Galois Representations Associated to Torsion Classes in the Cohomology of Hilbert Modular Varieties

Mathematical Applications of Science Fiction We construct continuous Galois representations associated totorsion eigenclasses in the cohomology of Hilbert modularvarieties over a totally real field 𝐹. By passing to the infinitelevel perfectoid Shimura variety and utilizing the Hodge-Tateperiod map 𝜋HT, we analyze the cohomology of the structuresheaf on the adic space. We establish a torsion Jacquet-Langlandscorrespondence and prove that for every maximal ideal 𝔪 of theHecke algebra acting on the torsion cohomology with coefficientsin Fℓ , there exists a semisimple continuous representation 𝜌𝔪 :Gal(𝐹/𝐹) → GL2 (Fℓ ) characterized by the Hecke eigenvalues.This extends results of Scholze and Calegari-Geraghty to theHilbert modular setting without assumptions on the weight orthe solvability of the base change.