A Program for Geometric Arithmetic

As stated above, (A) is aimed at establishing a Non-Abelian Class Field Theory. The starting point here is the following classical result: Over a compact Riemann surface, a line bundle is of degree zero if and only if it is flat, i.e., induced from a representation of fundamental group of the Riemann surface. Clearly, being a bridge connecting divisor classes and fundamental groups, this result may be viewed as and is indeed a central piece of the classical (abelian) class field theory. (See e.g., [Hilbert] and [Weil].) Thus it is then only natural to give a non-abelian generalization of it in order to offer a non-abelian class field theory. This was first done by Weil. In his fundamental paper on generalization of abelian functions [Weil1], Weil showed that over a compact Riemann surface, a vector bundle is of degree zero if and only if it is induced from a representation of fundamental group of the surface. Thus far, two new aspects naturally emerge. That is, unitary representations and non-compact Riemann surfaces, reflecting finite quotients of Galois groups and ramifications in Class Fields Theory, CFT for short, respectively: In a (complex) representation class of a finite group, there always exists a unitary one, while a discussion for compact Riemann surfaces results only unramified CFT. Thus mathematics demands new results to couple with them. As it is well-known that to this end we then have (i) Mumford’s stability of vector bundles in terms of intersection; (ii) Narasimhan-Seshadri’s correspondence; and (iii) Seshadri’s parabolic analog of (i) and (ii). That is to say, now the above result of Weil is further refined to the follows: Over (punctured) Riemann surfaces, (Seshadri) equivalence classes of semi-stable parabolic bundles of parabolic degree zero correspond naturally in one-to-one to isomorphism classes of unitary representations of fundamental groups.