Modular-Form Operator Universal Tool for Spectral Encoding of Modular Forms and Automorphic Representations

Key Points

• To create a framework for encoding modular forms and automorphic representations into spectral operators.
• Constructed H_f from modular forms f(τ) expressed as a series of Fourier coefficients.
• Introduced a Weight-Scaling Correspondence relating modular weight to fractal dimension.
• Utilized Hecke-Spectral Isomorphism to link Hecke eigenvalues with operator eigenvalues.
• Applied boundary conditions to encode level structures.
• Established a correspondence between modular weight and fractal dimension.
• Demonstrated spectral properties reflecting the arithmetic of Fourier coefficients.
• Applied findings to significant concepts like Ramanujan bounds and the Langlands program.

Abstract