Then. \ nl /. ~ L,~Z/m,. =0 otherwise. by orthogonality relations for characters on finite groups. Hence. Q~,6(t)= I-I-q+ 1. Er'EZr ~;£)1 [I'=1 ~"J'~z,", e q. IU=I rot\ ¢,>o.
Math. Ann. 216, 233--244 (1975) @ by Springer-Verlag 1975
Poincar6 Polynomials of the Variety of Stable Bundles Desate Usha V. and S. Ramanan
§1. In this note, we indicate a few improvements to [3]. Let X be an irreducible, non-singular projective algebraic curve defined over a finite field Fq with q elements, of characteristic p. Let SL(n, d) be the coarse moduli scheme of isomorphism classes of stable vector bundles of rank n and determinant isomorphic to an lFq-rational line bundle L of degree d. [We will assume that (n, d) = 1.] By replacing lFq by a finite extension if necessary, we may assume that SL(n, d) is defined over IFq. As one might expect, it is then indeed true that the IFq-rational points of SL(n, d) are precisely the stable vector bundles on X defined over IFq (see [3]). By the Weil conjectures, it is easy to write down the Poincar6 polynomials of SL(n, d) once the number of its lFq-rational points is known. In order to compute the latter, one first notices that the fact that the Tamagawa number of SL(n) is 1 can be interpreted as follows. 1
1
Proposition 1.1. ~ ~ A u t E _ ( q - 1) q(,2 _ 1)(o- 1)(x(2)." .(x(n) ' the summation extending over the isomorphism classes of vector bundles of rank n, degree d with a fixed determinant. Here ~x denotes the zeta function of the curve X. Proof The correspondence between X-lattices and vector bundles, together with the fact that the Tamagawa number of SL(n) is one leads to the formula (see [3, §2.3] or [2, §2]) 1
~x ~ {x- lkxc~SL(n, K)}
= q{,~ - ~*{o- ,){x(2)...
~x(n ) =_-z,
the summation runs over double cosets in k\~SL(n, A)/SL(n, K), K, k denoting respectively the function field of X and the stabiliser in SL(n, ,4) of a fixed lattice. Interpreting x - I kxc~SL(n, K) as the group of automorphisms of the vector bundle E x determined by x, which have determinant one, the L.H.S. of the above formula reduces to 1
(1.2)
~x ~ {AutE x with detl}"
To complete the proof of the proposition we now need the following simple lemma which generalises the Lemma t of [2]. The second-named author would like to thank the "Sonderforschungsbereich Rir Theoretische Mathematik" of the University of Bonn for its hospitality during the preparation of this paper.
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U.V. Desale and S. Ramanan
Lemma 1.3. Let E o be a vector bundle of rank n with a fixed determinant. Then there exist exactly #(iFq*/ImAutEo) double cosets in k\SL(n, A)/SL(n, K) which determine a bundle isomorphic to Eo, where ImAutE o denotes the image of AutE o in IF* under the determinant mapping. Using the lemma, the expression (1.2) reduces to
(1.4)
# {F*/ImAutE} ~E # {AutE with detl}"
In view of the exact sequence 0--,AutE with detl--*AutE d--~IF* ~ I F * / I m A u t E ~ 0 (1.4) becomes ~E
q-1 # AutE
where the summation runs over the isomorphism classes of vector bundles of rk n, with a fixed determinant. Since stable bundles admit only scalar automorphisms, Proposition 1.1 reduces the problem of computing the number ~L(n, d) of IFq-valued points of SL(n, d) to one of computing the IFq-aUtomorphisms of non-semistable bundles.
Proposition 1.5. A n y vector bundle E on X admits a unique flag
0=EoCE, C...CEk=E satisfying
i) Ei/E i_ 1 is semistable for i= 1..... k ii) #(Ei/E , _ 1) > p(Ei + x/E~) for i = 1..... k - 1, where #(F) is the rational number degF/rkF. Equivalently, 1) Ei/E i_ 1 is semistable for i = 1..... k 2) For any subbundle F o r e containing Ei we have ll(el/Ei- 1)> It(F/Ei- 1),
i= 1..... k .
For proof, see I-3, 1.3.9-1 and also [-8, 2.5.1]. Definition 1.6. The length of the unique flag corresponding to E is the length of E and is denoted by l(E). We denote by di(E), ri(E), ~i(E), the numbers deg(Ei/E_ 1), rank (Ei/E i_ 1) and ~(Ei/E i_ 1) respectively. Proposition 1.5 has two consequences. Firstly, since such a flag is unique, we see by Gatois descent that the flag is actually defined over IFq, when E is IFqrational. Secondly, this enables one to set up an induction on rank, in order to compute the number
1 ilL(n, d) = ~ Esemistable ~ AutE"
Poincar6 Polynomialsof the Variety of Stable Bundles
235
When n and d are coprime, since semistabte bundles are stable, we have (q - DilL(n, d)=~L(n, d). The induction to compute ilL(n, d) is done in [3] working again with the adele group. We will show in this note that once Proposition 1.1 is admitted, one could carry the induction forward only with vector bundles and this seems to lead not only to greater clarity, but also to better results. The key step in our approach is the following
Proposition 1.7. i) ilL(n, d) is independent of L, and hence may be written simply
fl(n, d). 1
ii) Denote by tic(n1 ..... nk) the sum ~ ~ A u t E ' where the summation extends over isomorphism classes of bundles E of rank n, determinant L and of length k with ri(E) = ni. Then flL(nl ..... rig)= g" #e Jg21 ~1~=1 fl(ni, d~) , Z.,qZQ.:~:) where the summation extends over (d I ..... dk)eZ k with ~i di = d and dl/nt >... > dk/n k, J ~ denotes the number oflFq-valued points q the Jacobian of X, and Z Ida. ..dk) denotes ~i Sup#(N), the supremum being taken over N
all no E~ we
subbundles N of M. In view of the proof of the Proposition 4.4 [6], there exist nonzero homomorphisms of E~ to M so that every automorphism of E keeps invariant and hence goes down to an automorphism of the quotient M. Thus get a map AutE~AutE~ x AutM
with kernel I + H°(X, Hom(M, E0). Also, AutE 1 x AutM acts on //~(X, Hom(M, E~)) i.e. on equivalence classes of extensions of M by E~ and two bundles given by such extensions are isomorphic if and only if they belong to same orbit under this action. The isotropy subgroup at E is precisely the image of AutE under the above map and hence is isomorphic to AutE/I + Ho(X, Hom(M, E0. Therefore we get f l L ( n l ""
"nk)=2E,,M ZExtens .... orMbye, 4~AutE.
1 #(elts in the orbit of E)
1 = ~E~,M # (AutE1 x AutM)q x(M*®E° where x(M*®EO denotes the Euler characteristic of M * ® E v The summation extends over all pairs of bundles (El, M) where E 1 is semistable of rkn~, M is of
236
U . V . D e s a l e a n d S. R a m a n a n
length ( k - 1) and has determinant equal to detE®detE~- 1 and
#(E1)>pa(M)>...>pk_l(M
) and
ri(M)=ni+ 1,
i = 1 ..... k - 1 .
Denoting by jd, the variety of isomorphism classes of line bundles of degd I on X
[nl n - n l 1 j in the notation of the proposition,
and noticing that z ( M * ® L ) = Z \ d l d _ d l we have
fiL(nl""nk)= 2d, sZ
d,>e~ tlI
2re.,~. flL®r-t(n2""nk) 2detE,=r qX(]:;-;:)
1
~AutE1 "
n2
Applying the induction to fiE®r -,(n2 ..... rig) and noticing that the last summation is simply fi,(n, d), since X \did_all] +Z dz...d k - Z \dl...dk }, the R.H.S. of the above equality simplifies to the required expression in the R.H.S. of the equality in (ii) of the proposition. Thus flE(na...nk) are independent of L and since
fiL(n,d) =
Z, q--1
EfiL(nt...nk )
(the summation being over all ordered tuples (nl...nk) with ~ = a ni =n and k=> 2), it follows that fiE(n, d) is independent of L. As mentioned earlier, as a consequence of the Weil conjectures and Artin comparison Theorems (see [3]), we have the following recipe to compute the Poincar6 polynomial Pn,d(t) of Sc(n , d), when n and d are coprime. The zeta function of X is given by [4, Chapter VI, §3] 1-12_° , (1--~oiq -s) ~x(S) = (1 -- q - ")( 1 -- q 1 - ,)" Then the substitution e ) i - - ' - t and q ~ t 2 in the expression for fi(n, d). ( q - 1 ) gives P.,d(t). From Propositions 1.1 and 1.7 we obtain
q(nzfi(,.d~-
I ) ( o - 1)
( q - 1)
_ q(,2_ 1)(0- 1) (q--l)
(X(2)...(x(n)- ~:C,,=, fiE(n, ..... rig) k> 2
r "VI ¢1 e) ~k- 1 [I~= 1 fi(ni, di) (x(2)'"~x(n)-/-"2k'>==z"''~t-" 2 qY(~',i]')
where the last summation is taken over di6Z , n~eN with ~ d i = d
dt -
-
/'/1
d2
dk
n2
nk
such that
> - - . . . > --. The last step uses the fact that the number of lFq-valued points
in the Jacobian of X is given by I-I2=0~ (1 -o)~). Since tensoring by a line bundle of degm gives a bijection of the set of isomorphism classes of semistable vector bundles of rkn, degd with the set of isomorphism classes of semistable vector bundles of rkn, deg(d+mn) and Aut E ~ A u t E ® L , for any line bundle L, we see that fl(n,d) depends only on the
Poincar6 Polynomials of the Variety of Stable Bundles
237
congruence class of d modulo n. This enables us to express fl(n, d) in the form q(n 2 - 1)(o- 1)
(1.8)
fl(n, d)=
( q - 1) -Z
(x(2)(x(3)...(x(n)
H i (1 - wl)k l qy, 2. Q~,~ are numbers independent of the genus g of the curve, gwen by (1.9)
1 Q~,~= Zq£, . . . > --. nl
nk
It is quite convenient to write the above expressions in the language of partitions. Definition 1.10. A partion (respectively an ordered partition) ~z of length k of a positive integer n is a (k + 1)-tuple (respectively an ordered ( k + 1)-tuple (nl ..... rig+ 1) such that ~ + 1 1 nl = n. A separation H of a partition n = (n 1..... rig+ 1) is a partition of length less than or equal to the length of n obtained by bracketing the n;'s into smaller partitions. The smaller component partitions will be called the separates of H with respect to ~. For example, if ~ =(nl...nT), then H = (n 1 + n 2 -f-n 3, n 4 +ns, n6, nT) is a separation of n with separates (nl, n2, n3), (ha, ns), (n6) and (nO. Notation 1.11. r(rc)=length of ~. Let 2(~)= ~ i < j n~nj. Note that 2(~) does not depend on the order of the partition 7r. Let z,=~,...z,~+, and r~(n)(~>z~=the power of ~x(i) in the product z~. ri(~) are determined as follows: Arrange the entries of ~ so that nl
dz/n2 >... > dr+ 1/nr+ 1. Let F...dini+ 1 - di+ 1hi, i = 1..... r. Using the condition ~ + 11di = d, di's are obtained in terms of Ffs as: dn~ Let L~ denote the affine form on the R.H.S. of (2.2). In the new coordinates F/'s the expression (2.1) for Q~,~ takes the form (2.3)
r~ , ~"1- x ; ~ , ¢~,~ "~n,6 ---VZ..
where the summation runs over (F)E:g ~ such that Fj>0, Lj(r-)-c~j is integral, nj
for all j and
Choose integers mt such that mt
~ L,~Z/m,
=0
\
nl /
is an integral affine form, l = 1..... r. Then
otherwise.
by orthogonality relations for characters on finite groups. Hence Q~,6(t)= I-I-q+ 1
I U = I rot\
--
1
Er'EZr ~ ; £ ) 1
[I'=1 ~"J'~z,", e
¢,>o
q 1
l-ITm,
E(jt)e[l~(Z/mt) ¢ 2~ij~(dn- ~ " ) H kr= l EF=I ~ t 2qkre- 2rti(~ljtfl~jF _
where fl~,denotes the coefficient of Fk in Ldnt. Computing the summation over F and substituting for/3~, from (2.2) we obtain
1 v , + ~, where A~ = z.,t:
5,j,,~llr+l,Z/m,, e =
'
/t
(Z, i :
),)"
,i~=,
[(
240
U.V. Desale and S. Ramanan
The general formula looks c u m b e r s o m e but simplifies considerably in particular cases. We give one example which is of interest viz. n = (1, 1...,1) where r(n) = n - 1. In case n = ( 1 , 1..... 1), 6i...OVi. N o t i n g that dr+l integer ~ d i integer Vi= 1..... r and mr+l can be chosen to be n, we can replace the set of conditions {Lift) is an integer Vi} by a single condition viz. Lr+ 1(~1...72) is an integer. Hence putting Jl...0V1 = 1..... r and replacing jr + 1 by j, mr+ 1 by n and m i by 1, i = 1..... r, the formula (2.5) reduces in this case to the simple formula (2.6)
1
__.d
.
. 2nv--~i. .
e2~-1J~I~'---~ { t 2 a " - O e ~ " J -
Q~,,...1),6(t)= ~ ~ z / .
1}-1
Another Method for Calculation oJ Q~,6 F o r ( b i ) ~ I ~ + ~ Z/(ni), we choose a representative for each 6i~Z/ni lying between 0 and n i - 1, which we again denote by (5i. Since di=bi(ni), di can be written as (2.7)
di=6i+qini,
(2.8)
Let
qi~Z.
mi=qz-qi+l
i = 1 ..... r.
T a k i n g q~+ 1 = b and solving back for qi's we get
q~= ~,j>=i mj=b,
i= 1,...,r.
Hence, given (mi)~Z ~ we can obtain (q/)~2U +1 satisfying the condition ~i di=d iff (2.9)
~-1
mi(~j -
(2.10)
di+ 1
-
n~
n~+l
iff m~>
~.i 6i)
m o d u l o n.
(~i+ 1
(~i
n~+l
nl
. Hence, substituting in (2.1), we get
Q~,6(t)=t-2Y,~jo,.~-6~.,) ~t-2E~- ..... ,
the s u m m a t i o n running over all (mi)eZ ~ satisfying mi>
6i+1
6i,
ni + 1
ni
E [ : I rni(~j 5. -
J(t)
The expression on the R.H.S. of(2.5) shows that Q~,6 are of the form [ L (1 - 2it i) with 2i~c and f(t) a polynomial. Hence the same statement is true for ~o~ as can be seen by induction. Therefore a term in the expression for the Poincar6 polynomial, corresponding to a partition ~ is of order at least t 2~). Claim: F o r Tc= (nl ..... ni), (~i ni >=5), we have (5.2)
2(~)>2(n-2)
exceptfor
~=(1, n-1),
n=(2, n-2).
The statement is obvious for r = 2 . F o r rc=(1,1, n - 2 ) , 2 ( T r ) = 2 ( n - 2 ) + l . 7r = ( n l , n2, n3) with no two of the ni's equal to one, we have
For
2(~) = nl(n 2 + n3) + n2n 3 >=2(n 2 + n 3 -- 2 ) + (n 2 + n3)n 1 (by statement for r = 2) = 2(n - 2) + (n 2 + n 3 - 2)n l > 2 ( n - 2) as n 2 + n 3 > 2 .
Poincar6 Polynomials of the Variety of Stable Bundles
243
The claim now follows easily by induction on r: We have for r > 3 , 2(~)=n r Z ~
n,+
El d > ~.
Clearly (n-1)(g-1)+d 6 and also for n=5, d = l . F o r d = 2, 1, n = 5, the R.H.S. of (5.10) is of order t 212{"-2)~g- t)+ 1] and 2 ( n - 2)(9 - 1)+
1 > ( n - 1)(g- 1)+d. Thus (3.6)...
(n-1)(g-1)+d5).
The formulae (1.13), (3.2), (3.4) and (3.5) give the Theorem 3. The dimensions of the l-adic cohomology groups of SL(n, d) for low dimensions are given by
/
ai
for 0 _ < i < 2 [ ( n - 1)(g- 1 ) + d ]
dimH'(S, QO= a,+bi, for 2 ~ n - 1)(g- 1 ) + d ]
