Non-Abelian L-Functions For Number Fields

In this paper we introduce non-abelian zeta functions and more generally non-abelian L-functions for number fields, based on geo-arithmetical cohomology, geo-arithmetical truncation and Langlands’ theory of Eisenstein series. More precisely, in Chapter I, we start with a new yet natural geo-arithmetical cohomology and a geo-arithmetical stability in order to define genuine non-abelian zeta functions for number fields. Then, in Chapter II, using examples for the field of rationals, we explain a relation between these non-abelian zeta functions and Eisenstein series, and point out where non-abelian contributions of such non-abelian zetas come. After that, in Chapter III, first, we give a more general geo-arithmetical truncation, and compare it with Arthur’s analytic truncation. Then, we define our general non-abelian L-functions as the integration of Eisesntein series associated to L-automorphic forms over the moduli spaces obtained from just mentioned general geo-arithmetic truncations, and to expose basic properties for these non-abelian L-functions. In Chapter IV, we, along with the line of Rankin-Selberg(-Langlands-Arthur) method, calculate the so-called abelian part of our non-abelian L-functions. Finally, in an Appendix, we establish a non-abelian class field theory for function fields over complex numbers to justify our approach to non-abelian Reciprocity Law.