Higher-rank zeta functions for elliptic curves
Lin Wenga,1and Don Zagierb,c,1
aGraduate School of Mathematics, Kyushu University, 819-0395 Fukuoka, Japan;bMax Planck Institute for Mathematics, 53111 Bonn, Germany;
andcMathematics Section, International Centre for Theoretical Physics, 34151 Trieste, Italy
Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved January 3, 2020 (received for review July 19, 2019)
In earlier work by L.W., a nonabelian zeta function was defined for any smooth curveXover a finite fieldFqand any integern≥1 by
ζX/Fq,n(s)=X
[V]
|H0(X,V)r{0}|
|Aut(V)| q−deg(V)s ( (s)>1),
where the sum is over isomorphism classes ofFq-rational semistable vector bundlesVof ranknonXwith degree divisible byn. This function, which agrees with the usual Artin zeta function ofX/Fqifn=1, is a rational function ofq−swith denominator (1−q−ns)(1− qn−ns) and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet series
X/Fq(s)=X
[V]
1
|Aut(V)|q−rank(V)s ( (s)>0),
where the sum is now over isomorphism classes of Fq-rational semistable vector bundles V of degree 0 on X, is equal to Q∞
k=1ζX/Fq(s+k), and use this fact to prove the Riemann hypothesis forζX,n(s) for alln.
elliptic curves over finite fields|semistable bundles|higher-rank zeta functions|Riemann hypothesis
L
etXbe a smooth projective curve of genusgover a finite fieldFq. For all positive integersn, a “nonabelian ranknzeta function”ofX/Fqwas defined in refs. 1 and 2 by
ζX,n(s) =ζX/Fq,n(s) = X
d≡0 (modn)
X
[V]∈MX,n(d)
qh0(X,V)−1
|Aut(V)| q−ds, [1]
whereMX,n(d)denotes the moduli stack ofFq-rational semistable vector bundles of ranknand degreedonX, andAut(V)and h0(X,V)denote the automorphism group ofV and the dimension of its space of global sections. In other words, if we define, for anyd, theαandβinvariants ofX/Fqby
αX,n(d) = X
[V]∈MX,n(d)
qh0(X,V)−1
|Aut(V)| , βX,n(d) = X
[V]∈MX,n(d)
1
|Aut(V)|, [2]
thenζX,n(s)is the generating function of the numbersαX,n(mn): ζX,n(s) = X
d≡0 (modn)
αX,n(d)td =
∞
X
m=0
αX,n(mn)Tm, [3]
wheret=q−s,T=tn=Q−swithQ=qn. Note thatqh0(X,V)−1 =|Hom( X,V)r{0}|, soαX,n(d)counts the number of isomor- phism classes of pairs consisting of a semistable vector bundleV of ranknand degreedtogether with an embedding X,→V, while the beta invariants were studied in the case(d,n) = 1(when semistability coincides with stability) by Harder and Narasimhan (3) in their famous work on moduli spaces of vector bundles. For the zeta function, on the other hand, it is crucial (for reasons indicated briefly in ref. 4, remark 1) to restrict to the opposite casen|d.
Significance
The Riemann zeta function and the Riemann hypothesis concerning its zeros are perhaps the most famous function and most famous conjecture in all of number theory. Almost 100 years ago, Artin defined an analog, the “Artin zeta function,” for a curve (one-dimensional variety) over a finite field. In 2005, L.W. found a whole series of zeta functions for curves over finite fields, including the original Artin zeta function. The analog of the (still unsolved) classical Riemann hypothesis was proved for the Artin L function by Andre Weil, but the corresponding statement for Weng’s higher zeta functions is still open. In this paper it is proved for the simplest nontrivial case, when the curve has genus one (elliptic curve).
Author contributions: L.W. and D.Z. performed research and 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 February 18, 2020.y
MATHEMATICS
The following properties ofζX,n(s)were shown in ref. 2, using Riemann–Roch, duality, and vanishing for semistable bundles:
Theorem. DefineζX,n(s)for alln≥1byEq.1. Then
1) The functionζX,1(s)equalsζX(s),the Artin zeta function ofX/Fq. 2) There exists a degree2gpolynomialPX,n(T)∈ [T]such that
ζX,n(s) = PX,n(T)
(1−T)(1−QT). [4]
3) The functionζX,nsatisfies the functional equation
ζX,n(1−s) =Q(g−1)(2s−1)·ζX,n(s). [5]
The following conjecture, verified in many examples, was also proposed in ref. 2.
Conjecture (Riemann Hypothesis). IfζX,n(s) = 0,then (s) =12.
Of course in the classical casen= 1this is a famous theorem of Weil.
A further indication that the higher zeta functions defined by Eq. 1are natural objects is that they turn out to coincide with the special caseG=SLn,P=Pn−1,1of the zeta functionsζXG,P(s)defined in refs. 2 and 5 for suitable reductive algebraic groups Gand parabolic subgroupsP. This equality, which was conjectured in ref. 2, is proved in the companion paper (4). In this paper we concentrate on the case of elliptic curvesX=E, i.e.,g= 1, where we can give much more complete results. In this case we haveαE,n(d) = (qd−1)βE,n(d)ford>0(becauseh0(V)−h1(V) =d by the Riemann–Roch theorem andh1(V)vanishes) and βE,n(mn) =βE,n(0)for allm(because tensoring with a line bundle of degree 1 gives an isomorphism between the sets of rankn semistable vector bundles of degreedand degreed+nfor anyd∈Z), so the zeta function Eq.3reduces simply to
ζE,n(s) =αE,n(0) +βE,n(0) (Q−1)T
(1−T)(1−QT). [6]
We will give explicit formulas and generating functions forαE,n(0)andβE,n(0)and prove the Riemann hypothesis forζE,n(s). The key fact isTheorem 5, which gives an elegant expression for the generating functionP
βE,n(0)q−ns as a product of the shifts of the zeta function ofE.
1. Statement of Main Results
From now onE denotes an elliptic curve overFq. By an “Atiyah bundle” overE we mean any direct sum of the vector bundles I1,I2,. . . overE defined by Atiyah in ref. 6:I1= E is the trivial line bundle andIk fork≥2is the unique (up to isomorphism) nontrivial extension ofIk−1byI1. Forn≥1letαAtE,n(0)andβEAt,n(0)be the numbers defined as in ref. 1, but with the summations now ranging only over Atiyah bundles. We show the following:
Theorem 1. Forn≥1we have
αAtE,n(0) =X
m
qm1+m2+··· −1ε(m1)ε(m2)· · ·
qN(m1,m2,...) , [7]
βE,nAt (0) =X
m
ε(m1)ε(m2)· · ·
qN(m1,m2,...) , [8]
where the sum is over all partitionsn=m1+ 2m2+ 3m3+· · · ofnand ε(m) = qm2
|GLm(Fq)| = qm(m+1)/2
(qm−1)(qm−1−1)· · ·(q−1), [9]
N(m1,m2,. . .) = X
k,`≥1
mkm`min(k,`). [10]
From this we deduce the following simple formulas forαAtE,n(0)andβE,nAt (0)(“special counting miracle”) by a direct combinatorial argument:
Theorem 2. Forn≥0we have
αAtE,n+1(0) =βE,nAt (0) =q−nε(n). [11]
This in turn is used together with considerations of algebraic structures of semistable bundles of degree 0 to obtain the following intrinsic relation betweenαandβinvariants (“general counting miracle”):
Theorem 3. For alln≥0we have
αE,n+1(0) =βE,n(0). [12]
We mention thatTheorem 3has been generalized to curves of arbitrary genus by Sugahara.
The above results, whose proofs are given in Section 2, show that the higher-rank zeta functions for elliptic curves are completely determined by their beta invariants. To understand the latter, we first use results of Harder and Narasimhan (3), Desale and Ramanan (7), and Zagier (8) to get an explicit formula for βE,n(0) in terms of special values of the Artin zeta function ofE/Fq:
Theorem 4. Forn≥1we have
βE,n(0) =
n
X
k=1
(−1)k−1 X
n1+···+nk=n n1,...,nk>0
vEn1· · ·vEnk
(qn1+n2−1)· · ·(qnk−1+nk−1), [13]
where the numbersvEn (n>0)are defined by
vE,n=ζE∗(1)ζE(2)· · ·ζE(n), ζE∗(1) = lim
s→1(1−q1−s)ζE(s). [14]
InSection 3, we will use Eq.13and a fairly complicated combinatorial calculation to establish the following simple formula for the generating Dirichlet series of the invariantsβE,n(0):
Theorem 5. Define a Dirichlet series E(s) = E/Fq(s)for (s)>0by
E/Fq(s) =
∞
X
n=0
βE,n(0)q−ns =X
[V]
1
|Aut(V)|q−rank(V)s, [15]
where the second sum is over isomorphism classes ofFq-rational semistable vector bundlesV of degree0onE.Then
E/Fq(s) =
∞
Y
k=1
ζE/Fq(s+k). [16]
This formula will then be used inSection 4to prove the following estimate:
Proposition 6. Forn≥2we have the inequalities
1< βE,n(0)
βE,n−1(0) < qn/2+ 1
qn/2−1. [17]
(In fact, we prove a stronger estimate; see Eq.52.) Combining these bounds with Eqs.6and12, we deduce the following:
Theorem 7. The Riemann hypothesis is true for elliptic curves.
2. Calculation of theαandβInvariants of Elliptic Curves
In this section we give explicit formulas forαE,n(mn)andβE,n(mn)for an elliptic curveE/Fq. By what was already explained in the Introduction, it suffices to do this form= 0. We will proveTheorems 1–4as stated inSection 1.
A. Automorphisms of Atiyah Bundles..LetV be an Atiyah bundle of ranknoverE as defined inSection 1. ThenV can be uniquely written in the form
V ∼=M
k≥1
Ik⊕mk [18]
for integersmk≥0withP
k≥1kmk=n, soTheorem 1follows from the following proposition:
Proposition 8. ForV/Eas inEq.18,we haveh0(V) =P
k≥1mkand
|Aut(V)|=qN(m1,m2,...) Y
k≥1
q−mk2|GLmk(Fq)|, [19]
where N(m1,m2,. . .)is defined as inEq.10.
Proof:By theorem 8 of ref. 6, for any integersk,`≥1we have
Ik⊗I` ∼= M
|k−`|<m<k+`
m≡k+`−1(mod2)
Im
and consequently, sinceIkis self-dual,
dim Hom(Ik,I`) =h0(Ik∨⊗I`) =h0(Ik⊗I`) = min(k,`). [20]
MATHEMATICS
This can be seen explicitly as follows. The bundleIk has a realization given locally away from0∈E byk-tuples of regular functions and near 0 byk-tuples(f1,. . .,fk), wheref1 and allzfi−fi−1 (2≤i≤k) are regular, wherez is a local parameter at 0. (In other words,fi is allowed to have a pole of orderi−1at 0 and, for instance, the residue off2 equals the value off1 at 0.) The space Hom(Ik,I`)is then spanned by the maps
(f1,. . .,fk) 7→ (0,. . ., 0
| {z }
`−s
,f1,. . .,fs) ( 1≤s≤min(k,`) ).
As a special case of Eq.20we haveh0(Ik) = 1for allk, making the first statement ofProposition 8obvious. We prove the second one in several steps.
1) In the special caseV=Ik, we have
|Aut(Ik)|=qk−1(q−1).
Indeed, from the above description, any endomorphism ofIkhas the form
ϕ=
a1 a2 · · · ak−1 ak
0 a1 · · · ak−2 ak−1
... ... . .. ... ... 0 0 · · · a1 a2
0 0 · · · 0 a1
[21]
for someai∈Fq, andϕis an isomorphism if and only ifa16= 0.
2) More generally, for any positive integerskandmwe have
|Aut(Ik⊕m)|=q(k−1)m2|GLm(Fq)|, [22]
because the automorphisms ofIk⊕mhave the same form as Eq.21, but with eachainow being anm×mmatrix overFqand witha1
invertible.
3) Finally, ifV is a general Atiyah bundle as in Eq.18, then
|Aut(V)|= Y
k6=`
|Hom(Ik,I`)|mkm` ·Y
k
|Aut(Ik⊕mk)|, and Eq.19then follows from Eqs.20and22.
B. Proof ofTheorem 2. Introduce the three generating functions
(x) =
∞
X
n=1
αAtE,n(0)xn−1, (x) =
∞
X
n=0
βE,nAt (0)xn, (x) =
∞
X
n=0
q−nε(n)xn, where the notations are as inTheorem 2. We must show that they coincide.
There is a bijection between terminating sequences(m1,m2,. . .)of nonnegative integers and monotone decreasing terminating sequences(p1,p2,. . .)of nonnegative integers, given by settingpk=P
`≥km`,mk=pk−pk+1. Under this correspondence, we have N(m1,m2,. . .) =
∞
X
k=1
k mk(mk+ 2mk+1+ 2mk+2+· · ·)
=
∞
X
k=1
k(pk−pk+1)(pk+pk+1) =
∞
X
k=1
pk2.
HenceTheorem 1shows that (x)and (x)can be given as (x) = 1
x
∞
X
p=1
(qp−1) p(x), (x) =
∞
X
p=0
p(x), [23]
with p(x)∈ (q)[[x]]defined by
p(x) = X
p=p1≥p2≥...≥0
∞
Y
k=1
ε(pk−pk+1)q−p2kxpk or, equivalently (seth=p2on the right-hand side), by the recursive formulas
0(x) = 1, qp2x−p p(x) =
p
X
h=0
ε(p−h) h(x). [24]
We claim that the solution of this recursion is given by
p(x) =
p
Y
j=1
qx
(qj−1)(qj−x) (p≥0). [25]
To prove this, we denote the right-hand side of Eq.25byBp(x)and show thatBp(x)satisfies the same recursion Eq.24as p(x), i.e., that we have
B(xb ,t) =
∞
X
n=0
ε(n)tn
!
B(x,t) = (qt)B(x,t), [26]
whereB(x,t)andB(xb ,t)are the two generating series defined by
B(x,t) =
∞
X
p=0
Bp(x)tp, B(xb ,t) =
∞
X
p=0
qp2x−pBp(x)tp.
But this is now fairly easy. The definition ofBp(x)gives the formulas
(1−x)Bp(qx) = (qp−x)Bp(x), (1−x)(qp−1)Bp(qx) =qx Bp−1(x), which translate into the four generating series identities
(1−x)B(qx,t) =B(x,qt)−xB(x,t),
(1−x) (B(qx,qt)−B(qx,t)) =qxt B(x,t), [27]
and
(1−x)B(qx,b qt) =B(xb ,qt)−xB(xb ,t), (1−x)
B(qx,b qt)−B(qxb ,t)
=qtB(xb ,qt). [28]
Now using the identity (t) = (1−t) (qt), which follows from (qt)− (t) =X
n≥1
qn(n−1)/2tn
(q−1)· · ·(qn−1−1) =t (qt),
we find from Eq.27that (qt)B(x,t)satisfies the same two recursions Eq.28asB(xb ,t)and hence that these two power series are equal. This proves Eq.26and hence also Eq.25and lets us rewrite Eq.23as
(x) = 1
x (B(x,q)−B(x, 1)), (x) =B(x, 1).
Substitutingt=q−1 into the sum of the two equations Eq.27 now gives(1−x) (qx) = (x) and hence (x) = (x), and then substitutingt= 1into the first equation of Eq.27gives (x) = (x). This completes the proof.
C. Proof of Theorem 3. LetVbe a semistable vector bundle of ranknand degree 0 overE/Fq. By the classification of indecomposable bundles on elliptic curves defined over algebraic closed fields given by Atiyah (6), we know that there are no stable bundles of rank
≥2and degree 0 overE:=E⊗FqFq. Consequently, overE, the graded bundleG(V)associated to a Jordan–H¨older filtration ofV decomposes as
G(V) =L1⊕L2⊕ · · · ⊕Ln
for some line bundlesLiof degree 0 onE. (For basics of Jordan–H¨older filtrations and their associated graded bundles for semistable bundles, see, e.g., ref. 1.) SinceLineed not be defined overFq, usually it is a bit complicated to classifyV overE. (This classification problem depends on the arithmetic of the curveEand, specifically, on the number ofFq-rational torsion points of order≤nonE.) Instead of doing this, we first note that to get a nontrivial contribution toαinvariants, we must haveh0(V)6= 0. Guided by this, we regroup the bundles appearing in the summation defining theαinvariant in the following way. Assume (after renumbering) that Lj∼= Efor1≤j≤iandLj6∼= E fori<j≤n. Since there are no nontrivial extensions of E byLj forLj6∼= E, we can uniquely decomposeV asU⊕W, whereU andWareFq-rational semistable bundles of degree 0 overEwith
G(U)∼= ⊕iE , G(W)∼=Li+1⊕ · · · ⊕Ln, Lj E.
Thenh0(E,V) =h0(E,U)andAut(V)∼= Aut(U)×Aut(W)(because there are no nontrivial homomorphisms among EandLj), andUandW range independently over bundles with the properties listed above. Hence
αE,n(0) =
n
X
i=1
α∗E,iβE∗,n−i,
MATHEMATICS
whereα∗E,iandβE,k∗ are the modifiedαandβinvariants defined by
α∗E,i = X
Usemistable G(U)∼= ⊕iE
qh0(E,U)−1
|Aut(U)| , βE,k∗ = X
Wsemistable G(W)∼=L1⊕···⊕Lk
deg(Lj) = 0,Lj E
1
|Aut(W)|.
But, by using Atiyah’s classification of indecomposable bundles on elliptic curves defined over algebraic closed fields again, we know that the bundlesU in the sum definingα∗E,iare precisely the Atiyah bundlesU=L
kIk⊕mk withP
kmk=i. Henceα∗E,i=αAtE,i(0).
By the same argument, of course, we haveβE,n(0) =Pn
i=0βAtE,i(0)βE,n−i∗ . (Note that this time the summation starts ati= 0, whereas forαE,n(0)we started ati= 1becauseα∗E,0= 0.)Theorem 3now follows immediately fromTheorem 2.
D. Proof ofTheorem 4. In this subsection, in whichX is again a curve of arbitrary genusg≥1, we combine results of refs. 3, 7, and 8 to give a closed formula forβX,n(0)for alln≥1.
The invariantβX,n(d)is periodic indof periodnby the same argument as given forg= 1in the Introduction. We renormalize slightly by setting
βbX,n(d) =q−(g−1)n(n−1)/2
βX,n(d), ζbX(s) =q(g−1)sζX(s) [29]
(note that these agree withβX,n(d)andζX(s)in the case wheng= 1), because this gives a simpler functional equationbζX(1−s) = ζbX(s)and will also lead to a formula inTheorem 9that has no explicit dependence ong. We also define
bvXn=ζbX∗(1)bζX(2)· · ·ζbX(n), ζbX∗(1) = lim
s→1(1−q1−s)ζbX(s) [30]
instead of Eq.14. Then the work of Harder and Narasimhan (3) and Desale and Ramanan (7) implies the following relation, involving an infinite summation:
Theorem. Forn≥1and anyd∈Zwe have X
k≥1
X
n1+···+nk=n n1,...,nk>0
X
d1+···+dk=d d1
n1>···>dk nk
q−
P
i<j(dinj−djni) k
Y
j=1
βbX,nj(dj) =bvXn. [31]
ThisTheoremis stated in ref. 7, p. 236 (beginning 9 lines from the bottom), except that there the authors useβandζinstead ofβb andζband write the equation in the slightly different formβbX,n(d) =bvXn− (sum over terms withk≥2in Eq.31) to make it clear that this equation gives a recursive determination of allβbX,n(d). This recursion relation was inverted in ref. 8. We state the result here in detail since in that paper only a corollary (namely, the application to the calculation of the Betti numbers of the moduli space MX,n(d)) was written out explicitly. The following theorem, however, is an immediate consequence of Eq.31and theorem 2 of ref.
8. Note that here the sum is finite!
Theorem 9. Forn≥1and anyd∈Zwe have
βbX,n(d) =X
k≥1
(−1)k−1 X
n1+···+nk=n n1,...,nk>0
k
Y
j=1
bvXnj ·
k−1
Y
j=1
q(nj+nj+1){d(n1+···+nj)/n}
q(nj+nj+1) −1 ,
where{t}fort∈Rdenotes the fractional part oft.
Theorem 4is just the special cased= 0andX=E, sinceβbE,n=βE,n. 3. The Generating Series of the Beta Invariants
A. Explicit Formulas. We keep all notations as in the Introduction andSection 1. Recall that the Artin zeta function ofE and its renormalized special value ats= 1as defined by Eq.14are given by
ζE/Fq(s) = 1−aq−s +q1−2s
(1−q−s)(1−q1−s), ζE/F∗ q(1) = |E(Fq)|
q−1 , wherea∈Zis defined by|E(Fq)|=q−a+ 1and satisfies|a| ≤2√
q. For convenience, from now on we write simplyβninstead of βE,n(0). Note thatβndepends only onqandaand belongs to (q)[a].
The closed formula forβngiven inTheorem 4has O(2n)terms. In this subsection we give several alternative expressions, including closed formulas withp(n) =O(ec
√n
)terms and with O(n3)terms, the generating series formula Eq.16, and a recursion permitting the calculation ofβ1,. . .,βnin O(n) steps. The proofs of these relations are given in the rest of the section.
We begin by calculating the first few values of βn from Eq.13. Here we note that there is considerable cancellation and we always have
βn ∈ 1
(qn−1)· · ·(q−1)Z[a,q], [32]
even though the least common denominator of the denominators of the terms in Eq.13is much greater; e.g., the first few values of βnare given by
β0 = 1 , β1=v1 = q−a+ 1 q−1 , β2 =v2 − v12
q2−1 = (q3−aq+ 1)(q−a+ 1)
(q2−1)(q−1)2 − (q−a+ 1)2
(q2−1)(q−1)2 = (q−a+ 1)(q2+q−a) (q2−1)(q−1) , β3 =v3 −2 v1v2
q3−1 + v13
(q2−1)2 = (q−a+ 1)(q5+q4−(a−2)q3−(2a−1)q2−(a+ 1)q+a2) (q3−1)(q2−1)(q−1) . Some more experimentation shows that in fact much more is true, namely
β1 =w1, β2 = w12 +w2
2 , β3 = w13 + 3w1w2+ 2w3
6 , β4 = w14 + 6w12w2+ 8w1w3 + 3w22 + 6w4
24 , . . .,
where the numberswm=wm,E=wm(a,q) (m≥1)are defined by wm =ζE/∗ Fqm(1) = (αm−1)(αm−1)
qm−1 (α+α=a,αα=q). [33]
These special cases suggest that the following theorem should hold:
Theorem 10.The numbersβn=βE,n(0)are given by
βn = X
n1,n2,···≥0, n1+2n2+···=n
w1n1w2n2· · ·
1n12n2· · ·n1!n2!· · · (n≥0), [34]
where the numberswm=wE,m(m≥1)are defined byEq.33.
Eq.34is the promised formula expressingβnas a sum ofp(n) =O(eπ
√2n/3
)rather than O(2n)terms.
To proceed further, we introduce the generating function
B(x) =BE/Fq(x) =B(x;a,q) =
∞
X
n=0
βnxn. [35]
Then Eq.34is equivalent to the formula
B(x) = exp
∞
X
m=1
wm
xm m
! , and substituting forwmfrom Eq.33we find
B(qx) B(x) = exp
∞
X
m=1
(qm−1)wm
xm m
!
= exp
∞
X
m=1
qm−αm−αm+ 1
m xm
!
= (1−αx)(1−αx)
(1−qx)(1−x) = 1−ax+qx2
(1−x)(1−qx). [36]
This in turn can be rewritten in three different ways, each of which is equivalent toTheorem 10. The first one is obtained by replacing x byx/qk in Eq.36 and taking the product over allk≥1to get the following multiplicative formula for the generating function B(x;a,q):
Theorem 11.The generating functionB(x;a,q)defined inEq.35has the product expansion
B(x;a,q) =
∞
Y
k=1
1−aq−kx +q1−2kx2
(1−q−kx)(1−q1−kx). [37]
Theorem 11is clearly equivalent toTheorem 5ofSection 1, by settingx=q−s.
For the second way, we recall the “q-Pochhammer symbol”(x;q)n, defined forx,q∈CasQn−1
m=0(1−qmx). This also makes sense forn=∞if|q|<1. Since ourqhas absolute value greater rather than less than 1, we replace it by its inverse. Then the calculation in Eq.36is just a version of the “quantum dilogarithm identity”
∞
X
m=1
xm
m(qm−1) = X
m,r≥1
q−rmxm
m = log 1
(q−1x;q−1)∞
(|q|>1 )
MATHEMATICS
(we refer to ref. 9, pp. 28–31, for a review of the quantum dilogarithm), and Eq.37says simply
B(x;a,q) = (q−1αx;q−1)∞(q−1αx;q−1)∞
(q−1x;q−1)∞(x;q−1)∞
. [38]
Together with the standard power series expansions of(x;q)∞and1/(x;q)∞as given in the survey paper (9) just quoted, this implies the following result, which is the abovementioned closed formula forβnwith O(n3)terms.
Theorem 12.The numbersβn=βE,n(0)are given by the sum
βn(E/Fq) = X
n1,n2,n3,n4≥0 n1+n2+n3+n4=n
(−1)n1+n2q(n1 +12 )+(n22)αn3αn4 (q;q)n1(q;q)n2(q;q)n3(q;q)n4
(n≥0),
whereαandαare defined as inEq.33.
Finally, multiplying both sides of Eq.36by their common denominator and comparing coefficients ofxn, we obtain the following:
Theorem 13.The numbersβnsatisfy,and are uniquely determined by,the recursion relation
(qn−1)βn = (qn+qn−1−a)βn−1−(qn−1−q)βn−2 [39]
together with the initial conditionsβ0= 1andβ−1= 0.
Theorem 13gives the simplest algorithm for computingβn of all of the formulas we have given, since, as already mentioned, it calculates eachβnin time O(1) from its predecessors and hence requires time only O(n)to calculate all of the numbersβ1,. . .,βn. We also remark that Eq.39immediately implies the assertion Eq.32by induction onn.
B. Proof ofTheorem 5: First Part. In the previous subsection we formulated four theorems, found experimentally, each of which was equivalent to the others and toTheorem 5. Of these, the simplest by far is the recursion relation Eq.39. Unfortunately, we were not able to find a direct proof that the numbers defined by Eq.13satisfy this recursion, and the proof ofTheorem 5that we give will be indirect and fairly complicated.
There are two main ideas. The first one is to replace the “closed formula” Eq.13forβnby a recursive formula (thus in some sense undoing the calculation in ref. 8 that led to that formula, which began with the recursion Eq.31and then inverted it). To do this, we break up the sum Eq.13intonpieces according to the value of the lastni; i.e., we decomposeβnas
β0 =b0(0), βn =
n
X
m=1
β(m)n (n≥1), [40]
whereβn(m)=β(m)n (E/Fq) =βn(m)(a,q)is the partial sum defined by β(m)n =
∞
X
k=1
X
n1,...,nk−1≥1,nk=m n1+···+nk=n
(−1)k−1vn1· · ·vnk
(qn1+n2−1)· · ·(qnk−1+nk−1) (n≥m≥1).
Denoting the last-but-one variablenk−1in this sum bypwheneverkis at least 2, we find
βn(m) =vm·
1 ifm=n,
−
n−m
P
p=1
βn−m(p)
qm+p−1 if 1≤m≤n−1,
which defines all of the numbersβ(m)n (and hence also all of the numbersβn) by recursion. Multiplying this formula by xn and summing over alln≥0, we find that the generating functions
B(m)(x) =B(m)E/Fq(x) =B(m)(x;a,q) =
∞
X
n=0
βn(m)xn [41]
of theβn(m)(observe that the sum here actually starts atn=m, so thatB(m)(x) =O(xm), and also thatB(0)(x)≡1) satisfy the identity
B(m)(x) =vmxm 1−
∞
X
p=1
B(p)(x) qm+p−1
!
(m≥1). [42]
A natural strategy of proof would therefore be to guess a closed formula for the individual seriesB(m)(x)that satisfies the same recursion and that gives Eq.37when summed overm≥0. Unfortunately, we were not able to do this here, so that we have to argue indirectly. The second idea is therefore to prove the identity for special values of the parametera. Since the desired formula Eq.37is
equivalent to the recursion Eq.39, which is an identity among polynomials inaand therefore is true if it can be verified for infinitely many values of the argumentafor eachn, it is enough to prove the identity Eq.37only for the special values
a =ak :=qk+1+q−k (k∈Z, k≥0). [43]
We denote byβn,k andβ(m)n,k the specializations ofβn andβn(m) to this value of a and byBk(x)andB(m)k (x) the corresponding generating series. Then Eq.42specializes to the identity
B(m)k (x) =vm,kxm 1−
∞
X
p=1
B(p)k (x) qm+p−1
!
(m≥1,k≥0), [44]
wherevm,k denotes the specialization ofvm=vm(a,q)toa=ak, so that if we can guess some other collection of numbersβen,k(m) whose generating functions satisfy the same identity, then we automatically haveβn,k(m)=βen,k(m). The reason for looking at the special value Eq.43is that Eq.37for this value ofasays that the generating functionBk(x)(k≥0) is given by
Bk(x) =
∞
Y
r=1
1−q−k−rx
1−qk+1−rx (1−q−rx)(1−q1−rx) =
k
Y
j=1
1−qjx
1−q−jx [45]
(in particular, it is a rational function ofx) and also that the numbersvm,kare given by
vm,k =
(−1)m−1q(m2)−km (q)k+m
(q)m(q)m−1(q)k−m if1≤m≤k,
0 ifm>k,
[46]
as one checks easily. (Here and for the rest of the section we use the notation(x)n for the q-Pochhammer symbol (1−x)(1− qx)· · ·(1−qn−1x), with(x)0= 1.) After a considerable amount of computer experimentation, we found that the generating function B(m)k (x)is given by the following closed formula:
Proposition 14. Fork≥0andm≥1the generating functionB(m)k (x) =B(m)(x;ak,q)is a rational function ofx,equal to0ifm>kand otherwise given by
B(m)k (x) = (−1)m−1 (q)m+k
(q)k(q)m−1
xmYk(m)(x)
Dk(x) , [47]
whereDk(x)∈Z[q,x]is defined by the product expansion
Dk(x) =
k
Y
j=1
qj−x
and whereYk(m)(x)∈Z[q,x]is the polynomial of degreek−mdefined by
Yk(m)(x) =
k−m
X
r=0
q(r+12 )+(k−m−r+12 ) k r
k k−m−r
xr = Coefficient ofTk−min
k
Y
j=1
1 +qjT
1 +qjTx
. [48]
The symbol k
r
used in Eq.48is theq-binomial coefficient (q)k
(q)r(q)k−r
, which occurs in the following two well-knownqversions of the binomial theorem
k
X
r=0
(−1)rq(r2) k r
xr = (x)k,
∞
X
r=0
k+r−1 r
xr = 1 (x)k
, [49]
wherekdenotes an integer≥0. The equality of the two expressions in Eq.48follows from the first of these formulas.
The proof ofProposition 14is given in the next subsection. Here we show that it implies our main identity Eq.37. For this, as we have already explained, it suffices to show that the sum overm≥1of the rational functions Eq.47coincides with the right-hand side of Eq.45. Combining Eq.47with the second part of Eq.48and the second part of Eq.49, we find
1 x
Dk(x) 1−qk+1
∞
X
m=1
B(m)k (x) =
k
X
m=1
(−x)m−1 k+m
k+ 1
Yk(m)(x) = Coefficient ofTk−1in Qk
j=1 1 +qjT (1 +xT)(1 +qk+1xT). But by comparing poles and residues (partial fractions decomposition), we see that
Qk
j=1 1 +qjT (1 +xT)(1 +qk+1xT)=
Qk
j=1 1−qjx−1 (1−qk+1)(1 +xT)+
Qk
j=1 1−qj−k−1x−1
(1−q−k−1)(1 +qk+1xT)+Pk−2(T),
MATHEMATICS
wherePk−2(T)is a polynomial of degree≤k−2inT. It follows that
∞
X
m=1
B(m)k (x) = (−x)k Dk(x) −
k
Y
j=1
(1−qjx−1) +qk(k+1)
k
Y
j=1
(1−qj−k−1x−1)
!
=−1 +
k
Y
j=1
1−qjx
1−q−jx =−1 +Bk(x), where the final equality is Eq.45. SinceB(0)k (x) = 1, this completes the proof that the sum of the functionsB(m)(x)defined recursively by Eq.42coincides with the right-hand side of Eq.37and hence, by what has already been said, completes the proof ofTheorem 11.
C. Completion of the Proof. It remains to proveProposition 14. For this purpose we reverse the order of the logic, taking Eq.47(with Yk(m)(x)defined as 0 ifm>k) as the definition of the power seriesB(m)k (x)for allm≥1andk≥0and then proving that these power series satisfy the identity Eq.44. Inserting Eqs.45,47, and48into Eq.44, we see after multiplying both sides by a common factor that the identity to be proved is
qkm−(m2)Yk(m)(x) = k
m
Dk(x) + (q)k+1
(q)m(q)k−m k
X
p=1
k+p k+ 1
Yk(p)(x) (−x)p
qm+p−1. [50]
But by a simpler version of the same partial fraction argument as the one that was used above we see thatDk(x)is the coefficient ofTk in(1 +xT)−1Qk
j=1(1 +qjT), and one also sees without difficulty thatYk(m)(x)equals(qk+1x)−mtimes the coefficient of Tk+min the same productQk
j=1(1−qjT)(1−qjTx)as the one used in the original definition Eq.48, so that the left-hand side of Eq.50can be written, using the first equation in Eq.49, as
qkm−(m2)Yk(m)(x) =Coefficient ofTkin
k
Y
j=1
1 +qjT
·
k−m
X
s=0
q(s+12 )+ms k m+s
(xT)s.
The identity Eq.50then follows immediately from the following lemma by replacingx byxT, multiplying both sides byQk j=1(1 + qjT), and comparing the coefficients ofTkon both sides.
Lemma 15. For fixedk≥0andm≥1,define two power seriesF1(x)andF2(x)by F1(x) = (−qx)k
∞
X
p=1
k+p k+ 1
(−x)p qm+p−1, F2(x) =
k m
(1 +x)−1−
k−m
X
s=0
q(s+12 )+ms k m+s
xs.
Then
F2(x) =− (q)k+1
(q)m(q)k−m
F1(x). [51]
Proof:The power seriesF1andF2satisfy the functional equations
(1 +qx)qmF1(qx)−(1 +qk+1x)F1(x) = (−qx)k+1
∞
X
p=1
k+p p−1
(−x)p = −x 1 +x (here we have used the second equation of Eq.49) and
(1 +qx)qmF2(qx)−(1 +qk+1x)F2(x)
= k
m qm−1 +qk+1x 1 +x
−
k−m
X
s=0
q(s+12 )+ms k m+s
n
qm+s−1
−
qk−m−s−1
qs+1+mxo xs
= k
m
1−qk+1 x
1 +x −(1−qm)
+
1−qkk+mX
s=0
k−1 m+s−1
q(s+12 )+msxs
−
1−qkk+m−1X
s=0
k−1 m+s
q(s+22 )+m(s+1)xs+1
= (q)k+1
(q)m(q)k−m
x
1 +x (telescoping series).
Together these imply Eq.51, since it is easily seen that a power seriesF(x)satisfying(1 +qx)qmF(qx) = (1 +qk+1x)F(x)for some integersk≥0andm≥1must vanish identically.
This completes the proof ofLemma 15,Proposition 14, and hence also ofTheorem 5.
4. The Riemann Hypothesis
A. Proof ofProposition 6. We can use the recursion relation Eq.39to give an easy inductive proof of the inequality Eq.17, which, as we will see in a moment, implies the Riemann hypothesis for our zeta functions. Indeed, Eq.17holds forn= 2since
β2
β1
= q2+q−a
q2−1 = 1 + N q2−1,
whereN=q−a+ 1 =|E(Fq)|satisfies0<N<2q+ 2, and ifn≥3and we assume by induction onnthat Eq.17holds forn−1, then Eq.39gives
βn
βn−1
> qn+qn−1−a −(qn−1−q)
qn−1 = 1 + N
qn−1 >1 and
(qn−1)
qn/2+ 1 qn/2−1 − βn
βn−1
=
qn/2+ 1
2 − qn+qn−1−a
+ qn−1−qβn−2
βn−1
>2qn/2+ 1−qn−1−(q+ 1) + qn−1−qq(n−1)/2−1 q(n−1)/2+ 1
= 2(qn−1−qn/2)(q1/2−1) q(n−1)/2+ 1 >0 (where we have again used only|a|<q+ 1, and not the stronger estimate|a| ≤2√
qgiven by the usual Riemann hypothesis ofE/Fq), completing the proof of Eq.17by induction.
In fact, the estimates Eq.17are quite wasteful, and by a more careful analysis one finds that
βn
βn−1
= 1 + (n−1)(q−a+ 1)−c(q)
qn +O
n2 q2n−2
[52]
uniformly asqn→ ∞, wherec(q) = 2 + 3(a−2)/q +· · · is independent ofn. (Recall thata=O(√
q).) We also remark that the bounds Eq.17together with the initial valueβ0= 1give upper and lower estimates for eachβn. In particular, we have the uniform estimate
βn = 1 +O(1/√
q), [53]
where the implicit constant is universal and can be taken, e.g., to be 3.
B. Proof of the Riemann Hypothesis. By Eqs.6and12, the polynomialPE,n(T)appearing in Eq.4is given by 1
αE,n(0)PE,n(T) = 1−
(Q+ 1)−(Q−1) βE,n(0) βE,n−1(0)
T+QT2, [54]
and by the inequalities Eq.17the coefficient ofTin the second factor lies between−2and2√
Q.Theorem 7follows immediately.
Note that this argument gives much more than just the Riemann hypothesis, for which we would only need that the coefficient of Tis between−2√
Qand2√
Q. In fact, inserting Eq.52into Eq.54, we see that the reciprocal roots ofPE,n(T), divided byqn/2, are not uniformly distributed on the unit circle, but are actually very near toiand−ifornlarge. In a related direction, we mention that, since eachβE,n(a)is completely determined byn,q, anda, the usual Sato–Tate distribution property for the roots of the local zeta functions of the reductionsE¯/Fpat varying primespof an elliptic curveEdefined over implies a corresponding explicit Sato–Tate distribution for the roots of the higher zeta functionsζE¯/Fp,n asp varies withnfixed and also, after a suitable renormalization, asn→ ∞.
5. Complements
The most important consequence ofTheorem 5, of course, is the Riemann hypothesis for the higher-rank zeta functionsζE,n(s), but Theorem 5has several other corollaries that seem to be of independent interest. We end the paper by listing some of these.
The first statement concerns the analytic continuation and functional equation of the Dirichlet series E/Fq(s)defined by Eq.15.
Corollary 16. The function E/Fq(s)continues meromorphically to the entire complex plane and satisfies the functional equation
E(s−1) =ζE(s) E(s). [55]
Proof:The meromorphic continuation is obvious from Eq.16, sinceζE(s)is meromorphic and tends rapidly to 1 as (s)→+∞. The functional equation Eq.55then follows tautologically from Eq.16.
Corollary 17. The meromorphic function defined by
±
E/Fq(s) = E/Fq(s) E/Fq(−s) [56]
is invariant unders7→s+ 1.
