functional equations of L- functions, GL2, Representation-theoretic methods; automorphic representations over local and global fields, GL1, automorphic representation, Zeta functions and \(L\)-functions of number fields, Representations of Lie and linear algebraic groups over global fields and adèle rings, adele ring, Linear algebraic groups over adèles and other rings and schemes, Langlands-Weil conjectures, nonabelian class field theory