4
JEFFREY
D.
ACHTER
THEOREM 3.1. Let g be a natural number, let N be an odd natural number, and
let lF be a sufficiently large finite field in which IGSp29 (Z/N)I is invertible. Then
l
l{f(x) E 1i2g+2(1F): Jac(C9,t)(JF)[N] ~ L}l _ ( IJFI
N
L)l
l1i2g+2(1F)I a g, ' '
2(2g
+
1)!1GSp29 (Z/N)I
~
.
PROOF. Consider the etale sheaf :FN
=
Jac(C9)[N]
--+
1i2g+2, which corre-
sponds (after fixing a basepoint fj) to a representation
p :
n
1
(1i2g+2, fj)
--+
Aut(:F
N,iJ)
~
GL29 (Z/N). Computing the proportion of points
f
E 1i29+2(1F) for which
Jac(C9,t)[N]
~Lis
the same as computing the proportion of points
f
E
1i29+2(JF)
for which ker(p(Fr
f,JF) -
id)
~
L; the latter task is accomplished using Katz's the-
orem 2.1. In the notation of (2.1),
ageom ~
Sp29 (Z/N).
If N
is prime, this is
attributed to an unpublished work of J.K. Yu (see [8, 2.4]) and is proved in [2,
Thm. 3.4] and in [13, Thm. 4.1]. The case where
N
is a prime power follows
formally from this [2, Cor. 3.5], while the case of general
N
(still prime to the
characteristic) follows from Goursat's lemma [17, Cor. 2.6].
By Lemma 2.2, forB in (2.2) one may take IGSp29 (Z/N)IB2, where B2 is any
number such that for every etale Galois cover ¢ : Y
--+
1i29+2 with deg ¢ invertible
in lF, ac(Y, Q£)
:-:=:;
deg
¢ ·
B2.
We provide an explicit value for B2 as follows. Let z1
, · · · ,
z29+2 be coordinates
on affine space
A29+2,
and let H29+2
=
A
29+2
-
Uiij ( Zi
=
Zj)
be the complement
of the fat diagonal.
It
is an irreducible, etale 829+2-cover of 1i29+2; the geometric
fiber over
f
E
1i2g+2 (JF) is the set of labelings of the roots of
f.
Let
¢ : Y
--+
1i29+2 be an etale Galois cover with automorphism group
G.
Trivially one has ac(Y, Q£)
:-:=:;
acQ'"
XH.
29
+2
H2g+2, Q£). Let r be the number of con-
nected components of Y
X11.
29
+2
1i2g+2· Each component is isomorphic to a K-cover
Z of H2g+2, where K is a subgroup of G of index r. By Lemma 3.3, ac(Z, Q£)
:-:=:;
IKI · ac(H2g+2, Q£). Since H2g+2 is smooth, ac(H2g+2, Q£)
=
a(H2g+2, Q£). The
calculation a(H29+2, Q£)
=
(2g
+
1)!, originally due to Arnol'd [4], may also be
recovered from equation (3.2) below. D
The space 1i29+2 introduced in the proof of Theorem 3.1 is the braid arrange-
ment studied by Arnol' d. In Lemma 3.3 we prove a case of [17, Prop. 4.5] optimized
for hyperplane arrangements.
Let
T--+ S
be a Galois cover of smooth varieties, and let
SH
C
S
be a suitably
generic hyperplane section. The strategy of [17, Prop. 4.5], which is a refinement of
the argument of [15], is to relate the Betti numbers
ofT
to those of
TH
=
T
x
s
S
H.
This inductive method produces explicit upper bounds for the Betti numbers
ofT,
but they tend to be pessimistically large. For example, the bounds obtained from
[17] for the Betti numbers of 1in are much larger than
n
n.
In the special case where
S
is a hyperplane arrangement, we can replace the
(abstract) Lefschetz theorem with an explicit calculation of
a(S,
Q£)-
a(SH,
Q£).
Moreover,
SH
is itself a hyperplane arrangement, so that we may use induction on
the dimension of S without leaving the class of hyperplane arrangements.
Let
V / k
be an n-dimensional vector space over an algebraically closed field,
and let
A
be a finite collection of hyperplanes in
V.
The complement of this
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