Differential Algebraic Closure Framework for Modular Forms: Explicit Constructions and Combinatorial Structures
shifa liu · Zenodo (CERN European Organization for Nuclear Research) · 2025
This paper establishes a rigorous differential algebraic framework for the explicit construction of modular forms and their Fourier coefficients. We introduce the modular forms closure KM, a differentially closed field extension that systematically contains solutions to a broad class of modular form problems.Within this closure, we derive explicit analytic expressions for Fourier coefficients, Hecke eigenvalues,and modular forms of various weights and levels.We provide complete constructive foundations with rigorous proofs, derive combinatorial expressions with explicit connections to symmetric function theory, present detailed algorithmic implementations with complexity analysis, and situate our results within classical modular form theory. Numerical experiments across diverse modular forms confirm spectral convergence and demonstrate the necessity of combinatorial corrections for higher-weight forms.This work establishes that explicit analytic expressions exist in KM for a significant class of modular form problems, providing a new algebraic perspective on modular-theoretic solvability while maintaining consistency with classical complexity results.