Galois-Representation Encoder Universal Tool for Spectral Encoding of Galois Representations and Arithmetic Symmetries

Key Points

• The aim is to develop a framework for encoding Galois representations into spectral operators, enhancing their application in arithmetic geometry.
• Introduced Frobenius-Spectral Correspondence for encoding eigenvalues.
• Developed Ramification Encoding using singular potentials at bad primes.
• Preserved ℓ-adic structure through fractal scaling techniques.
• Established a correspondence between Frobenius eigenvalues and operator eigenvalues.
• Introduced tools that integrate Galois representations with arithmetic spectral problems.
• Demonstrated implications for the Artin conjecture and Serre's modularity.