Table Of ContentSIEGEL MODULAR FORMS OF DEGREE THREE
AND THE COHOMOLOGY OF LOCAL SYSTEMS
1
1 JONAS BERGSTRO¨M, CAREL FABER, ANDGERARDVAN DER GEER
0
2
Abstract. WegiveanexplicitconjecturalformulaforthemotivicEu-
v
ler characteristic of an arbitrary symplectic local system on themoduli
o
N space A3 of principally polarized abelian threefolds. The main term of
theformulaisaconjecturalmotiveofSiegelmodularforms ofacertain
2 type; the remaining terms admit a surprisingly simple description in
terms ofthemotivic Eulercharacteristics forlower genera. Theconjec-
] ture is based on extensive counts of curves of genus three and abelian
G
threefolds over finite fields. It provides a lot of new information about
A vector-valued Siegel modular forms of degree three, such as dimension
. formulas and traces of Hecke operators. We also use it to predict sev-
h
eral lifts from genus 1 to genus 3, as well as lifts from G and new
t 2
a congruences of Harder type.
m
[
2
1. Introduction
v
1
It has been known since the 1950’s that there is a close relationship be-
3
7 tween modular forms and the cohomology of local systems on modulispaces
3 of elliptic curves. The theorem of Eichler and Shimura expresses this rela-
.
8 tionship by exhibiting the vector space S of cusp forms of even weight
k+2
0 k+2 on SL(2,Z) as the (k +1,0)-part of the Hodge decomposition of the
1
first cohomology group of the local system V on the moduli space A of
1 k 1
v: elliptic curves. Here, Vk is the k-th symmetric power of the standard local
system V := R1π Q of rank 2, where π : E → A is the universal elliptic
i ∗ 1
X
curve.
r Deligne strengthened the Eichler-Shimura theorem by proving that, for a
a
prime p, the trace of the Hecke operator T(p) on S can be expressed in
k+2
termsofthetraceoftheFrobeniusmaponthe´etalecohomologyoftheℓ-adic
counterpart of V on the moduli A ⊗F of elliptic curves in characteristic
k 1 p
p. Using the Euler characteristic e (A ,V ) = (−1)i[Hi(A ,V )], we can
c 1 k c 1 k
formulate this concisely (in a suitable Grothendieck group) as
P
e (A ,V ) =−S[k+2]−1,
c 1 k
where S[k+2] is the motive associated (by Scholl) to S and where the
k+2
term −1 is the contribution coming from the Eisenstein series.
The moduli space A of principally polarized abelian varieties carries the
g
local system V := R1π Q of rank 2g, where π : X → A is the universal
∗ g g
1991 Mathematics Subject Classification. 11F46, 11G18, 14G35, 14J15, 20B25.
1
2 JONASBERGSTRO¨M,CARELFABER,ANDGERARDVANDERGEER
principally polarized abelian variety. The local system V comes with a
symplectic form, compatible with the Weil pairing on the universal abelian
variety. For every irreducible representation of the symplectic group GSp ,
2g
described by its highest weight λ, one has a local system V , in such a way
λ
that V corresponds to the dual of the standard representation of GSp .
2g
Faltings and Chai have extended the work of Eichler and Shimura to a
relationship between the space S of Siegel modular cusp forms of degree
n(λ)
g and weight n(λ), and the cohomology of the local system V on A . Note
λ g
that the modular forms that appear are in general vector-valued and that
the scalar-valued ones only occur for the (most) singular highest weights.
The relationship leads us to believe that there should be a motivic equality
of the form
e (A ,V )= (−1)i[Hi(A ,V )] = (−1)g(g+1)/2S[n(λ)]+e (λ),
c g λ c g λ g,extra
generalizing theXone above for genus 1. The (conjectural) element S[n(λ)] of
theGrothendieck groupofmotives shouldbeassociated tothespaceS in
n(λ)
amannersimilartothecaseg =1. AsanelementoftheGrothendieckgroup
of ℓ-adic Galois representations, this means that the trace of a Frobenius
element F on S[n(λ)] should be equal to the trace of the Hecke operator
p
T(p) on S . Our ambition is to make the equality above explicit. To do
n(λ)
this, we need a method to find cohomological information.
The moduli space A is defined over the integers. According to an idea
g
of Weil, one can obtain information on the cohomology of a variety defined
over the integers by counting its numbers of points over finite fields. Recall
thatapointof A over F correspondsto acollection of isomorphismclasses
g q
of principally polarized abelian varieties over F that form one isomorphism
q
class over F ; such an isomorphism class [A] over F is counted with a factor
q q
1/|Aut (A)|. By counting abelian varieties over finite fields, one thus gets
Fq
cohomological information aboutA , and with an explicit formula as above,
g
one would also get information about Siegel modular forms.
In fact, if one has a list of all isomorphism classes of principally polarized
abelian varieties of dimension g over a fixed ground field F together with
q
the orders of their automorphism groups and the characteristic polynomials
of Frobenius acting on their first cohomology, then one can determine the
trace of Frobenius (for that prime power q) on the Euler characteristic of
the ℓ-adic version of V for all local systems V on A .
λ λ g
For g = 2, the moduli space A coincides with the moduli space Mct
2 2
of curves of genus 2 of compact type and this makes the above counting
feasible. Some years ago, the last two authors carried out the counting for
finite fields of small cardinality. Subsequently, they tried to interpret the
obtained traces of Frobenius on the Euler characteristic of V on A ⊗F in
λ 2 q
a motivic way and arrived at a precise conjecture for e (λ). That is, a
2,extra
precise conjecture on the relation between the trace of the Hecke operator
T(p) on a space of Siegel cusp forms of degree 2 and the trace of Frobenius
on the ℓ-adic ´etale cohomology of the corresponding local system on A ⊗
2
SIEGEL MODULAR FORMS OF DEGREE THREE 3
F , see [19]. This is more difficult than in the case of genus 1, due to
p
the more complicated contribution from the boundary of the moduli space
and to the presence of endoscopy. The conjecture for genus 2 has been
proved for regular local systems, but is still open for the non-regular case,
see Section 6. The conjecture and our calculations allow us to compute
the trace of the Hecke operator T(p) on all spaces of Siegel cusp forms of
degree 2, for all primes p ≤ 37. In [4], all three authors extended this work
to degree 2 modular forms of level 2. Inspired by the results in genus 2,
Harder formulated in [32] a conjecture on a congruence between genus 2
and genus 1 modular forms determined by critical values of L-functions.
Our aim in this article is to generalize the work above to genus 3, that is,
to give an explicit (conjectural) formula for the term e (λ) in terms of
3,extra
Euler characteristics of local systems for genus 1 and 2.
The conjecture is based on counts of points over finite fields, using the
closerelationshipbetweenA andthemodulispaceM ofcurvesofgenus3,
3 3
and a formula for the rank 1 part of the Eisenstein cohomology. Both in
genus 2 and genus 3, the motivic interpretation of the traces of Frobenius
is made easier by the experimental fact that there are no Siegel cusp forms
of low weight in level 1. Together, the conjecture and the calculations open
a window on Siegel modular forms of degree 3. We sketch some of the
(heuristic) results.
The numerical Euler characteristic
E (A ,V ):= (−1)idimHi(A ,V )
c 3 λ c 3 λ
has been calculated for the local sXystems V and is known for the correction
λ
term e (λ). This allows us to predict the dimension of the space of
3,extra
Siegel modular cusp forms of any given weight. So far, these dimensions are
only known for scalar-valued Siegel modular forms by work of Tsuyumine,
and our results agree with this. For g = 2, the dimensions of most spaces
of Siegel cusp forms were known earlier, so the dimension predictions are a
new feature in genus 3.
Assuming the conjecture and using our countings, we can calculate the
trace of T(p) on any space of Siegel modular forms S of degree 3 for
n(λ)
p ≤ 17. Moreover, if dimS = 1, we can compute the local spinor L-
n(λ)
factor at p = 2.
We make a precise conjecture on lifts from genus 1 to genus 3. In particu-
lar, for every triple (f,g,h) of elliptic cusp forms that are Hecke eigenforms
of weights a−c+3, b+3 and a+c+5, there should be a Siegel modular
cusp form F of weight (a−b,b−c,c+4) that is an eigenform for the Hecke
algebra with spinor L-function L(f ⊗g,s)L(f ⊗h,s−c−1).
We find strong evidence for the existence of Siegel modular forms of de-
gree 3 (and level 1) that are lifts from G , as predicted by Gross and Savin.
2
Finally, we are able to formulate conjectures of Harder type on congruences
between Siegel modular forms of degree 1 and degree 3. All these heuristic
findings provide strong consistency checks for our conjectures.
4 JONASBERGSTRO¨M,CARELFABER,ANDGERARDVANDERGEER
We hope that our results may help to make Siegel modular forms of
degree 3 as concrete as elliptic modular forms.
Our methods apply to some extent also to local systems on the moduli
space M and Teichmu¨ller modular forms; for more on this we refer to the
3
forthcoming article [5].
Afterreviewingthecaseofgenus1,weintroduceSiegelmodularformsand
give a short summary of the results of Faltings and Chai that we shall use.
Wethen discussthehypothetical motive attached to thespace ofcuspforms
of a given weight. In Section 6, we review and reformulate our results for
genus 2 in a form that is close to our generalization for genus 3. We present
our main conjectures in Section 7. The method of counting is explained in
Section8. WethendiscusstheevidenceforSiegelmodularformsoftypeG .
2
In the final section, we present our Harder-type conjectures on congruences
between degree 3 and degree 1 Siegel modular forms.
Acknowledgement
TheauthorsthankPierreDeligne, NeilDummigan, BenedictGross, Gu¨n-
ter Harder, Anton Mellit, and Don Zagier for their contributions, and the
Max Planck Institute for Mathematics in Bonn for hospitality and excellent
working conditions. We also thank Hidenori Katsurada for his remarks
about congruences.
2. Genus one
We start by reviewing the theorem in genus 1 by Eichler, Shimura and
Deligne that we wish to generalize to higher genera. This theorem describes
thecohomologyofcertainlocalsystemsonthemodulispaceofellipticcurves
in terms of elliptic modular forms.
Let π: X → A be the universal object over the moduli space of elliptic
1 1
curves. These spaces are smooth Deligne-Mumford stacks over Z.
First, we consider the analytic picture. Define a local system V := R1π C
∗
on A ⊗C. For any a ≥ 0 we put V := Syma(V). We wish to understand
1 a
thecompactlysupportedBetticohomology groupsHi(A ⊗C,V )withtheir
c 1 a
mixedHodgestructures. Definetheinnercohomology groupHi(A ⊗C,V )
! 1 a
as the image of the natural map
Hi(A ⊗C,V ) → Hi(A ⊗C,V ),
c 1 a 1 a
and the Eisenstein cohomology group Hi (A ⊗C,V ) as the kernel of the
Eis 1 a
same map. Since the element −Id of SL(2,Z) acts as −1 on V, all these
cohomology groups vanish if a is odd. We therefore assume from now on
that a is even.
The group SL(2,Z) acts on the complex upper half space H by z 7→
1
α(z) := (az +b)(cz +d)−1 for any z ∈ H and α = (a,b;c,d) ∈ SL(2,Z).
1
We note that A ⊗ C ∼= SL(2,Z)\H as analytic spaces. We define an
1 1
elliptic modular form of weight k as a holomorphic map f : H → C, such
1
that f is also holomorphic at the point in infinity and such that f(α(z)) =
SIEGEL MODULAR FORMS OF DEGREE THREE 5
(cz +d)kf(z) for all z ∈ H and α ∈ SL(2,Z). We call an elliptic modular
1
form a cusp form if it vanishes at the point in infinity. Let S be the
k
vector space of elliptic cusp forms of weight k and put s := dim S . The
k C k
Hecke algebra acts on S . It is generated by operators T(p) and T (p2)
k 1
for each prime p. These operators are simultaneously diagonalizable and
the eigenvectors will be called eigenforms. The eigenvalue for T(p) of an
eigenform f will be denoted by λ (f).
p
Example 2.1. The best known cusp form is probably ∆. It has weight 12
and can be defined by
∞ ∞
∆(q):= q (1−qn)24 = τ(n)qn,
n=1 n=1
Y X
where q := e2πiz and τ is the Ramanujan tau function. Since s = 1, ∆ is
12
an eigenform and λ (∆) = τ(p) for every prime p.
p
Similarly to the above, we define a bundle E := π Ω on A ⊗ C
∗ X1/A1 1
and for any a ≥ 0 we put E := Syma(E). The moduli space A can be
a 1
compactified by adding the point at infinity, giving A′ := A ∪{∞}. The
1 1
line bundles E can be extended to A′. We have the identification
a 1
H0 A′ ⊗C,E (−∞) ∼= S .
1 k k
The Eichler-Shimura theor(cid:0)em (see [11, Th´eor(cid:1)`eme 2.10]) then tells us that
S ⊕S ∼= H1(A ⊗C,V ),
a+2 a+2 ! 1 a
where S has Hodge type (a+1,0). If a > 0, then H1 (A ⊗C,V ) ∼= C
a+2 Eis 1 a
with Hodgetype (0,0), whereas H0(A ⊗C,V )and H2(A ⊗C,V ) vanish.
c 1 a c 1 a
In order to connect the arithmetic properties of elliptic modular forms
to these cohomology groups, we turn to ℓ-adic ´etale cohomology and we
redo our definitions of the local systems by putting V := R1π Q and
∗ ℓ
V := Syma(V), which are ℓ-adic local systems over A . For each prime
a 1
p, we define a geometric Hecke operator, also denoted by T(p), by using the
correspondence in characteristic p coming from the two natural projections
to A from the moduli space of cyclic p-isogenies between elliptic curves.
1
The operators T(p) act on Hi(A ⊗F ,V ) and Hi(A ⊗F ,V ). The geo-
c 1 p a 1 p a
metric action of T(p) on H1(A ⊗F ,V ) is then equal to the action of T(p)
! 1 p a
on S ⊕S , see [11, Prop. 3.19].
a+2 a+2
On the other hand, the cohomology groups Hi(A ⊗K,V ) come with an
c 1 a
actionofGal(K/K). Foranyprimep,thereisanaturalmapGal(Q /Q )→
p p
Gal(F /F ), and if p 6= ℓ, there is an isomorphism
p p
Hi(A ⊗F ,V ) → Hi(A ⊗Q ,V )
c 1 p a c 1 p a
of Gal(Q /Q )-representations. This isomorphism also holds for Eisenstein
p p
and inner cohomology. We define the (geometric) Frobenius map F ∈
q
6 JONASBERGSTRO¨M,CARELFABER,ANDGERARDVANDERGEER
Gal(F /F ) to be the inverse of x 7→ xq. The two actions are connected in
q q
the following way:
Tr F ,H1(A ⊗F ,V ) = Tr(T(p),S ),
p ! 1 p a a+2
see [11, Prop. 4.8].(cid:0)We also note that fo(cid:1)r a > 0,
Tr F ,H1 (A ⊗F ,V ) = 1
p Eis 1 p a
for all primes p 6= ℓ. (cid:0) (cid:1)
We can choose an injection Gal(Q /Q ) → Gal(Q/Q) and talk about
p p
FrobeniuselementsinGal(Q/Q)aselementsofGal(Q /Q )thataremapped
p p
to F ∈ Gal(F /F ). We will, for each prime p, choose such a Frobenius
p p p
element and it will by abuse of notation also be denoted by F .
p
Remark 2.2. The traces of F for all (unramified) primes p determine an ℓ-
p
adicGal(Q/Q)-representation uptosemi-simplification, see[10,Proposition
2.6].
For any p 6=ℓ, the two Gal(Q /Q )-representations Hi(A ⊗Q ,V ) and
p p c 1 p a
Hi(A ⊗Q,V ) are isomorphic, and the same holds for inner cohomology.
c 1 a
It follows that
Tr F ,H1(A ⊗Q,V ) = Tr(T(p),S ),
p ! 1 a a+2
and this then deter(cid:0)mines the Gal(Q/Q)(cid:1)-representation H1(A ⊗Q,V ) up
! 1 a
to semi-simplification.
For a ≥ 2, there is a construction by Scholl [44, Theorem 1.2.4] of a
corresponding Chow motive S[a + 2]. This motive is defined over Q, it
has rank 2s , its Betti realization has a pure Hodge structure with types
a+2
(a+1,0) and (0,a+1), and its ℓ-adic realization has the property that
Tr(F ,S[a+2]) = Tr(T(p),S ),
p a+2
for all primes p 6= ℓ. See also [9] for an alternative construction of S[a+2].
We can then rewrite the results above in terms of a motivic Euler char-
acteristic:
2
e (A ⊗Q,V ):= (−1)i[Hi(A ⊗Q,V )].
c 1 a c 1 a
i=0
X
Here follows the theorem that we wish to generalize to higher genera.
Theorem 2.3. For every even a > 0,
e (A ⊗Q,V ) = −S[a+2]−1.
c 1 a
Remark 2.4. As a result, one can compute Tr(T(p),S ) by means of
a+2
Tr(F ,e (A ⊗ F ,V )), which in turn can be found by counting elliptic
p c 1 p a
curves over F together with their number of points over F . Essentially,
p p
make a list of all elliptic curves defined over F up to F -isomorphism; de-
p p
termine for each E in this list |Aut (E)| and |E(F )|. Having done this,
Fp p
one can calculate Tr(T(p),S ) for all a > 0. But there are of course other
a+2
SIEGEL MODULAR FORMS OF DEGREE THREE 7
(possibly more efficient) ways of computing these numbers, see for instance
the tables of Stein [47].
For a = 0, we have e (A ⊗Q,V ) = L, where L is the so called Lefschetz
c 1 0
motive. The ℓ-adic realization of L equals the cyclotomic representation
Q (−1), which is one-dimensional and satisfies Tr(F ,Q (−1)) = q for all
ℓ q ℓ
prime powers q. For bookkeeping reasons, we want the formula in The-
orem 2.3 to hold also for a = 0, so we define S[2] := −L − 1. To be
consistent, we then also put s := −1.
2
For further use below, we note that in [44] there is actually a construc-
tion of a motive M for any eigenform f ∈ S . This motive will be de-
f k
fined over some number field and its ℓ-adic realization has the property that
Tr(F ,M )= Tr(T(p),f) = λ (f).
p f p
3. Siegel modular forms
In this section, we recall the notion of Siegel modular forms of degree g,
which are natural generalizations of the elliptic modular forms that occur
in genus 1. General references are [1], [21], [22], and [24].
The group Sp(2g,Z) acts on the Siegel upper half space
H := {z ∈ Mat(g×g) :zt = z,Im(z) > 0}
g
by τ 7→ α(τ) := (aτ +b)(cτ + d)−1 for any τ ∈ H and α = (a,b;c,d) ∈
g
Sp(2g,Z). Let W be a finite-dimensional complex vector space and ρ :
GL(g,C) → GL(W) an irreducible representation. A Siegel modular form
of degree g and weight ρ is a holomorphic map f : H → W such that
g
f(α(τ)) = ρ(cτ +d)f(τ) for all τ ∈ H and α ∈ Sp(2g,Z). When g = 1,
g
we also require f to be holomorphic at infinity. Let U be the standard
representation of GL(g,C). For each g-tuple (n ,...,n ) ∈ Ng, we define
1 g
U to be the irreducible representation of GL(g,C) of highest weight
(n1,...,ng)
in
Symn1(∧1U)⊗Symn2(∧2U)⊗...⊗Symng−1(∧g−1U)⊗(∧gU)ng.
It can be cut out by Schur functors. We have for instance U ⊗ ∧2U ∼=
U ⊕ U for g = 3. We let the cotangent bundle E := π Ω1
1,1,0 0,0,1 ∗ Xg/Ag
(the Hodge bundle) correspond to the standard representation of GL(g,C)
and using the above construction we get vector bundles E . For
(n1,...,ng)
g > 1, we can then identify the vector space of Siegel modular forms of
weight (n ,...,n ) with H0(A ⊗ C,E ). If we take any Faltings-
1 g g (n1,...,ng)
Chai toroidal compactification A′ of A and if we let D := A′ \A be the
g g g g
divisor at infinity, then we can define the vector space of Siegel modular
cusp forms, S , to be H0(A′ ⊗C,E (−D)). In other words, a
(n1,...,ng) g (n1,...,ng)
Siegel modularform is a cuspform if it vanishesalong thedivisor at infinity.
We will call a Siegel modular cusp form classical if it is scalar-valued, i.e.,
if n = n = ... = n = 0. Let us also put s := dim S .
1 2 g−1 (n1,...,ng) C (n1,...,ng)
8 JONASBERGSTRO¨M,CARELFABER,ANDGERARDVANDERGEER
The Hecke algebra, whose elements are called Hecke operators, acts on
the space S . It is a tensor product over all primes p of local Hecke
(n1,...,ng)
algebras that are generated by elements of the form T(p) and T (p2) for i =
i
1,...,g, see [1] or [22]. These operators are simultaneously diagonalizable
and we call the eigenvectors eigenforms with corresponding eigenvalues λ
p
and λ . To a Hecke eigenform we can then associate a homomorphism
i,p2
from the Hecke algebra to C.
The Satake isomorphism identifies the local Hecke algebra at any prime p
with the representation ring of GSpin (C), the dual group of GSp .
2g+1 2g
Thus, a Hecke eigenform f determines, for each prime p, a conjugacy class
s (f) in GSpin (C). If we fix a representation r of GSpin (C), we can
p 2g+1 2g+1
formanL-functionbyletting thelocal factor atpequalQ (p−s,f)−1, where
p
Q (X,f) is the characteristic polynomial det(1−r(s (f))X) of r(s (f)), see
p p p
[8, p.50]. In this article, we willlet r bethespin representation, from which
we get the so called spinor L-function L(f,s).
The local Hecke algebra of GSp can also beidentified with the elements
2g
ofthelocal Hecke algebraofthediagonaltorusinGSp thatarefixedbyits
2g
Weyl group. Using this, wecan associate to a Hecke eigenform f the(g+1)-
tupleofits Satake p-parameters(α (f),α (f),...,α (f)) ∈ Cg+1. Thelocal
0 1 g
factor L (s,f)ofthespinorL-function of f thenequals Q (p−s,f)−1, where
p p
g
Q (X,f) := 1−α (f)α (f)···α (f)X .
p 0 i1 ir
rY=0 1≤i1<i2Y<...<ir≤g(cid:0) (cid:1)
Note that the polynomial Q (X,f) has degree 2g and the coefficient of X is
p
equal to −λ (f).
p
Example 3.1. In [40, p. 314], Miyawaki constructed a non-zero classical
cusp form F ∈S . Define
12 0,0,12
E := {(x ,...,x ) ∈ R8 : 2x ∈Z,x −x ∈ Z,x +...+x ∈2Z},
8 1 8 i i j 1 8
which is the unique even unimodular lattice of rank 8. Let I be the 3×3
3
identity matrix and define Q to be the 3×8 matrix (I ,i·I ,0,0). If h·,·i
3 3
denotes the usual inner product on R8, then
8
F (τ) := Re det Q(v ,v ,v ) exp πiTr (hv ,v i)τ .
12 1 2 3 i j
v1,v2X,v3∈E8 (cid:16) (cid:0) (cid:1) (cid:17) (cid:16) (cid:0) (cid:1)(cid:17)
Sinces = 1,F isaneigenformandMiyawakicomputedtheeigenvalues
0,0,12 12
for T(2), T (4), T (4), and T (4), where the first equals −26·747. Based on
1 2 3
these computations, he conjectured for all primes p the equality
λ (F )= λ (∆) λ (f)+p9+p10 ,
p 12 p p
where f is an eigenform in S . Th(cid:0)is was proved by(cid:1)Ikeda, see further in
20
Section 7.3.
SIEGEL MODULAR FORMS OF DEGREE THREE 9
4. Cohomology of local systems
In this section we introduce the Euler characteristic that we would like
to compute and review results of Faltings and Chai on the cohomology in
question.
Let M be the moduli space of smooth curves of genus g and A the
g g
modulispaceofprincipallypolarizedabelian varieties of dimensiong. These
are smooth Deligne-Mumford stacks defined over Spec(Z).
Using the universal abelian variety π: X → A , we define a local system
g g
on A . It comes in a Betti version V := R1π Q on A ⊗Q and an ℓ-adic
g 0 ∗ g
version V := R1π Q on A ⊗Z[1/ℓ]. We will often denote both of them
ℓ ∗ ℓ g
simply by V.
For any[A] ∈ A , thestalk(V) isisomorphictoH1(A)(withcoefficients
g A
in Q or Q depending on the cohomology theory). Using the polarization
ℓ
and Poincar´e duality, we get a symplectic pairing V× V → Q(−1), where
0 0
Q(−1) is a Tate twist (and similarly, V × V → Q (−1), where Q (−1)
ℓ ℓ ℓ ℓ
corresponds to the cyclotomic character).
Let V be the contragredient of the standard representation of GSp ,
2g
which is isomorphic to the tensor product of the standard representation of
GSp and the inverse of the multiplier representation η, see [21, p. 224].
2g
We will consider partitions λ of length at most g and they will be written
in the form λ = (λ ≥ λ ≥ ... ≥ λ ≥ 0). For each such partition λ, we
1 2 g
definethe representation V of GSp to be the irreducible representation of
λ 2g
highest weight in
Symλ1−λ2(∧1V)⊗Symλ2−λ3(∧2V)⊗...⊗Symλg−1−λg(∧g−1V)⊗(∧gV)λg.
It can be cut out using Schur functors. We have for instance ∧2V ∼= V ⊕
1,1
V ⊗η−1 for genus 2. This gives all irreducible representations of GSp
0,0 2g
modulo tensoring with η.
Our local system V corresponds now to the equivariant bundle defined
by the contragredient of the standard representation of GSp . By applying
2g
the construction above to V, we define a local system V . For example, we
λ
have ∧2V ∼= V ⊕V (−1). If λ > ... >λ > 0, then this local system is
1,1 0,0 1 g
g
called regular. For a partition λ, we define |λ| := λ to be the weight
i=1 i
of λ.
P
We are interested in the motivic Euler characteristic
g(g+1)
(4.1) e (A ,V )= (−1)i[Hi(A ,V )].
c g λ c g λ
i=0
X
Inpractice, wewillbeconsideringthisexpressioneither intheGrothendieck
group of mixed Hodge structures (denoted K (MHS)), by taking the com-
0
pactlysupportedBetti cohomology of V onA ⊗C,orintheGrothendieck
0 λ g
groupofℓ-adicGal(Q/Q)-representations(denotedK (Gal )),bytakingthe
0 Q
compactly supported ℓ-adic ´etale cohomology of V on A ⊗Q.
ℓ λ g
10 JONASBERGSTRO¨M,CARELFABER,ANDGERARDVANDERGEER
Remark 4.1. Notethatfurthertensoringwithηdoesnotgivenewinteresting
local systems, sinceit correspondsto Tate twists, that is, Hi(A ,V (−j)) ∼=
c g λ
Hi(A ,V )(−j).
c g λ
Remark 4.2. The element −Id belongs to GSp and it acts as −1 on V.
2g
This has the consequence that H∗(A ,V ) = 0 if |λ| is odd. From now on,
c g λ
we therefore always assume that |λ| is even.
We also define the integer-valued Euler characteristic:
g(g+1)
E (A ,V )= (−1)idim Hi(A ⊗C, V ) .
c g λ c g 0 λ
i=0
X (cid:0) (cid:1)
4.1. Results of Faltings and Chai. The cohomology groups Hi(A ⊗
c g
C,V ) and Hi(A ⊗C,V ) carry mixed Hodge structures of weight ≤ |λ|+i
λ g λ
respectively ≥ |λ|+i, see [21, p. 233]. Since V is a sheaf of pure weight
ℓ λ
|λ|, the same weight claim holds for the ℓ-adic cohomology in the sense of
Deligne, see [14, Cor. 3.3.3, 3.3.4]. The steps in the Hodge filtration for the
cohomology groups are given by the sums of the elements of any of the 2g
subsets of {λ + 1,λ + 2,...,λ + g}. In genus 3, the explicit Hodge
g g−1 1
filtration for λ = (a ≥ b ≥ c≥ 0) is
F0 ⊇ Fc+1 ⊇ Fb+2 ⊇ Ft2 ⊇ Ft1 ⊇ Fa+c+4 ⊇ Fa+b+5 ⊇ Fa+b+c+6,
where t ≥ t and {t ,t } = {b +c+ 3,a + 3}. For H•(A ⊗C,V ) and
1 2 1 2 c 3 λ
a 6= b+c, the graded pieces in the Hodge filtration can be identified with
the following coherent cohomology groups:
F0/Fc+1 ∼=H•−0(A′ ⊗C,E (−D)),
3 b−c,a−b,−a
Fc+1/Fb+2 ∼=H•−1(A′ ⊗C,E (−D)),
3 b+c+2,a−b,−a
Fb+2/Fr1 ∼=H•−2(A′ ⊗C,E (−D)),
3 b+c+2,a−c+1,−a
Fa+3/Fr2 ∼=H•−3(A′ ⊗C,E (−D)),
3 a+c+3,b−c,1−b
Fb+c+3/Fr3 ∼=H•−3(A′ ⊗C,E (−D)),
3 b−c,a+c+3,−a
Fa+c+4/Fa+b+5 ∼=H•−4(A′ ⊗C,E (−D)),
3 a−c+1,b+c+2,1−b
Fa+b+5/Fa+b+c+6 ∼=H•−5(A′ ⊗C,E (−D)),
3 a−b,b+c+2,2−c
Fa+b+c+6 ∼=H•−6(A′ ⊗C,E (−D)),
3 a−b,b−c,c+4
where r ,r ,r depend in the obvious way on the ordering of b+c+3 and
1 2 3
a+3. Ifr = a+3= b+c+3,thentheaboveholds,exceptthatFr/Fa+c+4 ∼=
H•−3(A′ ⊗C,E (−D))⊕H•−3(A′ ⊗C,E (−D)).
3 a+c+3,b−c,1−b 3 b−c,a+c+3,−a
Notation 4.3. For any partition λ, we put
n(λ):= (λ −λ ,λ −λ ,...,λ −λ ,λ +g+1).
1 2 2 3 g−1 g g
