Spectral Fingerprints of Algebraic Cycles on K3 Surfaces: A Hodge-Theoretic Perspective

Abstract: This paper introduces a novel framework, the "Spectral Fingerprint Philosophy," for extracting intrinsic algebraic information from Hodge cycles, particularly divisors, on K3 surfaces. We depart from traditional geometric spectral analysis, which relies on the Laplace-Beltrami operator and faces significant computational challenges for Ricci-flat Kahler metrics. Instead, we leverage the powerful machinery of Hodge theory, variation of Hodge structures (VHS), and arithmetic geometry. We demonstrate how the "spectral fingerprint" of an algebraic cycle on a K3 surface is encoded in the algebraic coefficients of the Picard-Fuchs differential equations that govern the surface's periods. Furthermore, this fingerprint is also found in the arithmetic properties of these periods themselves, which frequently manifest as Fourier coefficients of automorphic forms on the moduli space of K3 surfaces. We provide theoretical proofs for these foundational claims, grounding the philosophy in established mathematical results. Through a preliminary exploration of genus-2 Riemann surfaces, utilizing the Frobenius endomorphism as an "arithmetic spectral operator," and a detailed case study of Kummer K3 surfaces, we present concrete results. These results illustrate how algebraic cycles lead to specific, algebraically defined constraints on this "spectrum," thereby offering a new perspective on the Hodge Conjecture.