero, a four-dimensional quasi-split unitary group of Witt index two contains a discrete, Zariski-dense free subgroup whose spectra all split over $F$. Over intrinsic fields of positive transcendence degree, we construct a single convex cocompact, Zariski-dense free subgroup of $\mathrm{SU}(3,1)$ for which, for every initial regular element and every normalized extraction, the Hermitian obstruction persists after adjoining any finite family of spectra and over the total spectral field.