Epilogue: Outlook to Number Theory

Abstract

Representations of groups show up in many places in Number Theory, for instance as representations of Galois groups. There seems to be no doubt that the relationship of the two topics culminates in the Langlands Program, which (roughly said) seeks to establish a correspondence between Galois and automorphic representations. Many eminent mathematicians have worked and work on this program, and it is now even interesting for physicists; see Frenkel’s Lectures on the Langlands Program and Conformal Field Theory [Fr]. We cannot dare to go into this here, but we will try to introduce at least some initial elements by presenting some representation spaces for the special groups we treated in our examples, consisting of, or at least containing, theta functions, and modular and automorphic forms. And we will also introduce the notions of zeta and L-functions, which ultimately are the foundations of the bridge between the Galois and automorphic representations. There are many useful books available. We can only cite some of them, which by now are classic: Representation Theory and Automorphic Functions by Gelfand, Graev and Pyatetskii-Shapiro [GGP], Automorphic Forms on GL(2) by Jacquet and Langlands [JL], Automorphic Forms on Adele Groups by Gelbart [Ge], Analytic Properties of Automorphic L-Functions by Gelbart and Shahidi [GS], Automorphic Forms and Representations by Bump [Bu], Theta Functions by Igusa [Ig], The Weil representation, Maslov index and Theta Series by Lion and Vergne [LV], Fourier Analysis on Number Fields by Ramakrishnan and Valenza [RV], and Mumford’s Tata Lectures on Theta I, II, and III [Mu].