Introduction to the Langlands Program

This article is an introduction to automorphic forms on the adeles of a linear reductive group over a number field. The first half is a summary of aspects of local and global class field theory, with emphasis on the local Weil group, the L functions of Artin and Hecke, and the role of Artin reciprocity in relating the two kinds of L functions. The first half serves as background for the second half, which discusses some structure theory for reductive groups, the definitions of automorphic and cusp forms, the Langlands L group, L functions, functoriality, and some conjectures. Much of the material in the second half may be regarded as a brief introduction to the Langlands program. There are ten sections: