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

Higher-rank zeta functions and SL _n -zeta functions for curves

Lin Weng^a,1and Don Zagier^b,1

aGraduate School of Mathematics, Kyushu University, 819-0395 Fukuoka, Japan; and^bMax Planck Institute for Mathematics, 53111 Bonn, Germany Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved January 15, 2020 (received for review July 19, 2019)

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 ranknover the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive groupGand its maximal parabolic subgroupP. It was conjectured that these two zeta functions coincide in the special case whenG=SLnandPis 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 ton=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.

nonabelian zeta function|curves over finite fields|special permutations|zeta functions|zeta functions forSL_n

I

ζX,n(s) =X

[V]

|H⁰(X,V)r{0}|

|Aut(V)| q^−deg(V)s (<(s)>1), [1]

where the sum is over the moduli stack ofF^q-rational semistable vector bundlesV of ranknonX with degree divisible byn. Using the Riemann–Roch, duality, and vanishing theorems for semistable bundles, it was shown thatζX,n(s)agrees with the usual Artin zeta functionζX(s)ofX/F^qifn= 1; that it has the formPX,n(T)/(1−T)(1−qⁿT)for some polynomialPX,n(T)of degree2gin T, wheregis the genus ofX andT=q^−ns; and that it satisfies the functional equation

ζbX,n(1−s) =ζbX,n(s) , where ζbX,n(s) :=q^n(g−1)s·ζX,n(s).

It was also conjectured thatζX,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 forζX,n(s)and a proof of the Riemann hypothesis were given for the case wheng= 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 functionζb_X^G,P(s) ofX/F^q, associated to a connected split algebraic reductive groupGand its maximal parabolic subgroupP. The precise definition, which is based on the theory of periods, is recalled inSection 2. In this paper, we are interested in the special case whenG=SLn

andP=Pn−1,1, the subgroup ofSLn consisting of matrices whose final row vanishes except for its last entry, and we then write simplyζb_X^SLn(s)forζb_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 functionsbζX,n(s)andζb_X^SLn(s)coincide for alln≥1.

Theorem 1should 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 forζb_X^SLn(s)established here with a formula forζbX,n(s)given in their paper. Specifically, the proof consists of three steps:

1) By analyzing the definition ofζb_X^G,P(s)forG=SLn,P=Pn−1,1, we will prove an explicit formula, givingζb_X^SLn(s)as a linear combination of the functionsζbX(ns−k)for0≤k<n with rational functions ofT as coefficients. The calculation is given in Sections 3–5.

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.

Author contributions: L.W. and D.Z. wrote the paper.y The authors declare no competing interest.y This article is a PNAS Direct Submission.y Published under thePNAS license.y

1To whom correspondence may be addressed. Email: [email protected] or [email protected] First published March 9, 2020.y

MATHEMATICS

2) In ref. 5, as recalled inSection 6, using the theory of Hall algebras and wall-crossing techniques, a formula forζbX,n(s)of the same general shape is proved.

3) A short calculation, given inSection 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 bundlesV of ranknand degree 0 on X (i.e., the number of suchVs, weighted by the inverse of the number of their automorphisms), and to make a proper normalization, we define numbersbvk(k≥1) inductively by

vbk = (

lims→1(1−q^1−s)ζbX(s) ifk = 1 ,

ζbX(k)bvk−1 ifk≥2 , [2]

whereζbX(s) =q^s(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 ranknzeta functions for elliptic curves over finite fields—we define rational functionsBk(x) (k≥0) either inductively by the formulas

Bk(x) =







1 ifk= 0 ,

k

P

m=1bvm

Bk−m(q^m)

1−q^mx ifk≥1 , [3]

or in closed form (ifk≥1) by

Bk(x) =

k

X

p=1

X

k₁,...,kp>0 k₁+···+k_p=k

bvk₁. . .bvkp

(1−q^k1+k2). . .(1−q^kp−1+kp)· 1

1−q^kpx. [4]

Then the formula that we will establish forζb_X^SLn(s)can be stated as follows:

Theorem 2. With the above notations,we have

ζb_X^SLn(s) =q(ⁿ₂)^(g−1)

n−1

X

k=0

Bk(q^ns−k)Bn−k−1(q^k+1−ns)ζbX(ns−k). [5]

Remarks:

1) In the definition Eq.1of the nonabelian zeta functionζX,n(s), vector bundles used are assumed to be of degrees divisible by the rankn. 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 ofTheorem 1for 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 1for the cases n= 2and 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)

LetGbe a connected split reductive algebraic group of rankrwith a fixed Borel subgroupBand associated maximal split torusT (over a base field). Denote by

V,h·,·i, Φ = Φ⁺∪Φ⁻, ∆ ={α1,. . .,αr},$:={$1,. . .,$r},W

the associated root system. That is,V is the real vector space defined as theRspan of rational characters ofT and, as usual, is equipped with a natural inner producth·,·i, with which we identifyV with its dualV^∗; andΦ⁺⊂V is the set of positive roots, Φ⁻:=−Φ⁺the set of negative roots,∆⊂V the set of simple roots,$⊂V the set of fundamental weights, andW the Weyl group.

By definition, the fundamental weights are characterized by the formulah$i,α^∨_j i=δij fori,j= 1, 2,. . .,r, where α^∨:=_hα,αi² α denotes the coroot of a rootα∈Φ. We also define the Weyl vectorρbyρ=¹₂P

α∈Φ⁺αand introduce a coordinate system onV (with respect to the base{$1,. . .,$r}ofV and the vectorρ) by writing an elementλ∈V in the form

λ=

r

X

j=1

(1−sj)$j =ρ−

r

X

j=1

sj$j,

thus fixing identifications ofV andV_C=V⊗_RCwithR^r andC^r. In addition, for each Weyl elementw∈W, we setΦw:= Φ⁺ ∩ w⁻¹Φ⁻, i.e., the collection of positive roots whosewimages are negative.

As usual, by a standard parabolic subgroup, we mean a parabolic subgroup ofGthat contains the Borel subgroupB. From Lie theory (e.g., ref. 9), there is a one-to-one correspondence between standard parabolic subgroupsP ofGand subsets∆P of∆. In particular, ifP is maximal, we may and will write∆P= ∆r{αp}for a certain uniquep=p(P)∈ {1,. . .,r}. For such a standard parabolic subgroupP, denote by VP theRspan of rational characters of the maximal split torusTP contained inP, byV_P^∗its

dual space and byΦP⊂VP the set of nontrivial characters ofTP occurring in the spaceV. Then, by standard theory of reductive groups (e.g., ref. 10),VP admits a canonical embedding inV (andVP^∗admits a canonical embedding inV^∗), which is known to be orthogonal to the fundamental weight$p, and henceΦPcan be viewed as a subset ofΦ. SetΦ⁺_P= Φ⁺ ∩ΦP,ρP=¹₂P

α∈Φ⁺ Pα, and cP= 2h$p−ρP,α^∨pi.

Now, letXbe an integral regular projective curve of genusgover a finite fieldF^q. In ref. 2, motivated by the study of zeta functions for number fields,^†for a connected split reductive algebraic groupGand its standard parabolic subgroupPas above (defined over the function field ofX), L.W. defined the period ofGforX by

ω^G_X(λ) := X

w∈W

1 Q

α∈∆(1−q^{−hwλ−ρ,α∨i}) Y

α∈Φ_w

ζbX(hλ,α^∨i) ζbX(hλ,α^∨i+ 1) and the period of(G,P)forX by

ω_X^G,P(s) := Reshλ−ρ,α^∨i= 0,α∈∆_PωX^G(λ)

s_p=s= Ress_r= 0· · ·Ress_p+1= 0Ress_p−1= 0· · ·Ress₁= 0ωX^G(λ) s_p=s,

wheres is a complex variable^‡ and where for the last equality we used the fact thathρ,α^∨i= 1 for allα∈∆and the relation thath$i,α^∨ji=δij for alli,j∈ {1,. . .,r}. As proved in refs. 2 and 11, the ordering of taking residues along singular hyperplanes hλ−ρ,α^∨i= 0forα∈∆Pdoes 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 ω^GX(λ) =X

w∈WTw(λ), where, for eachw∈W,

Tw(λ) := 1

Q

α∈∆(1−q^{−hwλ−ρ,α∨i}) Y

α∈Φ_w

ζbX(hλ,α^∨i) bζX(hλ,α^∨i+ 1).

The zeta function ofX associated to(G,P)will be defined in terms of the residueReshλ−ρ,α^∨i=0,α∈∆_PTw(λ).

We care only about those elementsw∈W(we call them special) that give nontrivial residues, namely, those satisfying the condition thatReshλ−ρ,α^∨i= 0,α∈∆_PTw(λ)6≡0. This can happen only if all singular hyperplanes are of one of the following two forms:

1) hwλ−ρ,α^∨i= 0for someα∈∆, giving a simple pole of the rational factorQ ¹

α∈∆(1−q−hwλ−ρ,α∨ i); 2) hλ,α^∨i= 1for someα∈Φw, giving a simple pole of the zeta factorζbX(hλ,α^∨i).

For specialw∈W and(k,h)∈Z², following ref. 11 (also ref. 2) we define

NP,w(k,h) := #{α∈w⁻¹Φ⁻:h$p,α^∨i=k,hρ,α^∨i=h}

MP(k,h) := max

wspecial(NP,w(k,h−1)−NP,w(k,h)).

=NP,w₀(k,h−1)−NP,w₀(k,h), [6]

wherew0is the longest element of the Weyl group and where the last equality is corollary 8.7 of ref. 12. Note thatMP(k,h) = 0for almost all but finitely many pairs of integers(k,h), so it makes sense to introduce the product

D_X^G,P(s) :=

∞

Y

k=0

∞

Y

h=2

ζbX(kn(s−1) +h)^MP(k,h). [7]

Following refs. 2 and 4, we define the zeta function ofXassociated to(G,P)by

ζb^G,P_X (s) :=q^{(g−1) dimNu(B)}·D^G,P(s)·ω_X^G,P(s). [8]

HereNu(B)denotes the nilpotent radical of the Borel subgroupBofG.

Remark: For special w∈W, even after taking residues, there are some zeta factors bζX(ks+h) left in the denominator of Reshλ−ρ,α^∨i= 0,α∈∆_PTw(λ). The reason for introducing the factor D_X^G,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 toSLn

From now on, we specialize to the case whenGis the special linear groupSLn andP is the maximal parabolic subgroupPn−1,1

consisting of matrices whose final row vanishes except for its last entry, corresponding to the ordered partition(n−1) + 1ofn. Our purpose is to study the zeta function ofX associated toSLn:

ζb_X^SLn(s) :=bζ_X^SLn,Pn−1,1(s). [9]

†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 ofω^G,P_X (and later ofζ^G,P_X ) being given bys=cp(sp−1) (=n(sp−1) in the special case (G,P)=(SLn,Pn−1,1)) rather thans=spas chosen here. With the normalization used here the functional equation relatessand 1−srather thansand−n−s.

MATHEMATICS

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 spaceRⁿ. The positive roots are given byΦ⁺:={ei−ej|1≤i<j≤n}, the simple roots by∆ ={α1:=

e1−e2,. . .,αn−1:=en−1−en}, and the Weyl vector byρ=Pn j=1

n+1−2j

2 ej. We identify the Weyl groupWwithSn, the symmetric group onnletters, by the assignmentw7→σw, wherew(ei−ej) =eσw(i)−eσw(j). For convenience, we also write the corresponding

∆P,Φ⁺_P,ρP,$P, andcPsimply as∆⁰,Φ⁰⁺,ρ⁰,$⁰, andc⁰, respectively. We have

∆⁰ ={α1,. . .,αn−2}, Φ⁰⁺ ={ei−ej: 1≤i<j≤n−1},

ρ⁰ =

n−1

X

j=1

n−2j

2 ej, $⁰=$n−1 = 1 n

n

X

j=1

ej −en.

In addition,hρ,αi= 1for allα∈∆, andα^∨=α, hρ,αi= 1for allα∈Φ⁺. Hence

ρ⁰ =ρ− n

2$⁰, c⁰= 2h$⁰−ρ⁰,αn−1i=n.

Accordingly, for positive rootsαij:=ei−ej∈Φ⁺, we have

hρ,αiji=j−i, h$⁰,αiji=δjn−δin, [10]

and, forλs:= (ns−n)$⁰+ρ,

hλs,αiji=







j−i ifi,j6=n, ns−i ifj=n,

−ns+j ifi=n.

[11]

To write down the zeta functionζb_X^SLn(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 ofSLn). Indeed, sincehλs−ρ,αn−1i=ns−n, and

λ→λlim_s

1−q^{−hλ−ρ,αi}

≡0 (∀α∈∆⁰), [12]

we have

ω^SL_X^n,Pn−1,1(s) = lim

λ→λ_s

Y

α∈∆⁰

(1−q^{−hλ−ρ,αi})·ω_X^SLn(λ)

!

. [13]

Recall thatω_X^SLn(λ) =P

w∈WTw(λ). Accordingly, to pin down the nonzero contributions for the terms appearing in the limit, we should consider, for a fixedw∈W, the limitlimλ→λ_s

Q

α∈∆⁰(1−q^{−hλ−ρ,αi})·Tw(λ)

or, equivalently, for a fixedσ∈Sn('W), the function

Lσ(s) = lim

λ→λ_s



 Q

α∈∆⁰(1−q^{−hλ−ρ,αi}) Q

β∈∆(1−q^{−hσλ−ρ,βi})

Y

α∈Φ⁺,σ(α)<0

ζbX(hλ,αi) ζbX(hλ,αi+ 1)



. [14]

For this limitLσ(s)to be nonzero, by Eq.12, there should be a complete cancellation of all of the factors(1−q^{−hλ−ρ,αi})in the numerator of the first term in Eq.14that vanish atλ=λswith either

1) factors

1−q^{−hσλs−ρ,βi}

appearing in the denominator of the first term in Eq.14or else

2) the poles atλ=λsof factorsζbX(hλ,αi)appearing in the numerator of the second term in Eq.14for whichhλs,αi= 1.

Sinceh·,·iisσinvariant, forα∈∆⁰, by Eq.10, hσλs−ρ,αi=hλs,σ⁻¹αi −1. Hence, forLσ(s)to have a nonzero contribution toω^(SL_X ^n,Pn−1,1)(s), the union of

Aσ:=

α∈∆⁰:σα∈∆ and Bσ:=

α∈∆⁰:σα <0 [15]

must be of cardinality n−2. Call such σ∈Sn special and denote the collection of special permutations by S⁰_n. Clearly, for σ∈Sn, we haveAσ∪Bσ⊂∆⁰, and Aσ∪Bσ= ∆⁰ if and only if σ∈S⁰_n. That is to say, the limitLσ(s) corresponding to the

permutationσ∈Sncan be nonzero only ifσis special, and in this case, we have∆⁰=AσtBσ. This then completes the proof of the following:

Lemma 3. With the notations above,

ω_X^SLn,Pn−1,1(s) = X

σ∈S⁰_n

Lσ(s). [16]

Hereσ∈S⁰nif and only ifAσ∪Bσ= ∆⁰.

The next lemma describesLσ(s)for special permutationsσ.

Lemma 4. Forσ∈S⁰n,set

Rσ(s) = Y

1≤k≤n−1 σ⁻¹α_k∈∆/ ⁰

1−q^{−hσλs−ρ,αki}

, ζb_σ^[n](s) = Y

1≤i≤n−1 σ(i)>σ(n)

ζbX(hλs,αini) ζbX(hλs,αini+ 1),

bζ_σ^[<n](s) :=







 Y

1≤k≤n−2 σ(k)>σ(k+1)

1−q^{−hλ−ρ,αki}

· Y

1≤i<j≤n−1 σ(i)>σ(j)

ζbX(hλ,αiji) ζbX(hλ,αiji+ 1)







 λ=λ_s

.

Then

Lσ(s) = 1

Rσ(s)·ζbσ^[n](s)·bζσ^[<n](s). [17]

Proof: This is obtained by regrouping the terms of Eq.14for special permutationσ∈S⁰_n, following the discussions above. We first cancel the terms in the numerator of the first factor in Eq.14forα∈Aσwith the corresponding terms in the denominator forβ=σα.

The first factor1/Rσ(s)in Eq.17is the value atλ=λσ of the product of the remaining termsβ∈∆rσAσin this denominator.

The second factorζbσ^[n](s)in Eq.17is the value atλ=λσ of the product of the terms in the second factor in Eq.14forα /∈Φ⁰⁺; i.e.,α=ei−en>0. The third factorζbσ^[<n](s)in Eq.17, which can also be written

ζb^[<n]_σ (s) =







 Y

α∈B_σ

(1−q^{−hλ−ρ,αi})· Y

α∈Φ⁰⁺

σ(α)<0

ζbX(hλ,αi) ζbX(hλ,αi+ 1)







 λ=λ_s

,

is obtained by collecting all of the remaining zeta factors and rational factors appearing in the numerator.

The terms occurring inζbσ^[<n](s)are of two types: Forα∈Bσwe must combine the quantities(1−q^{−hλ−ρ,αki})and ^{ζbX(hλ,αiji)}

ζb_X(hλ,α_iji+1)

before taking the limit asλ→λsbecause 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 toα∈Φ⁰⁺rBσ, we could simply substituteλ=λsinstead of taking a limit. We can say this differently as follows. By abuse of notation we write simplyζbX(1)for the limit ass→1of(1−q^1−s)bζX(s). (It should be writtenbv1, as defined in Eq.2, but the “bζX(1)” notation will let us write more uniform formulas.) Then the definition ofζbσ^[<n](s)can be rewritten using the first equation in Eq.11as

ζb_σ^[<n](s) =Y

k≥1

ζbX(k) ζbX(k+ 1)

!m_σ(k)

=Y

k≥1

ζbX(k)^nσ(k), [18]

where

mσ(k) = X

1≤i<j≤n−1 σ(i)>σ(j),j−i=k

1 = #{α∈Φ⁰⁺:σα <0,hρ,αi=k} [19]

and

nσ(k) =mσ(k)−mσ(k−1) , nσ(1) =mσ(1) = #Bσ. [20]

Eq.18gives an explicit formula for the third factor in Eq.17, which, as one sees, does not depend onsat all. The other two factors in Eq.17, which do depend ons, are computed later, inSection 5.

Lemmas 3and4calculate the third factorω^G,P_X (s)in the definition Eq.8ofbζ_X^G,P(s)in the special caseG=SLn,P=Pn−1,1, but since some of the numbersnσ(k)in Eq.18may 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 factorD^G,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 forD^G,P(s)and ζb_X^G,P(s)can be written explicitly as follows:

MATHEMATICS Lemma 5. We have

D^SLn,Pn−1,1(s) =

n−1

Y

k=2

ζbX(k)·bζX(ns) [21]

and

bζ_X^SLn(s) =q^{n(n−1)2 (g−1)}·D^SLn,Pn−1,1(s)·ω^(SL_X ^n,Pn−1,1)(s). [22]

Proof: In view of the definitions Eqs.7and8, we must show thatMP(k,h)equals 1 ifk= 0and2≤h<nork= 1andh=nand vanishes otherwise, which follows easily from Eq.6since herew0=

1 2 · · · n n n−1 · · · 1

.

4. Special Permutations

In this section we describe special permutations explicitly. Recall from Section 3that σ is special if and only ifAσtBσ= ∆⁰, where Aσ and Bσ are defined as in Eq. 15. This implies thatσ is special if and only if σ(i+ 1) =σ(i) + 1or σ(i+ 1)< σ(i) for all 1≤i≤n−2 (or equivalently, sinceσ is a permutation, if and onlyσ(i+ 1)≤σ(i) + 1for all 1≤i≤n−2). Denote by t1> . . . >tm the distinct values ofσ(i)−i for1≤i≤n−2and byIν(1≤ν≤m) the set ofi∈ {1,. . .,n−2}withσ(i)−i=tν. Thenσ mapsIν onto its image Iν⁰=σ(Iν)by translation bytν, and we haveS

Iν={1,. . .,n−1}andS

Iν⁰={1,. . .,n}r{a}, where a=σ(n)∈ {1,. . .,n}. It is easy to check thatI1<· · ·<Im (in the sense that all elements of Iν are less than all ele- ments ofIν+1if1≤ν≤m−1) andI1⁰>· · ·>Im⁰ (in the same sense). [Indeed, letAdenote the set of indexesi∈ {1,. . .,n−2}

withσ(i+ 1) =σ(i) + 1. Thenσ(i)−i is constant when we pass from anyi∈Atoi+ 1, so each setIν is a connected interval that is contained inAexcept for its right end-pointi0, which satisfiesσ(i0+ 1)< σ(i0), so thati0+ 1belongs to anIµsatisfying tµ<tν and henceµ > ν. But thenIµ contains a point that is bigger than one of the points ofIν 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 ofIµ lies to the right of all ofIν and that all ofIµ⁰ lies to the left of all of Iν⁰, proving the assertion.] These properties characterize spe- cial permutations and are illustrated in Fig. 1, in which the lengths of the intervalsIν withIν⁰ above (respectively below)a are denoted byk1,. . .,kp (resp. by`1,. . .,`r), so thatPp

i=1ki=n−a, Pr

j=1`j=a−1, andp+r=m. We denote the correspond- ing special permutation byσ(k1,. . .,kp;a;l1,. . .,lr)and also define two sequences of numbers0 =K0<K1<· · ·<Kp=n−aand 0 =L0<L1<· · ·<Lr=a−1by

Ki= k1+· · ·+ki (1≤i≤p), Lj=l1+· · ·+lj (1≤j≤r). [23]

Remark: Denote bySn,a (a= 1,. . .,n) the set of special permutations in Sn withσ(n) =a. From the above description we find thatSn,a∼=Xn−a×Xa−1, whereXK forK≥0is the set of ordered partitions ofK (decompositionsK=k1+· · ·+kp with allki≥1). Clearly the cardinality ofXK equals 1 ifK= 0(in which case onlyp= 0can occur) and2^K−1 ifK≥1(the ordered partitions ofKare in 1:1 correspondence with the subsets of{1,. . .,K−1}, each such subset dividing the interval[0,K]⊂Rinto intervals of positive integral length), so|Sn,a|equals2ⁿ⁻²fora∈ {1,n}and2ⁿ⁻³for1<a<n, and the whole setS⁰nhas cardinality 2ⁿ⁻³(n+ 2).

Fig. 1. The special permutationσ(k1,. . .,kp;a;l1,. . .,lr).

5. Proof ofTheorem 2

In this section, we use the characterization of special permutations given inSection 4to calculate the rational factorRσ(s)and the zeta factorsζbσ^[n](s)andζb^[<n]σ (s)appearing inLemma 4explicitly for special permutationsσ. We begin withRσ(s).

Lemma 6. For the special permutationσ=σ(k1,. . .,kp;a;l1,. . .,lr),the quantityRσ(s)defined in Lemma 4 is given by Rσ(s) = (1−q^k1+k2)· · ·(1−q^kp−1+kp)·(1−q^ns−n+a+kp)·(1−q^{−ns+n−a+l1+1})·(1−q^l1+l2)· · ·(1−q^lr−1+lr).

Proof: By definition,

Rσ(s) = Y

1≤k≤n−1 σ⁻¹(α_k)/∈∆⁰

1−q^{−hσλs−ρ,αki}

= Y

1≤k≤n−1 σ⁻¹(α_k)∈∆/ ⁰

1−q^{1−hλs,σ−1αki} .

For eachkoccurring in this product, writeσ⁻¹(αk) =ei−ej=:αij. Then the conditionαij∈/∆⁰says 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 ofσgiven in the last section. From that picture, we see that theks occurring in the product, in decreasing order, together with the corresponding values ofiandj, are given by the first three columns of the following table:

k i=σ⁻¹(k) j=σ⁻¹(k+ 1) 1− hλs,αiji

n−Kµ(1≤µ <p) Kµ+1 Kµ−1+ 1 kµ+kµ+1

a n Kp−1+ 1 ns−n+a+kp

a−1 n−a+l1 n −ns+n−a+l1+ 1

n−Lν(1≤ν <r) Lν+1 Lν−1+ 1 lν+lν+1

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

We next consider the zeta factorζbσ^[n](s).

Lemma 7. For the special permutationσ=σ(k1,. . .,kp;a;l1,. . .,lr),the zeta factorζbσ^[n](s)ofLσ(s)is given by

ζbσ^[n](s) = ζbX(ns−n+a) ζbX(ns) .

Lemma 7 implies in particular that to normalize bζσ^[n](s) we at least need to clear the denominator by multiplying by the zeta factorζbX(ns).

Proof: This is much easier. Fromλs= (ns−n)$+ρ, we gethλs,ei−eni=ns−i. Moreover, by Fig. 1 inSection 4, for the special permutationσ=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 ofζbσ^[n](s)given inLemma 4, we have

ζb_σ^[n](s) = Y

α=e_i−e_n,i≤n−1 σ(i)>σ(n)

ζbX(hλ,αi) bζX(hλ,αi+ 1)

λ=λs

=

n−a

Y

i=1

ζbX(ns−i)

ζbX(ns−i+ 1) = bζX(ns−n+a) ζbX(ns)

as asserted.

Finally, we treat the zeta factor ζbσ^[<n](s). However, with the normalization stated in Lemma 5, to obtain the group zeta function ζb_X^SLn(s), it suffices to investigate the product ζbσ^[<n](s)·Q

i≥2ζbX(i)⁻ⁿ⁽ⁱ⁾ or, equivalently, by Eq. 18, the product ζbX(1)^#BσQ

i≥2ζbX(i)^{nσ(i)−n(i)}, which we write asQ

i≥1bζX(i)^rσ(i)with rσ(k) =

( #Bσ ifk= 1, nσ(k)−n(k) ifk≥2,

where the numbersn(k)are defined, in analogy with the numbersnσ(k)inSection 3(Eqs.19and20), by m(k) = #{α >0 :hρ,αi=k}, n(k) =m(k)−m(k−1).

Clearlym(k) =n−kfor1≤k≤nandn(k) =−1for2≤k≤n.

Lemma 8. For the special permutationσ=σ(k1,. . .,kp;a;l1,. . .,lr),we have

Y

i≥1

ζbX(i)^rσ(i)=

p

Y

i=1

bvk_i·

r

Y

j=1

bvl_j. [24]

In particular,rσ(k)≥0.

MATHEMATICS Proof: This is based on a detailed analysis ofrσ(k). Obviously,

rσ(1) = #{α∈∆⁰:σα <0}= #{(i,i+ 1) : 1≤i≤n−2, σ(i)> σ(i+ 1)}.

Ifk≥2, by definition,

m(k)−mσ(k) = #{α >0 :hρ,αi=k} −#{α∈Φ⁰⁺:σα <0,hρ,αi=k}

= #{ei−en:hρ,αi=k}+ #{α∈Φ⁰⁺:σα >0,hρ,αi=k}

= 1 + #{α∈Φ⁰⁺:σα >0,hρ,αi=k},

since, by Eq.10,{ei−en:hρ,αi=k}={en−k−en}. Thus, by applying the characterization graph inSection 4for special permu- tationσ(k1,. . .,kp;a;l1,. . .,lr), we conclude thatα=αij∈Φ⁰⁺satisfyingσα >0(or equivalentlyα=αijsatisfyingi<j≤n−1 andσ(i)< σ(j)) if and only ifiandj belong to the same block, sayIµfor someµ, associated toσ(k1,. . .,kp;a;l1,. . .,lr), and also σ(j)∈Iµ(or equivalentlyj+ 1∈Iµ), since otherwiseσ(αij)<0.

Denote by(m(k)−mσ(k))µ(resp.rσ,µ(k)) the contribution tom(k)−mσ(k)(resp. torσ(k)) of the blockIµ. With the discussion above, we have

m(k)−mσ(k) =X

µ(m(k)−mσ(k))µ and rσ(k) =X

µrσ,µ(k).

Fix someµand letIµ:={a+ 1,a+ 2,. . .,a+b}witha,b∈Z^>0. Clearly, whenk= 1,rσ,µ(1) = #{(a+b−1,a+b)}= 1, since, for other(i,i+ 1)s,σ(i)< σ(i+ 1). Moreover, whenk≥2, by Eq.10and 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 fixedi(witha+ 1≤i<a+b),

#{(i,j) :a+ 1≤i<j<a+b, j=i+k}=

(1 i+k<a+b 0 i+k≥a+b.

Hence,(m(k)−mσ(k))µ=b−(k+ 1). This implies that for allk≥1rσ,µ(k) = (m(k−1)−mσ(k−1))µ−(m(k)−mσ(k))µ= 1.

Consequently,

Y

i≥1

ζbX(k)^rσ,µ(k)=ζbX(1)ζbX(2)· · ·ζbX(b).

Eq.24follows.

CombiningLemmas 5,6,7, and8, we get

ζb_X^SLn(s) q^{n(n−1)2 (g−1)}

=Y

i≥2

ζbX(i)⁻ⁿ⁽ⁱ⁾· lim

λ→λ_s



 Y

α∈∆_P

(1−q^{−hλ−ρ,α∨i})·ω^SL_Xⁿ(λ)





=

n

X

a=1

X

k₁,...,k_p>0 k1+···+k_p=n−a

bvk₁· · ·bvkp

(1−q^k1+k2). . .(1−q^kp−1+kp)· 1 1−q^ns−n+a+kp

×bζ(ns−n+a) X

l₁,...,l_r>0 l1+···+l_r=a−1

1

1−q^{−ns+n−a+1+l1} · bvl₁· · ·bvl_r

(1−q^l1+l2). . .(1−q^lr−1+lr).

This completes the proof ofTheorem 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 ranknnonabelian zeta functionζbX,n(s)found by Mozgovoy and Reineke, namely the following:

Theorem (theorem 7.2 of ref. 5). The functionbζX,n(s)is given by

bζX,n(s) =q(ⁿ₂)^(g−1)n−1X

h=1

X

n1 ,...,nh>0 n1 +···+nh=n−1

bvn₁· · ·bvn_h

Qh−1

j=1(1−q^nj+nj+1)

× ζbX(ns) 1−q^−ns+n1+1 +

h−1

X

i=1

(1−q^ni+ni+1)·ζbX(ns−(n1+· · ·+ni))

(1−q^{ns−(n1+···+ni−1)})(1−q^{−ns+n1+···+ni+1+1}) + ζbX(ns−n+ 1) 1−q^{ns−(n1+···+nk−1)}

!

. [25]

This already looks very similar toTheorem 2, and the precise equality of the two formulas will be verified inSection 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 toSection 7.

The first ingredient is that of semistable pairs and triples. Fix an integral regular projective curveX over a finite fieldF^q. By a pair (E,s)overX we mean a vector bundleEonX together with a global sectionsofE onX. Such pairs form anFq-linear category, a morphism(E,s)→(E⁰,s⁰)being an element(λ,f)∈Fq×HomX(E,E⁰)such thatf ◦s=λs⁰. A pair(E,s)is calledτsemistable (τ∈R) ifµ(F)≤τfor any subbundleF ofE andµ(E/F)≥τfor any subbundleF ofE withs∈H⁰(X,F). Here, as usual,µ(E) denotes the Mumford slope ofE. For(r,d)∈Z>0×Zwe denote byM^τ_X(r,d)the moduli stack ofτ-semistable pairs(E,s)of rank rand degreed. Ifτ=d/r, then this is the same as the usual slope semistability ofE, so if we writeMX(r,d)for the moduli space of semistable bundles of rankrand degreed, then (cf. corollary 3.7 of ref. 5)

X

(E,s)∈M^d/r_X (r,d)

1

#Aut(E,s) = 1 q−1

X

E∈M_X(r,d)

q^h0(X,E)−1

#AutE .

Next, we consider triplesE= (E0,E1,s)consisting of two coherent sheavesE0,E1onX and a morphisms:E1→E0. These triples form an abelian category which we denote byA. The tripleE= (E0,E1,s)is calledµτ semistable ifµτ(F)≤µτ(E)for any subobject FofE, where

µτ(E) := degE0+ degE1+τ·rankE1

rankE0+ rankE1

.

We also introduce χ(E,F) :=P2

k=0(−1)^kdim Ext^kA(E,F). It is known that χ(E,F) =χ(E0,F0) +χ(E1,F1)−χ(E1,F0), where as usual,χ(E,F) := dim Hom(E.F)−dim Ext¹(E,F). Forα= (r,d),β= (r⁰,d⁰)∈Z>0×Z, setχ(α) =d−(g−1)r and hα,βi:= 2(rd⁰−r⁰d). Similarly, forα= (α,v), β= (β,w)withv,w∈Z≥0we sethα,βi:=hα,βi −vχ(β) +wχ(α).

The next ingredients are Hall algebras and integration maps. LetK0( St_Fq)be the Grothendieck ring of finite-type stacks overFq

with affine stabilizers andLbe the Lefschetz motive. We introduce the coefficient ringR=K0( St_Fq)[L^±1/2]and define the quantum affine planeA0to be the completion of the algebraR[x1,x₂^±1]with the multiplication

x^α◦x^β := (−L^1/2)^hα,βix^α+β. (Here the completion is defined by requiring that forf=P

α∈N×Zfαx^α∈A⁰and anyt∈Rthere are only finitely many(r,d)with fr,d6= 0and _r+1^d <t.) If we further denote byA0the category of coherent sheaves onX and byH(A0)its associated Hall algebra, whose multiplication[E]◦[F]counts extensions fromExt¹(F,E), then we have a morphism of algebras

I: H(A0) −→ A⁰

E 7→ (−L^1/2)^χ(E,E)· ^x_[AutE]^ch(E) ,

which we call the integration map. Herech(E) := (rankE, degE). Similarly, if we introduce a second quantum affine planeAas the completion of the algebraR[x1,x₂^±1,x3]with the multiplication

x^α◦x^β := (−L^1/2)^hα,βix^α+β,

then we have an integration map on the Hall algebraH(A),

I:H(A) −→ A

E 7→ (−L^1/2)^χ(E,E)· ^x_[AutE]^cl(E) ,

where cl(E) := (rank E0, deg E0, rank E1). We haveI|H(A₀) =I. The map I is not an algebra morphism in general, but if Ext²(F,E) = 0, thenI(E ◦ F) =I(E)I(F).

The last and most important ingredient of the proof in ref. 5 is a wall-crossing formula. Forα= (r,d)∈Z^>0×Zandτ∈R, let

u(α) := (−L^−1/2)^χ(α,α)+d[MX(α)]

be the motivic class ofMX(α)counting semistable bundlesEonX withchE=α, and similarly set

MATHEMATICS

fτ(α) = (L−1)(−L^−1/2)^χ(α,α)+d[M^τX(α)].

We introduce the two generating series

uτ = 1 + X

µ(α)=τ

u(α)x^α∈A⁰, fτ =X

α

fτ(α)x^(α,1)∈A.

Then the ranknnonabelian zeta function forX can be expressed as ζX,n(s) = (q−1)X

k≥0

[MX(n,kn)]q^−sk =q^{n(n−1)2 (g−1)}X

k≥0

fk(n,kn)q^−ks.

We can also identify the moduli stackM^∞X(1,d)with the Hilbert schemeHilb^dX or withSym^dX, thedth symmetric product ofX. Consequently,

f∞:=x1x3

X

d≥0

[Sym^dX]x2^d =x1x3ZX(x2),

whereZX(t)is the Artin zeta function withζX(s) =ZX(q^−s). (This can be interpreted as the limiting special case offτ asτ→ ∞, since the condition of semistability with respect toτof a pair(E,s)in the limitτ→ ∞is equivalent to the requirement that coker(s) is finite.) Finally, set

u≥τ :=

→

Y

τ⁰≥τ

u_τ0,

where the product is taken in the decreasing slope order, and, for an elementg=P

αgαx^(α,1)∈A, set g|µ≤τ := X

µ(α)<τ

gαx^(α,1).

Then, using the theory of Hall algebras and wall-crossing techniques, the main result (theorem 5.4 of ref. 5) is the identity fτ= u⁻¹_>τ◦f∞◦u≥τ

_µ≤τ (τ∈R).

Eq.25is 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 ofTheorem 1 and Structure of the FunctionζX,n(s)

To complete the proof ofTheorem 1, we verify the term-by-term equality of the sums appearing in Eqs.5and25. Clearly, the factor q(ⁿ2)^(g−1)is the same in both cases. Both sums have the form of a linear combination ofζbX(ns−k)with0≤k≤n−1, so we have only to check the equality of the coefficients. The casek= 0is immediate: SinceB0(x)is identically 1, the coefficient ofbζX(ns)in the sum in Eq.5isBn−1(q^1−ns), which by Eq.4is identical with the coefficient ofζbX(ns)in the sum in Eq.25. (Setp=h,ki=nh+1−i.) The casek=n−1is exactly similar or can be deduced from the casek= 0by noticing that Eq.5is invariant underk→n−1−k, s→1−sand Eq.25undernj→nh+1−j,i→h−i, ands→1−s. If0<k<n−1, then the coefficient ofζbX(ns−k)in the sum in Eq.25can be rewritten as

X

0<i<h<n

X

n1 +···ni=k ni+1 +···+nh=n−1−k

bvn₁· · ·bvn_i

Qi−1

j=1(1−q^nj+nj+1)· 1

1−q^ns−k+ni · bvn_i+1· · ·bvn_h

Qh−1

j=i+1(1−q^nj+nj+1)· 1

1−q^{−ns+k+ni+1+1}

! ,

and since the summations over the tuples(n1,. . .,ni)with sumk and the tuples(ni+1,. . .,nh)with sumn−k−1are indepen- dent, this equalsBk(q^ns−k)Bn−k−1(q^k+1−ns) as required. This completes the comparison of Eqs.5and25and hence the proof ofTheorem 1.

We end this paper by looking briefly at the structure of the explicit formula for the higher-rank zeta functionζX,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 equationζbX,n(1−s) =ζbX,n(s), which, as we have already said, follows immediately from Eq. 5by interchangingk andn−k−1and using the known functional equationζbX(1−s) =ζbX(s). The other one concerned the form of ζX,n(s). Here it is more convenient to work with the variablest=q^−s andT=q^−ns=tⁿ, writingζX(s)andζX,n(s)asZX(t)and ZX,n(T), respectively, and similarlybζX(s) =ZbX(t)andζbX,n(s) =ZbX,n(T)withZbX(t) =t^1−gZX(t),ZbX,n(T) =T^1−gZX,n(T). It is well known thatZX(t)has the formP(t)/(1−t)(1−qt)whereP(t) =PX(t)is a polynomial of degree2g, and the assertion is thatZX,n(T), which from the definition Eq.1is just a power series inT, has the corresponding formPn(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 becomes

q⁻(ⁿ₂)^(g−1)

ZbX,n(T) =

n−1

X

k=0

Bk(q^−kT⁻¹)ZbX(q^kT)Bn−k−1(q^k+1T). [26]