Higher-rank zeta functions and SLn-zeta functions for curves

Lin Weng; Don Zagier

doi:10.1073/pnas.1912501117

Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved January 15, 2020 (received for review July 19, 2019)

Significance

Almost 100 years ago, Artin defined an analog of the famous Riemann zeta function for curves (one-dimensional varieties) over a finite field. In 2005, L.W. defined two different series of “higher zeta functions” for curves over finite fields that both generalized Artin’s zeta functions, one being defined geometrically and the other using advanced concepts from group representation theory, and conjectured that they always coincide. In this paper this conjecture is proved by giving a formula for one of the two series and showing that it agrees with the formula for the other series proved a few years ago by Sergey Mozgovoy and Markus Reineke.

Abstract

In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case when G=SLn and P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding to n=1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.

In refs. 1 and 2, a nonabelian zeta function ζX,n(s)=ζX/Fq,n(s) was defined for any smooth projective curve X over a finite field Fq and any integer n≥1 byζX,n(s)=∑[V]|H0(X,V) \{0}||Aut(V)| q−deg(V)s (R(s)>1),[1]where the sum is over the moduli stack of Fq-rational semistable vector bundles V of rank n on X with degree divisible by n. Using the Riemann–Roch, duality, and vanishing theorems for semistable bundles, it was shown that ζX,n(s) agrees with the usual Artin zeta function ζX(s) of X/Fq if n=1; that it has the form PX,n(T)/(1−T)(1−qnT) for some polynomial PX,n(T) of degree 2g in T, where g is the genus of X and T=q−ns; and that it satisfies the functional equationζ^X,n(1−s)=ζ^X,n(s) , where ζ^X,n(s) ≔ qn(g−1)s⋅ ζX,n(s).It was also conjectured that ζX,n(s) satisfies the Riemann hypothesis (i.e., that all of its zeros have real part 1/2). In a companion paper (3), explicit formulas for ζX,n(s) and a proof of the Riemann hypothesis were given for the case when g=1.

On the other hand, in refs. 2 and 4, a different approach to zeta functions for curves led to the so-called group zeta function ζ^X G,P(s) of X/Fq, associated to a connected split algebraic reductive group G and its maximal parabolic subgroup P. The precise definition, which is based on the theory of periods, is recalled in Section 2. In this paper, we are interested in the special case when G=SLn and P=Pn−1,1, the subgroup of SLn consisting of matrices whose final row vanishes except for its last entry, and we then write simply ζ^X SLn(s) for ζ^X G,P(s). Our main result is a proof of the following theorem, which was conjectured in ref. 2 (“special uniformity conjecture”).

Theorem 1.

The zeta functions ζ^X,n(s) and ζ^X SLn(s) coincide for all n≥1.

Theorem 1 should be regarded as a joint result of L.W. and D.Z. and of Sergey Mozgovoy and Markus Reineke (5), because the proof proceeds by comparing a formula for ζ^X SLn(s) established here with a formula for ζ^X,n(s) given in their paper. Specifically, the proof consists of three steps:

1) By analyzing the definition of ζ^X G,P(s) for G=SLn, P=Pn−1,1, we will prove an explicit formula, giving ζ^X SLn(s) as a linear combination of the functions ζ^X(ns−k) for 0≤k<n with rational functions of T as coefficients. The calculation is given in Sections 3–5.
2) In ref. 5, as recalled in Section 6, using the theory of Hall algebras and wall-crossing techniques, a formula for ζ^X,n(s) of the same general shape is proved.
3) A short calculation, given in Section 7, shows that the two formulas agree.

The explicit formula is not very complicated, and we can state it here. Motivated by the Siegel–Weil formula for the total mass of vector bundles V of rank n and degree 0 on X (i.e., the number of such Vs, weighted by the inverse of the number of their automorphisms), and to make a proper normalization, we define numbers v^k (k≥1) inductively byv^k=lims→1(1−q1−s) ζ^X(s) if k = 1 , ζ^X(k) v^k−1 if k≥2 ,[2]where ζ^X(s)=qs(g−1)ζX(s). Furthermore, as in ref. 3—where these functions were introduced for the purpose of writing down in a more structural way the nonabelian rank n zeta functions for elliptic curves over finite fields—we define rational functions Bk(x) (k≥0) either inductively by the formulasBk(x)= 1 if k=0 ,∑m=1kv^m Bk−m(qm)1−qmx if k≥1 ,[3]or in closed form (if k≥1) byBk(x)=∑p=1k∑k1,…,kp>0k1+⋯+kp=kv^k1…v^kp(1−qk1+k2)…(1−qkp−1+kp)⋅11−qkpx.[4]Then the formula that we will establish for ζ^X SLn(s) can be stated as follows:

Theorem 2.

With the above notations, we haveζ^X SLn(s)=qn2(g−1) ∑k=0n−1 Bk(qns−k) Bn−k−1(qk+1−ns) ζ^X(ns−k).[5]

Remarks:

1) In the definition Eq. 1 of the nonabelian zeta function ζX,n(s), vector bundles used are assumed to be of degrees divisible by the rank n. This definition is motivated by a work of Drinfeld (6) on counting supercuspidal representations in rank 2 and also because if we summed over all degrees as was originally done in ref. 1, then the functional equation would still hold but the Riemann hypothesis would not.
2) The analog of Theorem 1 for the case of number fields rather than function fields was proved by L.W. several years ago by totally different techniques, using the theory of Eisenstein series and Arthur trace formulas (combine the “global bridge” on p. 295 and the discussion on p. 305 of ref. 7 with the formulas on p. 284 of ref. 8 and on p. 197 of ref. 4).
3) A proof of Theorem 1 for the cases n=2 and n=3 was given in ref. 5, at a time when the current paper was still in the preprint stage.

2. Zeta Functions for (G,P)

Let G be a connected split reductive algebraic group of rank r with a fixed Borel subgroup B and associated maximal split torus T (over a base field). Denote byV, ⟨⋅,⋅⟩, Φ=Φ+∪Φ−, Δ={α1,…,αr}, ϖ≔{ϖ1,…,ϖr}, Wthe associated root system. That is, V is the real vector space defined as the R span of rational characters of T and, as usual, is equipped with a natural inner product ⟨⋅,⋅⟩, with which we identify V with its dual V*; and Φ+⊂V is the set of positive roots, Φ−≔−Φ+ the set of negative roots, Δ⊂V the set of simple roots, ϖ⊂V the set of fundamental weights, and W the Weyl group. By definition, the fundamental weights are characterized by the formula ⟨ϖi,αj∨⟩=δij for i,j=1,2,…,r, where α∨≔2⟨α,α⟩ α denotes the coroot of a root α∈Φ. We also define the Weyl vector ρ by ρ=12∑α∈Φ+α and introduce a coordinate system on V (with respect to the base {ϖ1,…,ϖr} of V and the vector ρ) by writing an element λ∈V in the formλ = ∑j=1r(1−sj)ϖj=ρ−∑j=1rsjϖj,thus fixing identifications of V and VC=V⊗RC with Rr and Cr. In addition, for each Weyl element w∈W, we set Φw≔Φ+ ∩ w−1Φ−, i.e., the collection of positive roots whose w images are negative.

As usual, by a standard parabolic subgroup, we mean a parabolic subgroup of G that contains the Borel subgroup B. From Lie theory (e.g., ref. 9), there is a one-to-one correspondence between standard parabolic subgroups P of G and subsets ΔP of Δ. In particular, if P is maximal, we may and will write ΔP=Δ\{αp} for a certain unique p=p(P)∈{1,…,r}. For such a standard parabolic subgroup P, denote by VP the R span of rational characters of the maximal split torus TP contained in P, by VP* its dual space and by ΦP⊂VP the set of nontrivial characters of TP occurring in the space V. Then, by standard theory of reductive groups (e.g., ref. 10), VP admits a canonical embedding in V (and VP* admits a canonical embedding in V*), which is known to be orthogonal to the fundamental weight ϖp, and hence ΦP can be viewed as a subset of Φ. Set ΦP+=Φ+ ∩ ΦP, ρP=12∑α∈ΦP+α, and cP=2⟨ϖp−ρP,αp∨⟩.

Now, let X be an integral regular projective curve of genus g over a finite field Fq. In ref. 2, motivated by the study of zeta functions for number fields,^† for a connected split reductive algebraic group G and its standard parabolic subgroup P as above (defined over the function field of X), L.W. defined the period of G for X byωXG(λ)≔∑w∈W1∏α∈Δ(1−q−⟨wλ−ρ,α∨⟩)∏α∈Φwζ^X(⟨λ,α∨⟩)ζ^X(⟨λ,α∨⟩+1)and the period of (G,P) for X byωXG,P(s)≔Res⟨λ−ρ, α∨⟩=0, α∈ΔPωXG(λ)sp=s=Ressr=0⋯Ressp+1=0Ressp−1=0⋯Ress1=0 ωXG(λ)sp=s,where s is a complex variable^‡ and where for the last equality we used the fact that ⟨ρ,α∨⟩=1 for all α∈Δ and the relation that ⟨ϖi,αj∨⟩=δij for all i, j∈{1,…,r}. As proved in refs. 2 and 11, the ordering of taking residues along singular hyperplanes ⟨λ−ρ,α∨⟩=0 for α∈ΔP does not affect the outcome, so that the definition is independent of the numbering of the simple roots.

To get the zeta function associated to (G,P) for X, certain normalizations should be made. For this purpose, write ωXG(λ)=∑w∈WTw(λ), where, for each w∈W,Tw(λ) ≔ 1∏α∈Δ(1−q−⟨wλ−ρ,α∨⟩)∏α∈Φwζ^X(⟨λ,α∨⟩)ζ^X(⟨λ,α∨⟩+1).The zeta function of X associated to (G,P) will be defined in terms of the residue Res⟨λ−ρ, α∨⟩=0, α∈ΔPTw(λ).

We care only about those elements w∈W (we call them special) that give nontrivial residues, namely, those satisfying the condition that Res⟨λ−ρ, α∨⟩=0, α∈ΔPTw(λ)≢0. This can happen only if all singular hyperplanes are of one of the following two forms:

1) ⟨wλ−ρ,α∨⟩=0 for some α∈Δ, giving a simple pole of the rational factor 1∏α∈Δ(1−q−⟨wλ−ρ,α∨⟩);
2) ⟨λ,α∨⟩=1 for some α∈Φw, giving a simple pole of the zeta factor ζ^X(⟨λ,α∨⟩).

For special w∈W and (k,h)∈Z2, following ref. 11 (also ref. 2) we defineNP,w(k,h)≔#{α∈w−1Φ− : ⟨ϖp,α∨⟩=k, ⟨ρ,α∨⟩=h}MP(k,h)≔maxw specialNP,w(k,h−1)−NP,w(k,h).=NP,w0(k,h−1)−NP,w0(k,h),[6]where w0 is the longest element of the Weyl group and where the last equality is corollary 8.7 of ref. 12. Note that MP(k,h)=0 for almost all but finitely many pairs of integers (k,h), so it makes sense to introduce the productDXG,P(s) ≔ ∏k=0∞ ∏h=2∞ζ^X(kn(s−1)+h)MP(k,h).[7]Following refs. 2 and 4, we define the zeta function of X associated to (G,P) byζ^XG,P(s) ≔ q(g−1)dimNu(B)⋅DG,P(s)⋅ωXG,P(s).[8]Here Nu(B) denotes the nilpotent radical of the Borel subgroup B of G.

Remark:

For special w∈W, even after taking residues, there are some zeta factors ζ^X(ks+h) left in the denominator of Res⟨λ−ρ, α∨⟩=0, α∈ΔPTw(λ). The reason for introducing the factor DXG,P(s) in our normalization of the zeta functions, based on formulas in refs. 2 and 11, is to clear up all of the zeta factors appearing in the denominators associated to special Weyl elements.

3. Specializing to SLn

From now on, we specialize to the case when G is the special linear group SLn and P is the maximal parabolic subgroup Pn−1,1 consisting of matrices whose final row vanishes except for its last entry, corresponding to the ordered partition (n−1)+1 of n. Our purpose is to study the zeta function of X associated to SLn:ζ^XSLn(s)≔ ζ^XSLn, Pn−1,1(s).[9]As usual, we realize the root system An−1 associated to SLn as follows. Denote by {e1,…,en} the standard orthonormal basis of the Euclidean space Rn. The positive roots are given by Φ+≔ {ei−ej∣1≤i<j≤n}, the simple roots by Δ={α1≔e1−e2,…,αn−1≔en−1−en}, and the Weyl vector by ρ=∑j=1nn+1−2j2 ej. We identify the Weyl group W with Sn, the symmetric group on n letters, by the assignment w↦σw, where w(ei−ej)=eσw(i)−eσw(j). For convenience, we also write the corresponding ΔP, ΦP+, ρP, ϖP, and cP simply as Δ′, Φ′+, ρ′,ϖ′, and c′, respectively. We haveΔ′={α1,…,αn−2}, Φ′+={ei−ej:1≤i<j≤n−1},ρ′=∑j=1n−1n−2j2 ej , ϖ′=ϖn−1=1n ∑j=1nej−en.In addition, ⟨ρ,α⟩=1 for all α∈Δ, and α∨=α, ⟨ρ,α⟩=1 for all α∈Φ+. Henceρ′=ρ−n2 ϖ′ , c′=2⟨ϖ′−ρ′,αn−1⟩=n.Accordingly, for positive roots αij≔ei−ej∈Φ+, we have⟨ρ,αij⟩=j−i, ⟨ϖ′,αij⟩=δjn−δin,[10]and, for λs≔(ns−n)ϖ′+ρ,⟨λs,αij⟩= j−i if i, j ≠ n,ns−i if j=n,−ns+j if i=n.[11]To write down the zeta function ζ^XSLn(s) explicitly, we express the multiple residues in the periods of (SLn,Pn−1,1) as a single limit, after multiplying by suitable vanishing factors (to the period of SLn). Indeed, since ⟨λs−ρ,αn−1⟩=ns−n, andlimλ→λs1−q−⟨λ−ρ,α⟩ ≡ 0 (∀ α∈Δ′),[12]we haveωXSLn, Pn−1,1(s) =limλ→λs∏α∈Δ′(1−q−⟨λ−ρ,α⟩)⋅ωXSLn(λ).[13]Recall that ωXSLn(λ)=∑w∈WTw(λ). Accordingly, to pin down the nonzero contributions for the terms appearing in the limit, we should consider, for a fixed w∈W, the limit limλ→λs∏α∈Δ′(1−q−⟨λ−ρ,α⟩)⋅Tw(λ) or, equivalently, for a fixed σ∈Sn(≃W), the functionLσ(s)=limλ→λs∏α∈Δ′(1−q−⟨λ−ρ,α⟩)∏β∈Δ(1−q−⟨σλ−ρ,β⟩) ∏α∈Φ+, σ(α)<0ζ^X(⟨λ,α⟩)ζ^X(⟨λ,α⟩+1).[14]For this limit Lσ(s) to be nonzero, by Eq. 12, there should be a complete cancellation of all of the factors (1−q−⟨λ−ρ,α⟩) in the numerator of the first term in Eq. 14 that vanish at λ=λs with either

1) factors 1−q−⟨σλs−ρ,β⟩ appearing in the denominator of the first term in Eq. 14 or else
2) the poles at λ=λs of factors ζ^X⟨λ,α⟩ appearing in the numerator of the second term in Eq. 14 for which ⟨λs,α⟩=1.

Since ⟨⋅ ,⋅⟩ is σ invariant, for α∈Δ′, by Eq. 10, ⟨σλs−ρ,α⟩=⟨λs,σ−1α⟩−1. Hence, for Lσ(s) to have a nonzero contribution to ωX(SLn, Pn−1,1)(s), the union ofAσ≔ α∈Δ′:σα∈Δ and Bσ≔ α∈Δ′:σα<0[15]must be of cardinality n−2. Call such σ∈Sn special and denote the collection of special permutations by Sn0. Clearly, for σ∈Sn, we have Aσ∪Bσ⊂Δ′, and Aσ∪Bσ=Δ′ if and only if σ∈Sn0. That is to say, the limit Lσ(s) corresponding to the permutation σ∈Sn can be nonzero only if σ is special, and in this case, we have Δ′=Aσ⊔Bσ. This then completes the proof of the following:

Lemma 3.

With the notations above,ωXSLn, Pn−1,1(s)=∑σ∈Sn0Lσ(s).[16]Here σ∈Sn0 if and only if Aσ∪Bσ=Δ′.

The next lemma describes Lσ(s) for special permutations σ.

Lemma 4.

For σ∈Sn0, setRσ(s) =∏1≤k≤n−1σ−1αk∉Δ′ 1−q−⟨σλs−ρ,αk⟩, ζ^σ[n](s) =∏1≤i≤n−1σ(i)>σ(n)ζ^X(⟨λs,αin⟩)ζ^X(⟨λs,αin⟩+1),ζ^σ[<n](s)≔∏1≤k≤n−2σ(k)>σ(k+1)1−q−⟨λ−ρ,αk⟩ ⋅∏1≤i<j≤n−1σ(i)>σ(j)ζ^X(⟨λ,αij⟩)ζ^X(⟨λ,αij⟩+1)λ=λs.ThenLσ(s)=1Rσ(s)⋅ζ^σ[n](s)⋅ζ^σ[<n](s).[17]

Proof:

This is obtained by regrouping the terms of Eq. 14 for special permutation σ∈Sn0, following the discussions above. We first cancel the terms in the numerator of the first factor in Eq. 14 for α∈Aσ with the corresponding terms in the denominator for β=σα. The first factor 1/Rσ(s) in Eq. 17 is the value at λ=λσ of the product of the remaining terms β∈Δ\σAσ in this denominator. The second factor ζ^σ[n](s) in Eq. 17 is the value at λ=λσ of the product of the terms in the second factor in Eq. 14 for α∉Φ′+; i.e., α=ei−en>0. The third factor ζ^σ[<n](s) in Eq. 17, which can also be writtenζ^σ[<n](s)= ∏α∈Bσ(1−q−⟨λ−ρ,α⟩)⋅∏α∈Φ′+σ(α)<0ζ^X(⟨λ,α⟩)ζ^X(⟨λ,α⟩+1)λ=λs,is obtained by collecting all of the remaining zeta factors and rational factors appearing in the numerator.□

The terms occurring in ζ^σ[<n](s) are of two types: For α∈Bσ we must combine the quantities (1−q−⟨λ−ρ,αk⟩) and ζ^X(⟨λ,αij⟩)ζ^X(⟨λ,αij⟩+1) before taking the limit as λ→λs because the first one has a zero and the second one has a pole, while in the remaining zeta quotients from the second term in Eq. 17, corresponding to α∈Φ′+ \ Bσ, we could simply substitute λ=λs instead of taking a limit. We can say this differently as follows. By abuse of notation we write simply ζ^X(1) for the limit as s→1 of (1−q1−s)ζ^X(s). (It should be written v^1, as defined in Eq. 2, but the “ζ^X(1)” notation will let us write more uniform formulas.) Then the definition of ζ^σ[<n](s) can be rewritten using the first equation in Eq. 11 asζ^σ[<n](s)=∏k≥1ζ^X(k)ζ^X(k+1)mσ(k)=∏k≥1ζ^X(k)nσ(k),[18]wheremσ(k)=∑1≤i<j≤n−1σ(i)>σ(j), j−i=k 1=#{α∈Φ′+:σα<0,⟨ρ,α⟩=k}[19]andnσ(k)=mσ(k)−mσ(k−1) , nσ(1) = mσ(1)=#Bσ.[20]Eq. 18 gives an explicit formula for the third factor in Eq. 17, which, as one sees, does not depend on s at all. The other two factors in Eq. 17, which do depend on s, are computed later, in Section 5.

Lemmas 3 and 4 calculate the third factor ωXG,P(s) in the definition Eq. 8 of ζ^XG,P(s) in the special case G=SLn, P=Pn−1,1, but since some of the numbers nσ(k) in Eq. 18 may be negative, the expression for this factor may still contain some zeta values in its denominator. These zeta values in the denominator will be canceled when we include the second factor DG,P(s) in Eq. 8. Our next task is therefore to evaluate this expression explicitly in the case (G,P)=(SLn, Pn−1,1). Then the formulas for DG,P(s) and ζ^XG,P(s) can be written explicitly as follows:

Lemma 5.

We haveDSLn,Pn−1,1(s)=∏k=2n−1ζ^X(k) ⋅ ζ^X(ns)[21]andζ^XSLn(s)=qn(n−1)2(g−1) ⋅ DSLn,Pn−1,1(s) ⋅ ωX(SLn,Pn−1,1)(s).[22]

Proof:

In view of the definitions Eqs. 7 and 8, we must show that MP(k,h) equals 1 if k=0 and 2≤h<n or k=1 and h=n and vanishes otherwise, which follows easily from Eq. 6 since here w0=1 2 ⋯ nn n−1 ⋯ 1.□

4. Special Permutations

In this section we describe special permutations explicitly. Recall from Section 3 that σ is special if and only if Aσ⊔Bσ=Δ′, where Aσ and Bσ are defined as in Eq. 15. This implies that σ is special if and only if σ(i+1)=σ(i)+1 or σ(i+1)<σ(i) for all 1≤i≤n−2 (or equivalently, since σ is a permutation, if and only σ(i+1)≤σ(i)+1 for all 1≤i≤n−2). Denote by t1>…>tm the distinct values of σ(i)−i for 1≤i≤n−2 and by Iν (1≤ν≤m) the set of i∈{1,…,n−2} with σ(i)−i=tν. Then σ maps Iν onto its image Iν′=σ(Iν) by translation by tν, and we have ⋃Iν={1,…,n−1} and ⋃Iν′={1,…,n}\{a}, where a=σ(n)∈{1,…,n}. It is easy to check that I1<⋯<Im (in the sense that all elements of Iν are less than all elements of Iν+1 if 1≤ν≤m−1) and I1′>⋯>Im′ (in the same sense). [Indeed, let A denote the set of indexes i∈{1,…,n−2} with σ(i+1)=σ(i)+1. Then σ(i)−i is constant when we pass from any i∈A to i+1, so each set Iν is a connected interval that is contained in A except for its right end-point i0, which satisfies σ(i0+1)<σ(i0), so that i0+1 belongs to an Iμ satisfying tμ<tν and hence μ>ν. But then Iμ contains a point that is bigger than one of the points of Iν and that has an image under σ that is smaller than the image of that point, and since all of these sets are connected intervals, this means that all of Iμ lies to the right of all of Iν and that all of Iμ′ lies to the left of all of Iν′, proving the assertion.] These properties characterize special permutations and are illustrated in Fig. 1, in which the lengths of the intervals Iν with Iν′ above (respectively below) a are denoted by k1,…,kp (resp. by ℓ1,…,ℓr), so that ∑i=1pki=n−a, ∑j=1rℓj=a−1, and p+r=m. We denote the corresponding special permutation by σ(k1,…,kp;a;l1,…,lr) and also define two sequences of numbers 0=K0<K1<⋯<Kp=n−a and 0=L0<L1<⋯<Lr=a−1 byKi= k1+⋯+ki (1≤i≤p), Lj= l1+⋯+lj (1≤j≤r).[23]

Fig. 1.

The special permutation σ(k1,…,kp;a;l1,…,lr).

Remark:

Denote by Sn,a (a=1,…,n) the set of special permutations in Sn with σ(n)=a. From the above description we find that Sn,a≅Xn−a×Xa−1, where XK for K≥0 is the set of ordered partitions of K (decompositions K=k1+⋯+kp with all ki≥1). Clearly the cardinality of XK equals 1 if K=0 (in which case only p=0 can occur) and 2K−1 if K≥1 (the ordered partitions of K are in 1:1 correspondence with the subsets of {1,…,K−1}, each such subset dividing the interval [0,K]⊂R into intervals of positive integral length), so |Sn,a| equals 2n−2 for a∈{1,n} and 2n−3 for 1<a<n, and the whole set Sn0 has cardinality 2n−3(n+2).

5. Proof of Theorem 2

In this section, we use the characterization of special permutations given in Section 4 to calculate the rational factor Rσ(s) and the zeta factors ζ^σ[n](s) and ζ^σ[<n](s) appearing in Lemma 4 explicitly for special permutations σ. We begin with Rσ(s).

Lemma 6.

For the special permutation σ=σ(k1,…,kp;a;l1,…,lr), the quantity Rσ(s) defined in Lemma 4 is given byRσ(s)=(1−qk1+k2)⋯(1−qkp−1+kp)⋅(1−qns−n+a+kp) ⋅ (1−q−ns+n−a+l1+1) ⋅ (1−ql1+l2)⋯(1−qlr−1+lr).

Proof:

By definition,Rσ(s) =∏1≤k≤n−1σ−1(αk)∉Δ′ 1−q−⟨σλs−ρ,αk⟩ =∏1≤k≤n−1σ−1(αk)∉Δ′ 1−q1−⟨λs,σ−1αk⟩.For each k occurring in this product, write σ−1(αk)=ei−ej=:αij. Then the condition αij∉Δ′ says that the points (i,σ(i)=k) and (j,σ(j)=k+1) do not belong to the same square block in the picture of the graph of σ given in the last section. From that picture, we see that the ks occurring in the product, in decreasing order, together with the corresponding values of i and j, are given by the first three columns of the following table:

View this table:

The fourth column follows from Eq. 11. The lemma follows.□

We next consider the zeta factor ζ^σ[n](s).

Lemma 7.

For the special permutation σ=σ(k1,…,kp;a;l1,…,lr), the zeta factor ζ^σ[n](s) of Lσ(s) is given byζ^σ[n](s)=ζ^X(ns−n+a)ζ^X(ns).Lemma 7 implies in particular that to normalize ζ^σ[n](s) we at least need to clear the denominator by multiplying by the zeta factor ζ^X(ns).

Proof:

This is much easier. From λs=(ns−n)ϖ+ρ, we get ⟨λs,ei−en⟩=ns−i. Moreover, by Fig. 1 in Section 4, for the special permutation σ=σ(k1,…,kp;a;l1,…,lr), we have{ei−en : 1≤i<n, σ(i)>σ(n)}={e1−en,e2−en,…,en−a−en}.Therefore, by the definition of ζ^σ[n](s) given in Lemma 4, we haveζ^σ[n](s)=∏α=ei−en, i≤n−1σ(i)>σ(n)ζ^X(⟨λ,α⟩)ζ^X(⟨λ,α⟩+1)λ=λs=∏i=1n−aζ^X(ns−i)ζ^X(ns−i+1)=ζ^X(ns−n+a)ζ^X(ns)as asserted.□

Finally, we treat the zeta factor ζ^σ[<n](s). However, with the normalization stated in Lemma 5, to obtain the group zeta function ζ^XSLn(s), it suffices to investigate the product ζ^σ[<n](s)⋅∏i≥2ζ^X(i)−n(i) or, equivalently, by Eq. 18, the product ζ^X(1)#Bσ∏i≥2ζ^X(i)nσ(i)−n(i), which we write as ∏i≥1ζ^X(i)rσ(i) withrσ(k)= # Bσ if k=1,nσ(k)−n(k) if k≥2,where the numbers n(k) are defined, in analogy with the numbers nσ(k) in Section 3 (Eqs. 19 and 20), bym(k)=#{α>0:⟨ρ,α⟩=k} , n(k) = m(k)−m(k−1).Clearly m(k)=n−k for 1≤k≤n and n(k)=−1 for 2≤k≤n.

Lemma 8.

For the special permutation σ=σ(k1,…,kp;a;l1,…,lr), we have∏i≥1ζ^X(i)rσ(i)=∏i=1pv^ki⋅∏j=1rv^lj.[24]In particular, rσ(k)≥0.

Proof:

This is based on a detailed analysis of rσ(k). Obviously,rσ(1)=#{α∈Δ′:σα<0}=#{(i,i+1):1≤i≤n−2, σ(i)>σ(i+1)}.If k≥2, by definition,m(k)−mσ(k)=#{α>0:⟨ρ,α⟩=k}−#{α∈Φ′+:σα<0,⟨ρ,α⟩=k}=#{ei−en:⟨ρ,α⟩=k}+#{α∈Φ′+:σα>0,⟨ρ,α⟩=k}=1+#{α∈Φ′+:σα>0,⟨ρ,α⟩=k},since, by Eq. 10, {ei−en:⟨ρ,α⟩=k}={en−k−en}. Thus, by applying the characterization graph in Section 4 for special permutation σ(k1,…,kp;a;l1,…,lr), we conclude that α=αij∈Φ′+ satisfying σα>0 (or equivalently α=αij satisfying i<j≤n−1 and σ(i)<σ(j)) if and only if i and j belong to the same block, say Iμ for some μ, associated to σ(k1,…,kp;a;l1,…,lr), and also σ(j)∈Iμ (or equivalently j+1∈Iμ), since otherwise σ(αij)<0.

Denote by (m(k)−mσ(k))μ (resp. rσ,μ(k)) the contribution to m(k)−mσ(k) (resp. to rσ(k)) of the block Iμ. With the discussion above, we havem(k)−mσ(k)=∑μ(m(k)−mσ(k))μ and rσ(k)=∑μrσ,μ(k).Fix some μ and let Iμ≔{a+1,a+2,…,a+b} with a, b∈Z>0. Clearly, when k=1, rσ,μ(1)=#{(a+b−1,a+b)}=1, since, for other (i,i+1) s, σ(i)<σ(i+1). Moreover, when k≥2, by Eq. 10 and the characterization of the graph again, we have(m(k)−mσ(k))μ=#(i,j): i, j+1∈Iμ, i<j, j=i+k=#(i,j): a+1≤i<j<a+b, j=i+k.Note that, for each fixed i (with a+1≤i<a+b),#(i,j): a+1≤i<j<a+b, j=i+k=1 i+k<a+b0 i+k≥a+b.Hence, (m(k)−mσ(k))μ = b−(k+1). This implies that for all k≥1 rσ,μ(k)=(m(k−1)−mσ(k−1))μ−(m(k)−mσ(k))μ=1. Consequently,∏i≥1ζ^X(k)rσ,μ(k)=ζ^X(1) ζ^X(2) ⋯ ζ^X(b).Eq. 24 follows.□

Combining Lemmas 5, 6, 7, and 8, we getζ^XSLn(s)qn(n−1)2(g−1) =∏i≥2ζ^X(i)−n(i)⋅limλ→λs∏α∈ΔP(1−q−⟨λ−ρ,α∨⟩)⋅ωXSLn(λ)=∑a=1n∑k1,…,kp>0k1+⋯+kp=n−av^k1⋯v^kp(1−qk1+k2)…(1−qkp−1+kp)⋅11−qns−n+a+kp × ζ^(ns−n+a)∑l1,…,lr>0l1+⋯+lr=a−111−q−ns+n−a+1+l1 ⋅ v^l1⋯v^lr(1−ql1+l2)…(1−qlr−1+lr).This completes the proof of Theorem 2.

6. The Theorem of Mozgovoy and Reineke

In the previous three sections we have given an explicit formula for the group zeta function associated to a curve over a finite field in the case (G,P)=(SLn,Pn−1,1). As explained in the Introduction, our main result (Theorem 1) will follow by comparing this formula with the explicit formula for the rank n nonabelian zeta function ζ^X,n(s) found by Mozgovoy and Reineke, namely the following:

Theorem (theorem 7.2 of ref. 5).

The function ζ^X,n(s) is given byζ^X,n(s)=qn2(g−1) ∑h=1n−1 ∑n1,…,nh>0n1+⋯+nh=n−1v^n1⋯v^nh∏j=1h−1(1−qnj+nj+1) ×ζ^X(ns)1−q−ns+n1+1 + ∑i=1h−1(1−qni+ni+1) ⋅ ζ^X(ns−(n1+⋯+ni))(1−qns−(n1+⋯+ni−1))(1−q−ns+n1+⋯+ni+1+1)+ζ^X(ns−n+1)1−qns−(n1+⋯+nk−1).[25]This already looks very similar to Theorem 2, and the precise equality of the two formulas will be verified in Section 7. But since the ideas leading to the expressions for the group zeta function and for the nonabelian zeta function are very different, and since the ideas of the proof in ref. 5 are very interesting, we include a brief account of their calculation for the benefit of the interested reader. A reader who is interested only in the proof of the main result, or who is already familiar with the paper (5), can skip this section and go immediately to Section 7.

The first ingredient is that of semistable pairs and triples. Fix an integral regular projective curve X over a finite field Fq. By a pair (E,s) over X we mean a vector bundle E on X together with a global section s of E on X. Such pairs form an Fq-linear category, a morphism (E,s)→(E′,s′) being an element (λ,f)∈Fq×HomX(E,E′) such that f○s=λ s′. A pair (E,s) is called τ semistable (τ∈R) if μ(F)≤τ for any subbundle F of E and μ(E/F)≥τ for any subbundle F of E with s∈H0(X,F). Here, as usual, μ(E) denotes the Mumford slope of E. For (r,d)∈Z>0×Z we denote by MXτ(r,d) the moduli stack of τ-semistable pairs (E,s) of rank r and degree d. If τ=d/r, then this is the same as the usual slope semistability of E, so if we write MX(r,d) for the moduli space of semistable bundles of rank r and degree d, then (cf. corollary 3.7 of ref. 5)∑(E,s)∈MXd/r(r,d)1#Aut(E,s)=1q−1∑E∈MX(r,d)qh0(X,E)−1#Aut E.Next, we consider triples E=(E0,E1,s) consisting of two coherent sheaves E0, E1 on X and a morphism s:E1→E0. These triples form an abelian category which we denote by A. The triple E=(E0,E1,s) is called μτ semistable if μτ(F)≤μτ(E) for any subobject F of E, whereμτ(E) ≔ degE0+degE1+τ⋅rank E1rank E0+rank E1.We also introduce χ(E,F)≔∑k=02(−1)kdimExtAk(E,F). It is known that χ(E,F)=χ(E0,F0)+χ(E1,F1)−χ(E1,F0), where as usual, χ(E,F)≔dimHom(E.F)−dimExt1(E,F). For α=(r,d), β=(r′,d′)∈Z>0×Z, set χ(α)=d−(g−1)r and ⟨α,β⟩≔2(rd′−r′d). Similarly, for α̲=(α,v), β̲=(β,w) with v, w∈Z≥0 we set ⟨α̲,β̲⟩≔⟨α,β⟩−v χ(β)+w χ(α).

The next ingredients are Hall algebras and integration maps. Let K0( StFq) be the Grothendieck ring of finite-type stacks over Fq with affine stabilizers and L be the Lefschetz motive. We introduce the coefficient ring R=K0( StFq)[L±1/2] and define the quantum affine plane A0 to be the completion of the algebra R[x1,x2±1] with the multiplicationxα○xβ ≔ (−L1/2)⟨α,β⟩xα+β.(Here the completion is defined by requiring that for f=∑α∈N×Zfαxα∈A0 and any t∈R there are only finitely many (r,d) with fr,d ≠ 0 and dr+1<t.) If we further denote by A0 the category of coherent sheaves on X and by H(A0) its associated Hall algebra, whose multiplication [E]○[F] counts extensions from Ext1(F,E), then we have a morphism of algebrasI: H(A0) →A0 E ↦ (−L1/2)χ(E,E)⋅ xch(E) [AutE],which we call the integration map. Here ch(E)≔(rank E,degE). Similarly, if we introduce a second quantum affine plane A as the completion of the algebra R[x1,x2±1,x3] with the multiplicationxα̲○xβ̲ ≔ (−L1/2)⟨α̲,β̲⟩xα̲+β̲,then we have an integration map on the Hall algebra H(A),I̲:H(A) →AE ↦ (−L1/2)χ(E,E)⋅ xcl(E) [AutE],where cl(E)≔(rank E0,deg E0,rank E1). We have I̲∣H(A0)=I. The map I̲ is not an algebra morphism in general, but if Ext2(F,E)=0, then I̲(E○F)=I̲(E)I̲(F).

The last and most important ingredient of the proof in ref. 5 is a wall-crossing formula. For α=(r,d)∈Z>0×Z and τ∈R, letu(α) ≔ (−L−1/2)χ(α,α)+d [MX(α)]be the motivic class of MX(α) counting semistable bundles E on X with ch E=α, and similarly setfτ(α)=(L−1)(−L−1/2)χ(α,α)+d[MXτ(α)].We introduce the two generating seriesuτ=1+∑μ(α)=τu(α) xα∈A0 , fτ=∑αfτ(α) x(α,1)∈A.Then the rank n nonabelian zeta function for X can be expressed asζX,n(s)=(q−1)∑k≥0[MX(n,kn)]q−sk=qn(n−1)2(g−1)∑k≥0fk(n,kn)q−ks.We can also identify the moduli stack MX∞(1,d) with the Hilbert scheme HilbdX or with SymdX, the dth symmetric product of X. Consequently,f∞ ≔ x1x3 ∑d≥0[SymdX] x2d=x1x3 ZX(x2),where ZX(t) is the Artin zeta function with ζX(s)=ZX(q−s). (This can be interpreted as the limiting special case of fτ as τ→∞, since the condition of semistability with respect to τ of a pair (E,s) in the limit τ→∞ is equivalent to the requirement that coker(s) is finite.) Finally, setu≥τ ≔ ∏τ′≥τ→uτ′,where the product is taken in the decreasing slope order, and, for an element g=∑αgαx(α,1)∈A, setg∣μ≤τ ≔ ∑μ(α)<τgαx(α,1).Then, using the theory of Hall algebras and wall-crossing techniques, the main result (theorem 5.4 of ref. 5) is the identityfτ=u>τ−1○f∞○u≥τ|μ≤τ (τ∈R).Eq. 25 is obtained from this basic formula by a somewhat involved combinatorial discussion, using a “Zagier-type formula” (i.e., one based on the combinatorics in ref. 13) for the motivic classes of moduli spaces of semistable bundles.

7. Proof of Theorem 1 and Structure of the Function ζX,n(s)

To complete the proof of Theorem 1, we verify the term-by-term equality of the sums appearing in Eqs. 5 and 25. Clearly, the factor qn2(g−1) is the same in both cases. Both sums have the form of a linear combination of ζ^X(ns−k) with 0≤k≤n−1, so we have only to check the equality of the coefficients. The case k=0 is immediate: Since B0(x) is identically 1, the coefficient of ζ^X(ns) in the sum in Eq. 5 is Bn−1(q1−ns), which by Eq. 4 is identical with the coefficient of ζ^X(ns) in the sum in Eq. 25. (Set p=h, ki=nh+1−i.) The case k=n−1 is exactly similar or can be deduced from the case k=0 by noticing that Eq. 5 is invariant under k→n−1−k, s→1−s and Eq. 25 under nj→nh+1−j, i→h−i, and s→1−s. If 0<k<n−1, then the coefficient of ζ^X(ns−k) in the sum in Eq. 25 can be rewritten as∑0<i<h<n ∑n1+⋯ni=kni+1+⋯+nh=n−1−kv^n1⋯v^ni∏j=1i−1(1−qnj+nj+1)⋅11−qns−k+ni ⋅ v^ni+1⋯v^nh∏j=i+1h−1(1−qnj+nj+1)⋅11−q−ns+k+ni+1+1,and since the summations over the tuples (n1,…,ni) with sum k and the tuples (ni+1,…,nh) with sum n−k−1 are independent, this equals Bk(qns−k)Bn−k−1(qk+1−ns) as required. This completes the comparison of Eqs. 5 and 25 and hence the proof of Theorem 1.

We end this paper by looking briefly at the structure of the explicit formula for the higher-rank zeta function ζX,n(s), and in particular we check that it implies the known properties of this zeta function as listed in the opening paragraph. One of these properties was the functional equation ζ^X,n(1−s)=ζ^X,n(s), which, as we have already said, follows immediately from Eq. 5 by interchanging k and n−k−1 and using the known functional equation ζ^X(1−s)=ζ^X(s). The other one concerned the form of ζX,n(s). Here it is more convenient to work with the variables t=q−s and T=q−ns=tn, writing ζX(s) and ζX,n(s) as ZX(t) and ZX,n(T), respectively, and similarly ζ^X(s)=Z^X(t) and ζ^X,n(s)=Z^X,n(T) with Z^X(t)=t1−gZX(t), Z^X,n(T)=T1−gZX,n(T). It is well known that ZX(t) has the form P(t)/(1−t)(1−qt) where P(t)=PX(t) is a polynomial of degree 2g, and the assertion is that ZX,n(T), which from the definition Eq. 1 is just a power series in T, has the corresponding form Pn(T)/(1−T)(1−qnT) where Pn(T)=PX,n(T) is again a polynomial of degree 2g. In these terms, the formula for the rank n zeta function becomesq−n2(g−1) Z^X,n(T)=∑k=0n−1 Bk(q−kT−1) Z^X(qkT) Bn−k−1(qk+1T).[26]From this it is clear that Z^X,n(T) is a rational function of T and grows at most like O(Tg−1) as T→∞ and like O(T1−g) as T→0, since the definition of the function Bk(x) shows that it is bounded at both 0 and ∞, so the only nontrivial assertion is that Z^X,n(T) has at most simple poles at T=1 and T=q−n and no other poles. From the definition of Bk(x) and the properties of Z^X(t) we see that every term in Eq. 26 has simple poles at T=1,q−1,…,q−n (the first factor has simple poles at q−i with 0≤i<k, the second one at i=k and i=k+1, and the third one at k+1<i≤n), so the only thing that needs to be checked is that the residues at q−i for 0<i<n sum to 0. Denote by Ri (0≤i≤n) the limiting value as T→q−i of the right-hand side of Eq. 26 multiplied by 1−qiT and by Ri,k the corresponding contribution from the kth term, so that Ri=∑k=0n−1Ri,k. Suppose that 0<i<n. Then for 0≤k≤i−2 we findRi,k=Bk(qi−k) Z^X(qk−i) v^i−k−1 Bn−i(qi−k−1)and for k=i−1 we findRi,i−1=Bi−1(q) v^1 Bn−i(1).Since Z^X(qk−i)v^i−k−1=v^i−k, these formulas can be written uniformly asRi,k=Bk(qi−k) v^i−k Bn−i(qi−k−1) (0≤k≤i−1).The formulas in the other two cases can be computed similarly, but this is not necessary since the abovementioned symmetry of the terms in Eq. 26 under (k,T)↦(n−1−k,q−nT−1) implies that Ri,k=−Rn−i,n−k−1 and hence Ri=Si−Sn−i with Si=∑k=0i−1Ri,k. But the formula just proved for Ri,k for 0≤k≤i−1 can be rewritten asRi,k=∑1≤s<r≤n∑n1,…,nr≥1n1+⋯+nr=nn1+⋯+ns−1=k, ns=i−kv^n1⋯v^nr(1−qn1+n2)⋯(1−qnr−1+nr),soSi=∑1≤s<r≤n∑n1,…,nr≥1n1+⋯+nr=nn1+⋯+ns=iv^n1⋯v^nr(1−qn1+n2)⋯(1−qnr−1+nr),which is visibly symmetric under i↦n−i by replacing nj by nr+1−j and s by r+1−s. This completes the proof of vanishing of Ri for 0<i<n, and by essentially the same calculation we also get the corresponding formulasRn=−R0=∑r=1n∑n1,…,nr≥1n1+⋯+nr=nv^n1⋯v^nr(1−qn1+n2)⋯(1−qnr−1+nr)for the two remaining coefficients Ri describing the poles of ζX,n(s).

Data Availability.

There are no data associated with this paper.

Acknowledgments

We thank Alexander Weisse of the Max Planck Institute for Mathematics in Bonn for the tikzpicture (Fig. 1) of special permutations given in Section 4. L.W. is partially supported by Japan Society for the Promotion of Science.

Footnotes

↵¹To whom correspondence may be addressed. Email: dbz{at}mpim-bonn.mpg.de or weng{at}math.kyushu-u.ac.jp.

Author contributions: L.W. and D.Z. wrote the paper.
The authors declare no competing interest.
This article is a PNAS Direct Submission.
↵^†For number fields, the analogs of the two functions to be introduced below are special kinds of Eisenstein periods, defined as integrals of Eisenstein series over moduli spaces of semistable lattices. For details, see ref. 4.
↵^‡We warn the reader that in refs. 4, 7, and 8 a different normalization is used, with the argument of ωXG,P (and later of ζXG,P) being given by s=cp(sp−1) ( =n(sp−1) in the special case (G,P)=(SLn,Pn−1,1)) rather than s=sp as chosen here. With the normalization used here the functional equation relates s and 1−s rather than s and −n−s.

Published under the PNAS license.

References

↵
1. L. Weng
, Non-abelian zeta functions for function fields. Am. J. Math. 127, 973–1017 (2005).
OpenUrl
↵
1. L. Weng
, Zeta functions for curves over finite fields. arXiv:1202.3183 (15 February 2012).
↵
1. L. Weng,
2. D. Zagier
, Higher-rank zeta functions for elliptic curves. Proc. Natl. Acad. Sci. U.S.A., doi:10.1073/pnas.1912023117 (2020).
OpenUrl Abstract/FREE Full Text
↵
1. I. Nakamura,
2. L. Weng
1. L. Weng
, “Symmetries and the Riemann hypothesis” in Algebraic and Arithmetic Structures of Moduli Spaces, I. Nakamura, L. Weng, Eds. (Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, Japan, 2010), vol. 58, pp. 173–223.
OpenUrl
↵
1. S. Mozgovoy,
2. M. Reineke
, Moduli spaces of stable pairs and non-abelian zeta functions of curves via wall-crossing. J. l’École Polytech. Math., 1, 117–146 (2014).
OpenUrl
↵
1. V. G. Drinfeld
, Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field. Funct. Anal. Appl. 15, 294–295 (1981).
OpenUrl
↵
1. L. Weng,
2. M. Kaneko
1. L. Weng
, “A geometric approach to L-functions” in The Conference on L-Functions, L. Weng, M. Kaneko, Eds. (World Scientific Publishing, Hackensack, NJ, 2007), pp. 219–370.
↵
1. I. Nakamura,
2. L. Weng
1. L. Weng
, “Stability and arithmetic” in Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007), I. Nakamura, L. Weng, Eds. (Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, Japan, 2010), vol. 58, pp. 225–359.
OpenUrl
↵
1. J. E. Humphreys
, Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics (Springer-Verlag, Berlin, Germany, 1972), vol. 9.
↵
1. J. Arthur,
2. D. Ellwood,
3. R. Kottwitz
1. J. Arthur
, “An introduction to the trace formula” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, J. Arthur, D. Ellwood, R. Kottwitz, Eds. (Proceedings of the Clay Mathematics Institute, American Mathematical Society, Providence, RI, 2005), vol. 4, pp. 1–263.
OpenUrl
↵
1. Y. Komori
, Functional equations of Weng’s zeta functions for (G,P)/Q. Am. J. Math. 135, 1019–1038 (2013).
OpenUrl
↵
1. H. Ki,
2. Y. Komori,
3. M. Suzuki
, On the zeros of Weng zeta functions for Chevalier groups. Manuscr. Math. 148, 119–176 (2015).
OpenUrl
↵
1. M. Teicher,
2. F. Hirzebruch
1. D. Zagier
, “Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula” in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry, M. Teicher, F. Hirzebruch, Eds. (Israel Mathematical Conference Proceedings, Bar-Ilan University, Ramat Gan, Isreal, 1996), vol. 9, pp. 445–462.
OpenUrl

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Submit

[1] ↵
L. Weng
, Non-abelian zeta functions for function fields. Am. J. Math. 127, 973–1017 (2005).
OpenUrl

[2] L. Weng

[3] ↵
L. Weng
, Zeta functions for curves over finite fields. arXiv:1202.3183 (15 February 2012).

[4] L. Weng

[5] ↵
L. Weng,
D. Zagier
, Higher-rank zeta functions for elliptic curves. Proc. Natl. Acad. Sci. U.S.A., doi:10.1073/pnas.1912023117 (2020).
OpenUrl Abstract/FREE Full Text

[6] L. Weng,

[7] D. Zagier

[8] ↵
I. Nakamura,
L. Weng
L. Weng
, “Symmetries and the Riemann hypothesis” in Algebraic and Arithmetic Structures of Moduli Spaces, I. Nakamura, L. Weng, Eds. (Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, Japan, 2010), vol. 58, pp. 173–223.
OpenUrl

[9] I. Nakamura,

[10] L. Weng

[11] L. Weng

[12] ↵
S. Mozgovoy,
M. Reineke
, Moduli spaces of stable pairs and non-abelian zeta functions of curves via wall-crossing. J. l’École Polytech. Math., 1, 117–146 (2014).
OpenUrl

[13] S. Mozgovoy,

[14] M. Reineke

[15] ↵
V. G. Drinfeld
, Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field. Funct. Anal. Appl. 15, 294–295 (1981).
OpenUrl

[16] V. G. Drinfeld

[17] ↵
L. Weng,
M. Kaneko
L. Weng
, “A geometric approach to L-functions” in The Conference on L-Functions, L. Weng, M. Kaneko, Eds. (World Scientific Publishing, Hackensack, NJ, 2007), pp. 219–370.

[18] L. Weng,

[19] M. Kaneko

[20] L. Weng

[21] ↵
I. Nakamura,
L. Weng
L. Weng
, “Stability and arithmetic” in Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007), I. Nakamura, L. Weng, Eds. (Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, Japan, 2010), vol. 58, pp. 225–359.
OpenUrl

[22] I. Nakamura,

[23] L. Weng

[24] L. Weng

[25] ↵
J. E. Humphreys
, Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics (Springer-Verlag, Berlin, Germany, 1972), vol. 9.

[26] J. E. Humphreys

[27] ↵
J. Arthur,
D. Ellwood,
R. Kottwitz
J. Arthur
, “An introduction to the trace formula” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, J. Arthur, D. Ellwood, R. Kottwitz, Eds. (Proceedings of the Clay Mathematics Institute, American Mathematical Society, Providence, RI, 2005), vol. 4, pp. 1–263.
OpenUrl

[28] J. Arthur,

[29] D. Ellwood,

[30] R. Kottwitz

[31] J. Arthur

[32] ↵
Y. Komori
, Functional equations of Weng’s zeta functions for (G,P)/Q. Am. J. Math. 135, 1019–1038 (2013).
OpenUrl

[33] Y. Komori

[34] ↵
H. Ki,
Y. Komori,
M. Suzuki
, On the zeros of Weng zeta functions for Chevalier groups. Manuscr. Math. 148, 119–176 (2015).
OpenUrl

[35] H. Ki,

[36] Y. Komori,

[37] M. Suzuki

[38] ↵
M. Teicher,
F. Hirzebruch
D. Zagier
, “Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula” in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry, M. Teicher, F. Hirzebruch, Eds. (Israel Mathematical Conference Proceedings, Bar-Ilan University, Ramat Gan, Isreal, 1996), vol. 9, pp. 445–462.
OpenUrl

[39] M. Teicher,

[40] F. Hirzebruch

[41] D. Zagier

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Higher-rank zeta functions and SL_n-zeta functions for curves

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