Non-Abelian Zeta Functions For Function Fields

In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. More precisely, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields, by a ‘weighted count’ on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. Then we define non-abelian L-functions for curves over finite fields using integrations of Eisenstein series associated to L-automorphic forms over certain generalized moduli spaces. In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. It consists of two chapters. More precisely, in Chapter I, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields. This is achieved by a ‘weighted count’ on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. We justify our construction by establishing basic properties for these new zetas such as functional equation and rationality, and show that if only line bundles are involved, our newly defined zetas coincide with Artin’s Zeta. All this, in particular, the rationality, then leads naturally to our definition of (global) non-abelian zeta functions (for curves defined over number fields), which themselves are justified by a convergence result. We end this chapter with a detailed study on rank two non-abelian zeta functions for genus two curves, based on what we call infinitesimal structures of Brill-Noether loci (and Weierstrass points). In Chapter II, we begin with a similar construction for the field of rationals to motivate what follows. In particular, we show that there is an intrinsic relation between our non-abelian zeta functions and Eisenstein series. Due to this, instead of introducing general non-abelian L-functions for curves defined over finite fields with more general test functions (as what Tate did in his Thesis for abelian L-functions), we then define non-abelian L-functions for curves over finite fields as integrations of Eisenstein series associated to L-automorphic forms over certain generalized moduli spaces. Here geometric truncations play a key role. Basic properties for these non-abelian L-functions, such as meromorphic continuation, functional equations and singularities, are established as well, based on the theory of Eisenstain series of Langlands and Morris. We end this chapter by establishing a closed formula for what we call the abelian parts of non-abelian Lfunctions associated with Eisenstein series for cusp forms, via the Rankin-Selberg method, motivated by a formula of Arthur and Langlands. This work is an integrated part of our vast yet still under developing Program for Geometric Arithmetic [We1], and is motivated by our new non-abelian L-functions for number fields [We2] in connection with non-abelian arithmetic aspects of global fields. Chapter I. Non-Abelian Zeta Functions This consists of two aspects: construction and justification. For the construction, we first introduce a new type of zeta functions for curves defined over finite fields using the corresponding moduli spaces of semi-stable vector bundles. We show that these new zeta functions are indeed rational and satisfy certain functional equation, based on vanishing theorem, (duality, Riemann-Roch theorem) for cohomologies of semistable vector bundles. Based on this, in particular, the rationality, we then introduce global non-abelian zeta functions for curves defined over number fields, via the Euler product formalism. Moreover, we establish a convergence result for our Euler products using the Clifford Lemma, an ugly yet quite explicit formula for 1 local non-abelian zeta functions, a result of (Harder-Narasimhan) Siegel about quadratic forms, and Weil’s theorem on Riemann Hypothesis for Artin zeta functions. As for the 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, global and local, coincide with the classical Artin zeta functions for curves defined over finite fields and Hasse-Weil zeta functions for curves defined over number fields respectively. Moreover, as concrete examples, we compute rank two zeta functions for genus two curves by studying Weierstrass points and non-abelian Brill-Noether loci in terms of what we call their infinitesimal structures. I.1. Local Non-Abelian Zeta Functions for Curves In this section, we introduce our non-abelian zeta functions for curves defined over finite fields. Basic properties for these non-abelian zeta functions, such as meromorphic extensions, rationality and functional equations, are established. 1.1. Moduli Spaces of Semi-Stable Bundles 1.1.1. Semi-Stable Bundles. Let C be a regular, reduced and irreducible projective curve defined over an algebraically closed field k̄. Then according to Mumford [Mu], a vector bundle V on C is called semi-stable (resp. stable) if for any proper subbundle V ′ of V , μ(V ) := d(V ) r(V ′) ≤ (resp. < ) d(V ) r(V ) =: μ(V ). Here d denotes the degree and r denotes the rank. Proposition. Let V be a vector bundle over C. Then (a) ([HN]) there exists a unique filtration of subbundles of V , the Harder-Narasimhan filtration of V , {0} = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vs−1 ⊂ Vs = V such that all Vi/Vi−1 are semi-stable and for 1 ≤ i ≤ s− 1, μ(Vi/Vi−1) > μ(Vi+1/Vi); (b) (see e.g. [Se]) if moreover V is semi-stable, there exists a filtration of subbundles of V , a Jordan-Hölder filtration of V , {0} = V t+1 ⊂ V t ⊂ · · · ⊂ V 1 ⊂ V 0 = V such that for all 0 ≤ i ≤ t, V /V i+1 is stable and μ(V /V ) = μ(V ). Moreover, the associated graded bundle Gr(V ) := ⊕i=0V /V , the (Jordan-Hölder) graded bundle of V , is determined uniquely by V . 1.1.2. Moduli Space of Stable Bundles. Following Seshadri, two semi-stable vector bundles V andW are called S-equivalent, if their associated Jordan-Hölder graded bundles are isomorphic, i.e., Gr(V ) ≃ Gr(W ). Applying Mumford’s general result on geometric invariant theory, Narasimhan and Seshadri proved the following Theorem. (See e.g. [NS] and [Se]) Let C be a regular, reduced, irreducible projective curve of genus g ≥ 2 defined over an algebraically closed field. Then over the set MC,r(d) (resp. MC,r(L)) of S-equivalence classes of rank r and degree d (resp. rank r and determinant L) semi-stable vector bundles over C, there is a natural normal, projective (r(g− 1)+1)-dimensional (resp. (r − 1)(g− 1)-dimensional) algebraic variety structure. Remark. In this paper, we always assume that the genus of g is at least 2. For elliptic curves, whose associated moduli spaces are very special, please see [We3]. 1.1.3. Rational Points. Now assume that C is defined over a finite field k. It makes sense to talk about k-rational bundles over C, i.e., bundles which are defined over k. Moreover, from geometric invariant theory, projective varieties MC,r(d) are defined over a certain finite extension of k; and if L itself is defined over k, the same holds for MC,r(L). Thus it makes sense to talk about k-rational points of these moduli spaces too. The relation between these two types of rationality is given by Harder-Narasimhan based on a discussion about Brauer groups: Proposition. ([HN]) Let C be a regular, reduced, irreducible projective curve of genus g ≥ 2 defined over a finite field k. Then there exists a finite field Fq such that for all d (resp. all k-rational line bundles L), 2 the subset of Fq-rational points of MC,r(d) (resp. MC,r(L)) consists exactly of all S-equivalence classes of Fq-rational bundles in MC,r(d) (resp. MC,r(L)). From now on, without loss of generality, we always assume that the finite fields Fq (with q elements) satisfy the property stated in the Proposition. Also for simplicity, we write MC,r(d) (resp. MC,r(L)) for MC,r(d)(Fq) (resp. MC,r(L)(Fq)), the subset of Fq-rational points, and call them moduli spaces by an abuse of notations. Clearly these sets are all finite. 1.2. Local Non-Abelian Zeta Functions 1.2.1. Definition. Let C be a regular, reduced, irreducible projective curve of genus g ≥ 2 defined over the finite field Fq with q elements. Define the rank r non-abelian zeta function ζC,r,Fq(s) of C by setting ζC,r,Fq(s) := ∑ V ∈[V ]∈MC,r(d),d≥0 q (C,V ) − 1 #Aut(V ) · (q) , Re(s) > 1. Proposition. With the same notation as above, ζC,1,Fq(s) is nothing but the classical Artin zeta function ζC(s) for curve C. That is to say, ζC,1,Fq(s) = ∑ D≥0 1 N(D)s =: ζC(s) Re(s) > 1. Here D runs over all effective divisors of C, and N(D) := q with d(ΣPnPP ) := ΣPnPd(P ). Proof. By definition, the classical Artin zeta function ([A], [Mo]) for C is given by ζC(s) := ∑ D≥0 1 N(D)s . Thus by first grouping effective divisors according to their rational equivalence classes D, then taking the sum on effective divisors in the same class, we obtain ζC(s) = ∑