The book by Hervé Jacquet and Langlands on presented a theory of automorphic forms for the general linear group , establishing among other things the Jacquet–Langlands correspondence showing that functoriality was capable of explaining very precisely how automorphic forms for related to those for quaternion algebras. This book applied the adelic trace formula for and quaternion algebras to do this. Subsequently, James Arthur, a student of Langlands while he was at Yale, successfully developed the trace formula for groups of higher rank. This has become a major tool in attacking functoriality in general, and in particular has been applied to demonstrating that the Hasse–Weil zeta functions of certain Shimura varieties are among the -functions arising from automorphic forms.