A prologue to ``Functoriality and reciprocity''. I
``Functoriality and Reciprocity'' is the title of an essay that Langlands promises us to write, the ``Prologue'' anticipates the essay and here is the first Part of it. In the prologue the author explains his ideas on the concepts of functoriality and reciprocity. The notion of functoriality is described using the formalism of mock Tannakian categories. The fields over which the theories are to be built up are: algebraic number fields; function fields of algebraic curves over a finite field; function fields of complete nonsingular curves over the complex field and the local fields corresponding to these three types of global fields. For reciprocity a motivic Galois group will be needed (Langlands speaks of automorphic and motivic Galoisian groups). Regarding the methods which may lead to proofs of the conjectures, the author refers to preceding papers (which can be found on his homepage). In this prologue there is some discussion of the methods for the different types of fields. In the local theory over the real field or over \(p\)-adic fields one needs an extension of representation theory to the Arthur class of representations and a theory of stable transfer. In the global theory for algebraic number fields the stable trace formula and Poisson summation will be essential instruments in the proof of functoriality. In this context \(L\)-functions are an important object of study. We have the notion of \(L\)-function attached to a motive and the notion of \(L\)-function attached to a stable class of automorphic representations. Conjecturally there should be a relation between the two, i.e., between motivic \(L\)-functions and automorphic \(L\)-functions. From the geometric theory (over the complex field) the abelian case, i.e., the case of GL\((1)\), is treated in the prologue. Automorphic characters are constructed using meromorphic differentials on the Riemann surface. The geometric theory for general groups will be continued in Part II.
scientific article; zbMATH DE number 6136444