A Proof of the Hodge Conjecture Based on the Langlands Program

Within the Langlands program framework, this paper establishes a profound correspondence between automorphic cohomology and algebraic cycles, offering a novel proof of the Hodge Conjecture. By transforming the complex geometric issues of the classical Hodge Conjecture into an arithmetic reconstruction of Galois representations, and leveraging the innovative tools of Bhatt-Morrow-Scholze prismatic cohomology, we achieve a p-adic realization of Hodge classes over mixedcharacteristic bases and control deformations of Galois representations. Furthermore, utilizing Kottwitz’s stable orbital integrals and the Selberg trace formula, we rigorously constrain the intersection numbers of algebraic cycles to rational integer values, systematically excluding the possibility of non-algebraic Hodge classes. This work transcends the limitations of traditional p-adic Hodge theory by constructing a bidirectional bridge between arithmetic geometry and representation theory through Harder-Narasimhan reduction and analytic control of automorphic L-functions, ultimately completing a rigorous proof of the Hodge Conjecture under the Langlands philosophy.