Independent Frobenius Coordinates and a Prescribed Formal-Local Spectral Map for Generic GSp4
- 1SAP
Abstract
On the strongly 7-generic ordinary Enns–Lee deformation chart for GSp4 (Qp) with fixed similitude, we prove that the reductions of two divided Frobenius units are algebraically independent and have linearly independent differentials at the reference point. The proof gives integral chart coordinates and an invertible formal change of parameters. Prescribing two generator values in Herzig’s freely presented weight Hecke algebra then defines an algebra map whose kernel is exactly the chosen central specialization. We call this a prescribed spectral map: compatibility with the spherical Hecke action on a Serre-weight multiplicity space is not proved. Under the stated deformation-comparison and congruent-patching inputs, the units descend to the fixed-similitude formal germ of the selected Emerton–Gee component; their extension to the full completion is a pullback along a reference-similitude retraction. Allowing the prescribed central value to vary with the unramified similitude gives an injective three-generator Laurent polynomial map on that germ, and the twist-compatible pushforward of the divided Frobenius units gives a second such map on the same germ with the same fixed-similitude specialization. A separate, source-relative calculation compares layer depth two with valuation exponent one for a specified finite-group intertwiner normalization. That weighted layer calculation is not needed for the coordinate-independence theorem; the same appendix’s principal-series multiplicity-freeness is a separate input to the descent argument. The published patched roof uses the same filtration data; its compatibility with the literal counting formula remains an explicit normalization obligation. No global-functions theorem or spectral Satake compatibility theorem is claimed.
Related
- Spectral Curves for Semisimple Conformal Frobenius Manifolds
- The maximum spectral radius of connected graphs with prescribed order and size
- Extremal Graphs with Prescribed Parameters for the Spectral Radius of Weighted Adjacency Matrices with Property P∗
- Prescribed spectral configurations under rank-one perturbations
- Prescribed Beurling densities in spectral eigen-subspaces
BibTeX
@misc{tao2026independent,
title = {{Independent Frobenius Coordinates and a Prescribed Formal-Local Spectral Map for Generic GSp4}},
author = {Lin Tao},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.20944/preprints202609.2468.v2},
url = {https://zenodo.org/records/23882305},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Independent Frobenius Coordinates and a Prescribed Formal-Local Spectral Map for Generic GSp4
AU - Lin Tao
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.20944/preprints202609.2468.v2
UR - https://zenodo.org/records/23882305
AB - On the strongly 7-generic ordinary Enns–Lee deformation chart for GSp4 (Qp) with fixed similitude, we prove that the reductions of two divided Frobenius units are algebraically independent and have linearly independent differentials at the reference point. The proof gives integral chart coordinates and an invertible formal change of parameters. Prescribing two generator values in Herzig’s freely presented weight Hecke algebra then defines an algebra map whose kernel is exactly the chosen central specialization. We call this a prescribed spectral map: compatibility with the spherical Hecke action on a Serre-weight multiplicity space is not proved. Under the stated deformation-comparison and congruent-patching inputs, the units descend to the fixed-similitude formal germ of the selected Emerton–Gee component; their extension to the full completion is a pullback along a reference-similitude retraction. Allowing the prescribed central value to vary with the unramified similitude gives an injective three-generator Laurent polynomial map on that germ, and the twist-compatible pushforward of the divided Frobenius units gives a second such map on the same germ with the same fixed-similitude specialization. A separate, source-relative calculation compares layer depth two with valuation exponent one for a specified finite-group intertwiner normalization. That weighted layer calculation is not needed for the coordinate-independence theorem; the same appendix’s principal-series multiplicity-freeness is a separate input to the descent argument. The published patched roof uses the same filtration data; its compatibility with the literal counting formula remains an explicit normalization obligation. No global-functions theorem or spectral Satake compatibility theorem is claimed.
ER -