# Automorphic Differential Algebraic Closure: ARigorous Framework for Explicit Parameterization of Automorphic Forms

Article · shifa liu

- Archive id: doi:10.5281/zenodo.18097012
- Published: 2025-12-30
- Venue: Zenodo (CERN European Organization for Nuclear Research)
- DOI: https://doi.org/10.5281/zenodo.18097012
- Citations: 0
- Source URL: https://doi.org/10.5281/zenodo.18097012
- Provenance: openalex

## Abstract

This paper establishes a comprehensive differential algebraic framework for the explicit analytic parameterization of automorphic forms. We construct a rigorously defined automorphic differential algebraic closure KAut and prove that local analytic parameterizations of automorphic forms can be expressed within this closure. The unified parameterization formula is derived through a synthesis of differential algebra, representation theory, and automorphic forms theory. We provide complete constructive proofs with detailed combinatorial analysis of automorphic correction coefficients, present comprehensive algorithms with rigorous complexity analysis, and validate the method through theoretical examples including modular forms, Maass forms, and Siegel modular forms. The work demonstrates that explicit analytic parameterizations exist in KAut for a broad class of automorphic forms and establishes fundamental connections between differential algebra, automorphic forms, and the Langlands program. Connections to contemporary developments including the geometric Langlands program, p-adic Hodge theory, and the Gan-Gross-Prasad conjectures are thoroughly explored, with explicit attention to Γ-equivariance conditions and convergence properties derived from spectral gap estimates.

_Abstract reproduced under CC-BY (publisher license, via OpenAlex)._

Topics: [Landmark Results in Pure Mathematics](https://space-kitty.com/topics/landmark-math-results)

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