Table Of ContentTheta operators on Siegel modular forms
and Galois representations
Alexandru Ghitza∗ and Angus McAndrew†
School of Mathematics and Statistics
University of Melbourne
6
1
January 11, 2016
0
2
n
a
J
8 We describe the effect of the differential operators defined by Bo¨cherer-
Nagaoka [1], Flander-Ghitza [4] and Yamauchi [12] on the Galois represen-
]
T tations (conjecturally) attached to Siegel modular eigenforms.
N
.
h 1 Introduction
t
a
m Relations between modularformsand Galois representations play acentral rolein modern
[ arithmetic geometry. The story starts in the 1970s with Deligne’s proof of the Ramanujan
conjecture and Serre and Swinnerton-Dyer’s pioneering work on congruences, continues in
1
v the 1990s with Wiles’s proof of the Shimura-Taniyama conjecture (and hence of Fermat’s
5
Last Theorem), and leads to the current flurry of activity around the proof of Serre’s
3
8 conjecture by Khare-Wintenberger and numerous other modularity results due to Taylor
1 and his collaborators.
0
While this paper is devoted to groups of higher rank, it is modelled on the situation for
.
1
the group GL . Let f be a Hecke eigenform (mod p). Deligne proved that there exists a
0 2
continuous group representation
6
1
: ρ : Gal(Q/Q) −→ GL (F )
v f 2 p
i
X that remembers the arithmetic content of f (namely, the Hecke eigenvalues). The algebra
r of modular forms (mod p) is endowed with a differential operator ϑ that almost commutes
a
with the Hecke operators: T ◦ϑ = ℓϑ◦T . A simple consequence is that
ℓ ℓ
ρ ∼= χ⊗ρ ,
ϑf f
where χ: Gal(Q/Q) → F× is the cyclotomic character (mod p).
p
Our interest is in Siegel modular forms (mod p), which arise from the group GSp
2g
(the group GL being the special case g = 1). Such forms conjecturally produce group
2
representations
ρ : Gal(Q/Q) −→ GSpin (F ),
f 2g+1 p
∗([email protected]) Research of the first author was supported by a Discovery Grant from the
AustralianResearchCouncilandanEarlyCareerResearcherGrantfromtheUniversityofMelbourne.
†([email protected]) Research of thesecond authorwas partially supportedbyan
Australian Postgraduate Award and theAlbert Shimmins Writing Up Award.
1
where GSpin is the Langlands dual of GSp . The question is how the operator ϑ
2g+1 2g
generalises, and how these generalisations relate to the representations ρ .
f
Our main result (Theorem 17) takes as input an operator ϑ satisfying a certain type of
commutation relation with Hecke operators and describes its effect on the Galois repre-
sentations attached to eigenforms. Operators to which Theorem 17 applies include those
defined by Flander-Ghitza [4] for arbitrary g and by Yamauchi [12] for g = 2. We also
analyse the operator ϑ on scalar-valued forms defined by B¨ocherer-Nagaoka [1]. This
BN
requires us to determine in Section 7 the commutation relation between ϑ and Hecke
BN
operators, which is a result of independent interest.
Acknowledgements
We thank Martin Weissman for many suggestions, including a much cleaner statement of
Theorem17. WealsothankArunRamandOlavRichter forseveral usefulcomments. The
project originated many years ago with Dick Gross’s suggestion that the relation between
ϑ and Galois representations is best understood via the Satake isomorphism.
2 The group of symplectic similitudes
We work with the algebraic group G = GSp of symplectic similitudes. For any commu-
2g
tative ring R, this is defined by
GSp (R) = M ∈ GL (R)| MJMt = η(M)J for some η(M) ∈ R× ,
2g 2g
0 I
where J = (cid:8) g . (cid:9)
−I 0
g
(cid:18) (cid:19)
The symplectic group Sp (R) is the kernel of η.
2g
Let T ⊂ G denote the maximal torus formed by diagonal matrices
t = t(u ,...,u ) = diag(u ,...,u ;u u−1,...,u u−1) with u ,...,u ∈ G .
1 g+1 1 g g+1 1 g+1 g 1 g+1 m
The character lattice X of G is
X = Hom(T,G )= Ze ⊕···⊕Ze ,
m 1 g+1
where
e (t) =u for j = 1,...,g+1.
j j
The cocharacter lattice X∨ of G is
X∨ = Hom(G ,T)= Zf ⊕···⊕Zf ,
m 1 g+1
where
f (u) = t(1,...,1,u,1,...,1) with u in the j-th spot, for j = 1,...,g+1
j
diag(1,...,1,u,1,...,1;1,...,1,u−1,1,...,1) if 1 ≤ j ≤ g
=
(diag(1,...,1;u,...,u) if j = g+1.
Under the natural pairing
h·,·i: X ×X∨ → Hom(G ,G )= Z
m m
2
given by composition, we have he ,f i = δ .
i j ij
We choose simple roots lying in the root basis ∆ = {α ,...,α }, with corresponding
1 g
coroots ∆∨ = {α∨,...,α∨}, where
1 g
α = e −e α∨ =f −f
1 1 2 1 1 2
α = e −e α∨ =f −f
2 2 3 2 2 3
. .
. .
. .
α = e −e α∨ =f −f
g−1 g−1 g g−1 g−1 g
α = 2e −e α∨ =f .
g g g+1 g g
The positive Weyl chamber is given by
P+ = {λ = a f +···+a f ∈ X∨ | hα ,λi ≥ 0 for j = 1,...,g}
1 1 g+1 g+1 j
= {λ = a f +···+a f ∈ X∨ | 2a ≥ 2a ≥ ··· ≥ 2a ≥ a }.
1 1 g+1 g+1 1 2 g g+1
There is a partial order on P+ given by µ ≤ λ if
g
µ ≤ λ if λ−µ = n α∨, where n ∈Z .
j j j ≥0
j=1
X
Proposition 1. If β∨ is a coroot of GSp , then η(β∨(ℓ)) =1.
2g
Proof. One observes this to be true for the elements α∨ above, and these form a (multi-
i
plicative) basis for the coroots.
The following consequence will play an important role in the proof of the main result
(Theorem 17).
Corollary2. Ifλ,µ ∈ X∨ arecoweightsofGSp suchthatλ ≥ µ, thenη(λ(ℓ)) = η(µ(ℓ)).
2g
Proof. Thisfollowssincethedefinitionoftheorderingisthatλandµdifferbycoroots.
The root datum of G is the quadruple (X,X∨,∆,∆∨). Let Gˆ denote the dual group
of G, i.e. the algebraic group whose root datum is (X∨,X,∆∨,∆). It is known that Gˆ is
isomorphic to GSpin , which we describe briefly in the following section.
2g+1
3 The spin group
Fix a commutative ring R in which 2 is invertible. Let C(R) be the R-algebra with
generators c ,c ,...,c subject to the relations
1 2 2g+1
c2 = 1 for all i,
i
c c = −c c for i 6= j.
i j j i
Then C(R) is a Clifford algebra of dimension 2g. It has a Z/2Z-grading with even part
C (R) (resp. odd part C (R)) spanned by monomials consisting of even (resp. odd) num-
0 1
bers of generators c . As a superalgebra, it is isomorphic to the simple superalgebra
i
a b
Q(2g)= b a : a,b ∈ M2g(R) .
(cid:26)(cid:18) (cid:19) (cid:27)
3
The parity automorphism of C(R) is the R-linear map γ: C(R)→ C(R) defined by
x if x is even
γ(x) =
(−x if x is odd.
Let M ⊂ C (R) be the R-span of {c ,...,c }. The special Clifford group1 is
1 1 2g+1
GSpin (R) = x ∈ C (R)× | γ(x)mx−1 ∈ M for all m ∈M .
2g+1 0
(cid:8) (cid:9)
4 The Satake isomorphism
The Satake isomorphism relates the local Hecke algebra of a reductive group G to the
representation ring of the dual group Gˆ. We follow the description of this relation given
in [6]. Our current interest is of course in the pair (G = GSp ,Gˆ = GSpin ), but the
2g 2g+1
setup applies to any pair of dual reductive groups.
There is a bijective correspondence
{λ ∈ P+} ←→ {irreducible representations V of Gˆ}.
λ
Fix a prime ℓ 6= p and consider the local Hecke algebra
f locally constant, compactly supported,
H = H(G(Q ),G(Z )) = f: G(Q ) −→ Z
ℓ ℓ ℓ ℓ G(Z )-bi-invariant
(cid:26) (cid:12) ℓ (cid:27)
(cid:12)
(cid:12)
with the convolution product
(cid:12)
(fg)(x) = f(t)g(t−1x)dt,
ZG(Qℓ)
wheretheHaar measuredt onG(Q )isnormalised sothatthemaximalcompact subgroup
ℓ
G(Z ) has volume 1.
ℓ
We work with the basis of H consisting of the characteristic functions
ℓ
c = char(G(Z )λ(ℓ)G(Z )) for λ ∈ P+.
λ ℓ ℓ
The Satake transform is a ring isomorphism
S : H ⊗Z[ℓ±1/2]−→ R(Gˆ)⊗Z[ℓ±1/2].
Z,ℓ ℓ
We tensor this with F :
p
S = S : H ⊗F −→ R(Gˆ)⊗F .
ℓ Fp,ℓ ℓ p p
If λ ∈ P+ then the image of c can be written2
λ
S (c ) = b (µ)ℓhρ,µiχ ,
ℓ λ λ µ
µ≤λ
X
where µ runs over the elements in P+ such that µ ≤ λ, b (µ)∈ F and b (λ) = 1.
λ p λ
1The special Clifford group is sometimes denoted CSpin or SΓ.
2g+1
2Theelement ρisthehalf-sum of thepositiveroots ofG,butwewill notneed toknowthis,onlythat it
is thesame in all theidentities related to theSatakeisomorphism S.
4
There is a similar identity expressing χ in terms of images of characteristic functions:
λ
χ = ℓ−hρ,λi d (µ)S (c ),
λ λ ℓ µ
µ≤λ
X
where d (µ) ∈ F and d (λ) = 1.
λ p λ
For future use, let us record a consequence of this identity:
(1) S−1(χ ) = ℓ−hρ,λi d (µ)c .
ℓ λ λ µ
µ≤λ
X
5 Siegel modular forms
Let N ≥ 3 and consider the moduli space A parametrising principally polarised g-
g,N
dimensional abelian schemes with principal level N structure. This is a scheme over
Z/NZ endowed with a rank g locally free sheaf E (the Hodge bundle). Let κ = (k ≥
1
··· ≥ k ) ∈ Zg bethehighest weight of arational representation of GL , andlet Eκ denote
g g
the locally free sheaf obtained by applying the representation to the transition functions
of E. Given a ring R where N is invertible, the space of Siegel modular forms of degree g,
weight κ and level N with coefficients in R is by definition3
M (N;R) = H0(A ⊗R,Eκ).
κ g,N
Inparticular,givenaprimep ∤ N,thespaceofSiegelmodularforms(modp)isM (N;F ).
κ p
Let
1
F(g) = n =(n ) ∈Mat Z : n ∈ Z,n symmetric semipositive definite .
ij g×g ii
2
(cid:26) (cid:18) (cid:19) (cid:27)
Any f ∈ M (N;R) has a Fourier expansion of the form
κ
f(q)= a(n)qn with a(n) ∈ R,
N
n∈F(g)
X
where, for n = (n ), we have
ij
1/N
qn = qnii q2nij .
N ii ij
i i<j
Y Y
If R = C, the space M (N;C) consists of holomorphic functions f: S → V such that
κ g κ
a b
f (az+b)(cz+d)−1 = κ(cz+d)f(z) for all ∈Γ(N) and all z ∈ S ,
c d g
(cid:18) (cid:19)
(cid:0) (cid:1)
where the principal congruence subgroup of level N, resp. the Siegel upper half space, are
given by
Γ(N) = ker Sp (Z) −→ Sp (Z/NZ)
2g 2g
a b a b I 0
= (cid:0) ∈ Sp (Z): (cid:1)≡ g (mod N)
c d 2g c d 0 I
g
(cid:26)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
S = z∈ Sym (C)| Im(z) positive definite .
g g
3An additional conditio(cid:8)n is required in thecase g=1, namely that t(cid:9)hesections extendto thecompacti-
fication of A1,N.
5
The Fourier expansion of f ∈ M (N;C) is
κ
f(q)= a(n)qnN, where qnN = e2NπiTr(nz).
n∈F(g)
X
A B
Given a power series in q as above and a matrix M = ∈ GSp (Q) with
N C D 2g
(cid:18) (cid:19)
positive determinant, we define the weight κ slash operator
M| f = η(M)Pκj−g(g+1)/2κ(Cz+D)−1f (Az+B)(Cz+D)−1 .
κ
Note that if f has coefficients a(n) lying in a ring R,(cid:0)then the coefficients o(cid:1)f M| f are in
κ
the ring extension R[ζ] generated by a root of unity of order N times the determinant of
M.
Let ℓ be a prime not dividing N and let K = GSp (Z ). Given a double coset KMK
2g ℓ
and its decomposition into right cosets
h
KMK = KM ,
i
i=1
a
we define an operator on M (N;C) by
κ
h
(KMK)(f) = M | f.
i κ
i=1
X
For arbitrary characteristic zero rings R, the same approach gives the action of KMK
on M (N;R). For other rings (e.g. of positive characteristic not dividing N), the slash
κ
operator must be replaced by certain isogenies on the moduli space of abelian varieties
(see [3, Section VII.3] for details). Note that the operator KMK does preserve the ring
of definition of the coefficients of f.
The various Hecke operators are special cases of (linear combinations of) KMK for
specific types of matrices M:
Operator Double coset description
I
T(ℓ) K g K
pI
g
(cid:18) (cid:19)
I
g−i
pI
Ti(ℓ2), 0≤ i ≤ g K i p2I K
g−i
pI
i
T(ℓ2) g T (ℓ)2
i=0 i
P
6 Galois representations attached to Siegel modular forms
A Hecke eigenform f defines a Hecke ℓ-eigensystem Ψ , which we think of as a ring
f,ℓ
homomorphism
Ψ : H ⊗F −→ F
f,ℓ ℓ p p
6
taking an operator T to its eigenvalue: T(f)=Ψ (T)f.
f,ℓ
Using the Satake isomorphism S , we can define a character ω : R(Gˆ)⊗F → F by
ℓ f p p
ω (χ ) = Ψ (S−1(χ )).
f λ f,ℓ ℓ λ
But the characters of the representation ring R(Gˆ)⊗F are indexed by the semi-simple
p
conjugacy classes in Gˆ(F ). Given such a class s, the corresponding character ω is given
p s
by
ω (χ ) = χ (s).
s λ λ
In particular, the character ω given above is indexed by some s ∈ Gˆ(F ), called the
f f,ℓ p
ℓ-Satake parameter of f.
Conjecture 3. Let f be a degree g Siegel modular form (mod p) of level N and suppose
f is a Hecke eigenform. There exists a semisimple continuous group representation
ρ : Gal(Q/Q) −→ Gˆ(F )
f p
that is unramified outside pN. If ℓ is a prime not dividing pN, then ρ (Frob )= s , the
f ℓ f,ℓ
ℓ-Satake parameter of f.
It is expected that ρ is odd (for the meaning of this condition in the general case
f
see [7]).
The case g = 1 of Conjecture 3 follows from a well-known result of Deligne (k ≥ 2) and
Deligne-Serre (k = 1):
Theorem 4 (Deligne, see [5, Proposition 11.1]). Let f = a qn ∈ M (Γ (N),ǫ;F ) be
n k 1 p
a normalised Hecke eigenform (mod p). Then there is a continuous, semisimple Galois
P
representation
ρ : Gal(Q/Q) −→ GL (F ),
f 2 p
that is unramified at all primes ℓ ∤Np and such that
charpolyρ (Frob )= X2−a X +ε(ℓ)ℓk−1
f ℓ ℓ
for all primes ℓ ∤Np.
Remark 5. A priori it is not clear how to reconcile the appearance of level Γ(N) in
Conjecture 3 with the level Γ (N) appearing in Deligne’s theorem. However, there is an
1
injective group homomorphism Γ (N2)→ Γ(N) given by
1
a b a Nb
7−→
c d 1c d
(cid:18) (cid:19) (cid:18)N (cid:19)
that induces a Hecke-equivariant embedding ι: M (Γ(N);R) ֒→ M (Γ (N2);R) whose
k k 1
effect on Fourier expansions is
a qn 7−→ a qn.
n N n
n n
X X
(Over C, the map is given by (ιf)(z) = f(Nz).)
The case g = 2 of Conjecture 3 follows from work4 of Laumon, Taylor and Weissauer.
4Laumon,TaylorandWeissauerconstructp-adicGaloisrepresentationsattachedtoSiegelmodulareigen-
forms in characteristic zero. The mod p version we are interested in here follows by first lifting a mod
p eigenform to characteristic zero and then reducing theresulting representation modulo p.
7
7 The theta operator of B¨ocherer-Nagaoka
We reinterpret an operator defined by B¨ocherer and Nagaoka in [1, Section 4], and deter-
mine its commutation relation with the Hecke operators.
Let R and S be g×g symmetric matrix variables and let x be an indeterminate. The
following defines polynomials P (R,S),...,P (R,S):
0 g
g
det(R+xS) = P (R,S)xj.
j
j=0
X
Let k ,k be integers such that 2k ≥ g. We define
1 2 i
g
2k −j 2k −g+j
Q(g) (R,S) = (−1)jj!(g−j)! 2 1 P (R,S).
k1,k2 g−j j j
j=0 (cid:18) (cid:19)(cid:18) (cid:19)
X
We define a matrix differential operator ∂ in the g(g+1)/2 variables q of the Fourier
q ij
expansion of a Siegel modular form:
∂ 1∂ ...
11 2 12
∂q = 12(1+δij)∂ij = 21 ∂...12 ∂2...2 ...... ,
(cid:18) (cid:19)
∂
gg
where we write ∂ = q ∂ .
ij ij ∂qij
We can use this to define a differential operator of order g
(g) (g)
D = Q (∂ ,∂ )
k1,k2 k1,k2 q1 q2
and a bilinear operator
(g)
[F,G] = D F(q )G(q ) .
k1,k2 1 2 q=q1=q2
(cid:12)
(cid:0) (cid:1)(cid:12)
Theorem 6 (Eholzer-Ibukiyama). Let Γ ⊂ Sp2g(R) be(cid:12)a discrete subgroup of finite covol-
ume. If F ∈ M (Γ;C) and G ∈ M (Γ;C) with 2k ≥ g, then [F,G] ∈ M (Γ;C).
k1 k2 i k1+k2+2
(g)
Proof. By the calculations of [2, Section 5.1], the polynomial Q is associated with an
k1,k2
invariant pluriharmonic polynomial of the correct type. Therefore, by [2, Theorem 2.3]
(itself a special case of [8, Theorem 2]), [F,G] is modular of weight k +k +2.
1 2
(g)
A crucial remark is that the polynomials P and Q have coefficients in Z, so that if
j k1,k2
F and G have Fourier coefficients in a ring A, then so does [F,G].
For the rest of this section, fix g > 1 and a prime p > g(g + 1)/2. Let Z denote
(p)
the subring of Q consisting of fractions with denominators not divisible by p. By [1,
Theorem 1], there exists a form H ∈ M (1;Z ) such that its reduction modulo p has
p−1 (p)
Fourier expansion H(q) = 1. We define an operator5
ϑ0 : M (N;Z ) → M (N;Z )
BN k (p) k+p+1 (p)
by setting ϑ0 (F) = [F,H].
BN
By reduction modulo p, we obtain the following slight reworking of [1, Theorem 4]:
5Note that, since p∤N, the local ring Z is a Z[1/N]-module.
(p)
8
6
Theorem 7. Suppose g > 1, p > g(g + 1)/2 and k > g + 1. There is a linear map
ϑ : M (N;F )→ M (N;F ) whose effect on q-expansions is given by:
BN k p k+p+1 p
1
if f(q)= a(n)qn then (ϑ f)(q) = det(n)a(n)qn.
N BN Ng N
n∈F(g) n∈F(g)
X X
Proof. Let f ∈ M (N;F ). Since k > g+1, [11, Theorem 1.3] says that f can be lifted to
k p
a form F ∈ M (N;Z ) such that F¯ = f. Let ϑ (f)∈ M (N;F ) be defined by
k (p) BN k+p+1 p
(−1)g
ϑ (f)= the reduction mod p of ϑ0 (F).
BN (g+1)! BN
(Note that we are only dividing by factors strictly smaller than p.)
It remains to check the statement about the effect on q-expansions. First note that
∂ (F(q )H(q )) = F(q )∂ (H(q ))
q2 1 2 1 q2 2
reduces to zero modulo p, since H has constant q-expansion modulo p. Note also that
(g)
if j ≥ 1 then P (R,S) is divisible by S, so these parts of the operator D do not
j k1,k2
contribute anything modulo p.
Therefore the only contribution comes from P (R,S) = det(R), so modulo p we obtain
0
(−1)g 2p−2
(ϑ f)(q) = g! det(∂ )F(q) = det(∂ )f(q),
BN q q
(g+1)! g
(cid:18) (cid:19)
which is easily seen to equal the expression in the statement.
We record for future use part of the argument in the above proof:
Corollary 8. As an operator on formal power series in characteristic zero, ϑ0 can be
BN
written as
ϑ0 = ϑ1 +pϑ2 ,
BN BN BN
where ϑ0,1,2: Z [[q ]] → Z [[q ]] and
BN (p) N (p) N
1
ϑ1 f = det(n)a(n)qn if f = a(n)qn.
BN Ng N N
n n
X X
Moreover,
(−1)g
ϑ (f)= the reduction mod p of ϑ1 (F).
BN (g+1)! BN
Remark 9. Eigenforms typically require going up to field extensions, so they are in
M (N;F ) rather than in M (N;F ). Hence, we will often work with ϑ ⊗ F (and
k p k p BN Fp p
abuse notation by simply referring to it as ϑ ).
BN
In order to study the interaction between ϑ and the Hecke operators, we will use the
BN
following right coset decomposition proved by Ryan and Shemanske:
6In level one, B¨ocherer and Nagaoka do not impose any restrictions on the weight k, and only need to
ensure that p ≥ g+3. This comes at the expense of getting an operator that is only defined on the
subspace of mod p Siegel modular forms of level 1 that are reductions of forms in characteristic zero.
9
Proposition 10 (Local version of [10, Proposition 2.10]). Let r ≥ 0 and let M ∈
GSp (Q ) satisfy η(M) = ℓr. There exists a g-tuple b = (b ,...,b ) ∈ Zg with
2g ℓ 1 g
r ≥ b ≥ ··· ≥ b ≥ 0
1 g
such that
ℓr·1−b 0
KMK = K K, where K = GSp (Z ).
0 ℓb 2g ℓ
(cid:18) (cid:19)
Moreover, KMK can be decomposed into right cosets of the form
ℓr(DT)−1 B
K .
0 D
(cid:18) (cid:19)
A B
Lemma 11. If M = ∈ GSp (Q ), then the following diagram commutes:
0 D 2g ℓ
(cid:18) (cid:19)
ϑ1
BN
Z [[q ]] Z [[q ]]
(p) N (p) N
M| M|
k k+p+1
Z [ζ][[q ]] Z [ζ][[q ]]
(p) N (p) N
det(M)ϑ1
BN
Proof. Since M is a symplectic similitude matrix, we have
A= η(M) DT −1.
(cid:0) (cid:1)
Suppose
F = a(n)qnN, where qnN = e2Nπi Tr(nz).
n
X
Using Tr(XY)= Tr(YX), we see that
(M| F)(z) = η(M)kg−g(g+1)/2det(D)−kF (Az+B)D−1
k
= η(M)kg−g(g+1)/2det(D)−k (cid:0) a(n)c(n)e2πiT(cid:1)r(nη(M)(DT)−1zD−1)/N
n
X
= η(M)kg−g(g+1)/2det(D)−k a(n)c(n)qn′,
N
n
X
where n′ = η(M)D−1n DT −1 and c(n) = e2πiTr(nBD−1)/N.
Note that
(cid:0) (cid:1)
det(n′) = η(M)gdet(D)−2det(n).
10
