Constructions of Non-Abelian Zeta Functions for Curves

In this paper, we initiate a geometrically oriented study of local and global non-abelian zeta functions for curves. This consists of two parts: construction and justification. For the construction, we first use moduli spaces of semi-stable bundles to introduce a new type of zeta functions for curves defined over finite fields. Then, we prove that these new zeta functions are indeed rational and satisfy the functional equation, based on vanishing theorem, duality, Riemann-Roch theorem for cohomology of semi-stable vector bundles. With all this, we next introduce certain global non-abelian zeta functions for curves defined over number fields, via the Euler product formalism. Finally we establish the convergence of our Euler products, from the Clifford Lemma, an ugly yet explicit formula for local nonabelian zeta functions, a result of (Harder-Narasimhan) Siegel about quadratic forms, and Weil’s theorem on Riemann Hypothesis of Artin zeta functions. As for justification, surely, we check that when only line bundles are involved, (so moduli spaces of semi-stable bundles are nothing but the standard Picard groups), our (new) zeta functions coincide with the classical Artin zeta functions for curves over finite fields and Hasse-Weil zeta functions for curves over number fields respectively. Moreover we compute the rank two zeta functions for genus two curves by studying the so-called non-abelian Brill-Noether loci and their infinitesimal structures. This is indeed a qutie interesting, and in general, should be a very important aspect of the theory: We not only need to precisely describe all of the Brill-Noether loci but the so-called associated infinitesmal structures attached to all Seshadri equivalence classes, in which Weierstrass points appear naturally. This work is motivated by our studies on new non-abelian zeta functions for number fields and Tamagawa measures associated to the Weil-Petersson metrics on moduli spaces of stable vector bundles. So related conjectures, or better, working hypothesis, are proposed. We hope that our non-abelian zeta functions, which do not really follow the present style of the theory of zeta functions, are acceptable and hence play a certain role in exploring the non-abelian aspect of arithmetic of curves.