Riemann-Roch Theorem, Stability and New Zeta Functions for Number Fields

In this paper, we introduce new non-abelian zeta functions for number fields and study their basic properties. Recall that for number fields, we have the classical Dedekind zeta functions. These functions are usually called abelian, since, following Artin, they are associated to one dimensional representations of Galois groups; moreover, following Tate and Iwasawa, they may be constructed as integrations over abelian spaces, i.e., GL1 over adelic space AF for F . Thus to define non-abelian versions of zeta functions for number fields, naturally, mathematicians use higher dimensional representations of Galois groups and/or algebraic groups. This turns to be extremely important and very fruitful. As a result, now we have the so-called Artin L-functions, automorphic Lfunctions, etc.. However in this paper, we are not going to touch any part of such a fascinating representation oriented number theoretical theory. Instead, we do it more geometrically. It consists of two aspects, i.e., the one for integrands and the one for integration domains, along with the pioneer works of Tata and Iwasawa. To construct quite satisfied integrands, we need a completed cohomology theory, form which RiemannRoch theorem holds. For this purpose, in Part I of this paper, for a number field F with KF a canonical element of degree log |∆F |, we first introduce an adelic version of vector bundles E over number fields; then, we define the 0-th cohomology h(F,E) and the 1-st cohomology h(F,E) for these vector bundles which satisfy the standard duality h(F,E) = h(F,E ⊗ KF ); and finally, we prove the following Riemann-Roch theorem for them: h(F,E) − h(F,E) = deg(E)− rank(E) 2 · log |∆F |.