Langlands Reciprocity: L-Functions, Automorphic Forms, and Diophantine Equations
LANGLANDS RECIPROCITY: L-FUNCTIONS, AUTOMORPHIC FORMS, AND DIOPHANTINE EQUATIONS MATTHEW EMERTON Abstract. This chapter gives a description of the theory of reciprocity laws in algebraic number theory and its relationship to the theory of L-functions. It begins with a historical overview of the work of Euler, Dirichlet, Riemann, Dedekind, Hecke, and Tate. It then proceeds through the development of class field theory, before turning to the theory of Artin L-functions, and then of general motivic L-functions. It culminates in a description of Langlands's very general reciprocity conjecture, relating arbitrary motivic L-functions to the automorphic L-functions that he defined. The reciprocity conjecture is closely bound up with another fundamental conjecture of Langlands, namely his functoriality conjecture, and we give some indication of the relationship between the two. We conclude with a brief discussion of some recent results related to reciprocity for curves of genus one and two. The goal of this chapter is to discuss the theory of ζ- and L-functions from the viewpoint of number theory, and to explain some of the relationships (both established and conjectured) with Langlands's theory of automorphic L-functions and functoriality. The key conjectural relationship, also due to Langlands, is that of reciprocity. The word reciprocity is a storied one in number theory, beginning with the celebrated theorem of quadratic reciprocity, discovered by Euler and Legendre, and famously proved by Gauss (several times over, in fact, and many more times since by others). The meaning of reciprocity, both in its precise technical usage and in its more general, intuitive sense, has evolved in a complicated way since it was coined (by Legendre) in the context of quadratic reciprocity in the 18th century.
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