Genus-Spectral Correspondence From Algebraic Curves to Operator Spectra

Universal correspondence between geometric genus of algebraic curves and spectral density bounds of associated differential operators. For curves C/ℚ with genus g, constructs canonical operators H_C whose eigenvalue distribution is constrained by g. Key result: genus g > 1 (hyperbolic) forces finite spectral capacity, providing operator-theoretic proofs of Faltings-type finiteness theorems. Unifies Diophantine geometry and spectral theory. Applications to Fermat-Catalan, abc conjecture, elliptic curves, modular curves.