COMPUTING GENUS 2 HILBERT-SIEGEL MODULAR
FORMS OVER Q(
5) VIA THE JACQUET-LANGLANDS
CORRESPONDENCE
CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
Abstract. In this paper we present an algorithm for computing Hecke
eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of
narrow class number one. We give some illustrative examples using the
quadratic field Q(
5). In those examples, we identify Hilbert-Siegel
eigenforms that are possible lifts from Hilbert eigenforms.
Introduction
Let F be a real quadratic field of narrow class number one and let B be the
unique (up to isomorphism) quaternion algebra over F which is ramified at
both archimedean places of F and unramified everywhere else. Let GU2(B)
be the unitary similitude group of B⊕2. This is the set of Q-rational points
of an algebraic group GB defined over Q. The group GB is an inner form of
G := ResF/Q(GSp4) such that G
B(R) is compact modulo its centre. (These
notions are reviewed at the beginning of Section 1.)
In this paper we develop an algorithm which computes automorphic forms
on GB in the following sense: given an idealN inOF and an integer k greater
than 2, the algorithm returns the Hecke eigensystems of all automorphic
forms f of level N and parallel weight k. More precisely, given a prime p in
OF , the algorithm returns the Hecke eigenvalues of f at p, and hence the
Euler factor Lp(f, s), for each eigenform f of level N and parallel weight k.
The algorithm is a generalization of the one developed in [D1 2005] to the
genus 2 case. Although we have only described the algorithm in the case of
a real quadratic field in this paper, it should be clear from our presentation
that it can be adapted to any totally real number field of narrow class
number one.
The Jacquet-Langlands Correspondence of the title refers to the conjec-
tural map JL : Π(GB) → Π(G) from automorphic representations of GB to
automorphic representations of G, which is injective, matches L-functions
and enjoys other properties compatible with the principle of functoriality;
Date: October 29, 2018.
1991 Mathematics Subject Classification. Primary: 11F41 (Hilbert and Hilbert-Siegel
modular forms).
Key words and phrases. Hilbert-Siegel modular forms, Jacquet-Langlands Correspon-
dence, Brandt matrices, Satake parameters.
http://arxiv.org/abs/0704.0011v3
2 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
in particular, the image of the Jacquet-Langlands Correspondence is to be
contained in the space of holomorphic automorphic representations. If we
admit this conjecture, then the algorithm above provides a way to produce
examples of cuspidal Hilbert-Siegel modular forms of genus 2 over F and
allows us to compute the L-factors of the corresponding automorphic repre-
sentations for arbitrary finite primes p of F .
In fact, we are also able to use these calculations to provide evidence for
the Jacquet-Langlands Correspondence itself by comparing the Euler factors
we find with those of known Hilbert-Siegel modular forms obtained by lifting.
This we do in the final section of the paper where we observe that some of
the Euler factors we compute match those of lifts of Hilbert modular forms,
for the primes we computed. Although this does not definitively establish
that these Hilbert-Siegel modular forms are indeed lifts, in principle one can
establish equality in this way, using an analogue of the Sturm bound.
The first systematic approach to Siegel modular forms from a computa-
tional viewpoint is due to Skoruppa [Sk 1992] who used Jacobi symbols to
generate spaces of such forms. His algorithm, which has been extensively
exploited by Ryan [R 2006], applies only to the case of full level structure.
More recently, Faber and van der Geer [FvdG1 2004] and [FvdG2 2004]
also produced examples of Siegel modular forms by counting points on hy-
perelliptic curves of genus 2; again their results are available only in the
full level structure case. The most substantial progress toward the com-
putation of Siegel modular forms for proper level structure is by Gunnells
[Gu 2000] who extended the theory of modular symbols to the symplectic
group Sp4/Q. However, this work does not see the cuspidal cohomology,
which is the only part of the cohomology which is relevant to arithmetic
geometric applications. To the best of our knowledge, there are no numer-
ical examples of Hilbert-Siegel modular forms for proper level structure in
the literature, with the exception of those produced from liftings of Hilbert
modular forms.
The outline of the paper is as follows. In Section 1 we recall the basic
properties of Hilbert-Siegel modular forms and algebraic automorphic forms
together with the Jacquet-Langlands Correspondence. In Section 2 we give
a detailed description of our algorithm. Finally, in Section 3 we present
numerical results for the quadratic field Q(
Acknowledgements. During the course of the preparation of this paper,
the second author had helpful email exchanges with several people includ-
ing Alexandru Ghitza, David Helm, Marc-Hubert Nicole, David Pollack,
Jacques Tilouine and Eric Urban. The authors wish to thank them all.
Also, we would like to thank William Stein for allowing us to use the SAGE
computer cluster at the University of Washington. And finally, the sec-
ond author would like to thank the PIMS institute for their postdoctoral
fellowship support, and the University of Calgary for its hospitality.
COMPUTING HILBERT-SIEGEL MODULAR FORMS 3
1. Hilbert-Siegel modular forms and the Jacquet-Langlands
correspondence
Throughout this paper, F denotes a real quadratic field of narrow class
number one. The two archimedean places of F and the real embeddings of
F will both be denoted v0 and v1. For every a ∈ F , we write a0 (resp. a1)
for the image of a under v0 (resp. v1). The ring of integers of F is denoted
by OF . For every prime ideal p in OF , the completion of F and OF at p
will be denoted by Fp and OFp , respectively.
Let B be the unique (up to isomorphism) totally definite quaternion al-
gebra over F which is unramified at all finite primes of F . We fix a maximal
order OB of B. Also, we choose a splitting field K/F of B that is Ga-
lois over Q and such that there exists an isomorphism j : OB ⊗Z OK ∼=
M2(OK)⊕M2(OK), where M2(A) denotes the ring of 2× 2-matrices with
entries from a ring A. For every finite prime p in F , we fix an isomorphism
Bp ∼= M2(Fp) which restricts to an isomorphism from OB, p onto M2(OFp ).
The algebraic group G = ResF/Q(GSp4) is defined as follows. For any
Q-algebra A, the set of A-rational points of G is given by
G(A) =
γ ∈ GL4(A⊗Q F )
t = νG(γ)J2
νG(γ) ∈ (A⊗Q F )×
where
−12 0
This group admits an integral model with A-rational points for every Z-
algebra A given by
GZ(A) =
γ ∈ GL4(A⊗Z OF )
t = νG(γ)J2
νG(γ) ∈ (A⊗Z OF )×
For any Q-algebra A, the conjugation on B extends in a natural way to the
matrix algebra M2(B ⊗Q A).
The algebraic group GB/Q is defined as follows. For any Q-algebra A,
the set of A-rational points of GB is given by
GB(A) =
γ ∈ M2(B ⊗Q A)
γγ̄t = νGB(γ)12
νGB (γ) ∈ (A⊗Q F )×
This group also admits an integral model with A-rational points for every
Z-algebra given by
GBZ (A) =
γ ∈ M2(OB ⊗Z A)
γγ̄t = νGB(γ)12
νGB (γ) ∈ (A⊗Z OF )×
The group GB/Q is an inner form of G/Q such that GB(R) is compact
modulo its center. Combining the isomorphism j (see above) with con-
jugation by a permutation matrix, we obtain an isomorphism GBZ (OK) ∼=
4 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
GZ(OK), which we fix from now on. For every prime ideal p in F , the split-
ting of GB at p amounts to the splitting of the quaternion algebra B at p;
we refer to [D1 2005] for further details.
By the choice of the quaternion algebra B, we have GB(Q̂) ∼= G(Q̂). (We
denote the finite adèles of Q (resp. Z) by Q̂ (resp. Ẑ)).
1.1. Hilbert-Siegel modular forms. We fix an integer k ≥ 3 and, for
simplicity, we restrict ourselves to Hilbert-Siegel modular forms of parallel
weight k. The real embeddings v0 and v1 of F extend to G(Q) = GSp4(F )
in a natural way. We denote by GSp+4 (F ) the subgroup of elements γ with
totally positive similitude factor νG(γ). We recall that the Siegel upper-half
plane of genus 2 is defined by
H2 = {γ ∈ GL2(C)
∣ γt = γ and Im(γ) is positive definite }.
We also recall that GSp+4 (F ) acts on H
(τ0, τ1) :=
(a0τ0 + b0)(c0τ0 + d0)
−1, (a1τ1 + b1)(c1τ1 + d1)
This induces an action on the space of functions f : H22 → C by
, f |kγ(τ) =
νG(γi)
det(ciτi + di)k
f(τ).
Let N be an ideal in OF and set
Γ0(N) =
∈ GSp+4 (OF )
∣ c ≡ 0(N)
A Hilbert-Siegel modular form of level N and parallel weight k is a
holomorphic function f : H22 → C such that
∀γ ∈ Γ0(N), f |kγ = f.
The space of Hilbert-Siegel modular forms of parallel weight k and level N
is denoted Mk(N). Each f ∈Mk(N) admits a Fourier expansion, which by
the Koecher principle takes the form
∀τ ∈ H22, f(τ) =
{Q}∪{0}
2πiTr(Qτ),
where Q ∈ M2(F ) runs over all symmetric totally positive and semi-definite
matrices. A Hilbert-Siegel modular forms f is a cusp form if, for all γ ∈
4 (F ), the constant term in the Fourier expansion of f |kγ is zero. The
space of Hilbert-Siegel cusp forms is denoted Sk(N).
COMPUTING HILBERT-SIEGEL MODULAR FORMS 5
1.2. The Hecke algebra. The space Sk(N) comes equipped with a Hecke
action, which we now recall. Take u ∈ GSp+4 (F ) ∩M4(OF ), and write the
finite disjoint union
Γ0(N)uΓ0(N) =
Γ0(N)ui.
Then the Hecke operator [Γ0(N)uΓ0(N)] on Sk(N) is given by
[Γ0(N)uΓ0(N)]f =
f |kui.
Let p be a prime ideal in OF and let πp be a totally positive generator of
p; let T1(p) and T2(p) be the Hecke operators corresponding to the double
Γ0(N)-cosets of the symplectic similitude matrices
1 0 0 0
0 1 0 0
0 0 πp 0
0 0 0 πp
1 0 0 0
0 πp 0 0
0 0 π2p 0
0 0 0 πp
respectively. (We remind the reader of the symplectic form J2 fixed at
the beginning of Section 1.) The Hecke algebra Tk(N) is the Z-algebra
generated by the operators T1(p) and T2(p), where p runs over all primes
not dividing N .
1.3. Algebraic Hilbert-Siegel autormorphic forms. We only consider
level structure of Siegel type. Namely, we define the compact open subgroup
U0(N) of G(Q̂) by
U0(N) =
GSp4(OFp )×
ep ),
where N =
p|N p
ep and
ep ) :=
∈ GSp4(OFp )
∣ c ≡ 0 mod pep
The weight representation is defined as follows. Let Lk be the repre-
sentation of GSp4(C) of highest weight (k− 3, k− 3). We let Vk = Lk ⊗Lk
and define the complex representation (ρk, Vk) by
ρk : G
B(R) −→ GL(Vk),
where the action on the first factor is via v0, and the action on the second
one is via v1.
The space of algebraic Hilbert-Siegel modular forms of weight k and
level N is given by
MBk (N) :=
f : GB(Q̂)/U0(N) → Vk
∣ ∀γ ∈ GB(Q), f |kγ = f
6 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
where f |kγ(x) = f(γx)γ, for all x ∈ GB(Q̂)/U0(N). When k = 3, we let
IBk (N) :=
f : GB(Q)\GB(Q̂)/U0(N) → C
∣ f is constant
Then, the space of algebraic Hilbert-Siegel cusp forms of weight k and
level N is defined by
SBk (N) :=
MBk (N) if k > 3,
MBk (N)/I
k (N) if k = 3.
The action of the Hecke algebra on SBk (N) is given as follows. For any
u ∈ G(Q̂), write the finite disjoint union
U0(N)uU0(N) =
uiU0(N),
and define
[U0(N)uU0(N)] : S
k (N) → SBk (N)
f 7→ f |k[U0(N)uU0(N)],
f |k[U0(N)uU0(N)](x) =
f(xui), x ∈ G(Q̂).
For any prime p ∤ N , let ̟p be a local uniformizer at p. The local Hecke alge-
bra at p is generated by the Hecke operators T1(p) and T2(p) corresponding
to the double U0(N)-cosets ∆1(p) and ∆2(p) of the matrices
1 0 0 0
0 1 0 0
0 0 ̟p 0
0 0 0 ̟p
1 0 0 0
0 ̟p 0 0
0 0 ̟2
0 0 0 ̟p
respectively. We let TBk (N) be the Hecke algebra generated by T1(p) and
T2(p) for all primes p ∤ N .
1.4. The Jacquet-Langlands Correspondence. The Hecke modules Sk(N)
and SBk (N) are related by the following conjecture known as the Jacquet-
Langlands Correspondence for symplectic similitude groups.
Conjecture 1. The Hecke algebras Tk(N) and T
k (N) are isomorphic and
there is a compatible isomorphism of Hecke modules
Sk(N)
∼−→ SBk (N).
It is common, but perhaps not entirely accurate, to attribute this con-
jecture to Jacquet-Langlands. To the best of our knowledge, the correspon-
dence in this form was first discussed by Ihara [Ih 1964] in the case F = Q. In
[Ib 1984], Ibukiyama provided some numerical evidence. On the other hand,
it is appropriate to refer to Conjecture 1 as the Jacquet-Langlands Corre-
spondence (for GSp(4)) since it is an analogue of the Jacquet-Langlands
COMPUTING HILBERT-SIEGEL MODULAR FORMS 7
Correspondence (for GL(2)) which relates automorphic representations of
the multiplicative group of a quaternion algebra with certain automorphic
representations of GL(2) (see [JL 1970]). Both correspondences are, in turn,
special consequences of the principle of functoriality, as expounded by Lang-
lands. Finally, it appears that Conjecture 1 may soon be a theorem due to
the work of [So 2008] and the forthcoming book by James Arthur on auto-
morphic representations of classical groups.
2. The Algorithm
In this section, we present the algorithm we used in order to compute
the Hecke module of (algebraic) Hilbert-Siegel modular forms. The main
assumption in this section is that the class number of the principal genus
of GB is 1. (We refer to [D3 2007] to see how one can relax this condition
on the class number.) We recall that since B is totally definite, GB satis-
fies Proposition 1.4 in Gross [Gr 1999]. Thus the group GB(R) is compact
modulo its centre, and Γ = GB(Z)/O×F is finite.
For any prime p in F , let Fp = OF /p be the residue field at p and define
the reduction map
M2(OB, p) → M4(Fp)
g 7→ g̃,
where we use the splitting of OB,p that was fixed at the beginning of Sec-
tion 1. Now, choose a totally positive generator πp of p and put
Θ1(p) := Γ\
u ∈ M2(OB)
∣ uūt = πp12and rank(g̃) = 2
Θ2(p) := Γ\
u ∈ M2(OB)
∣ uūt = π2
12 and rank(g̃) = 1
We let H20(N) = G(Ẑ)/U0(N). Then the group Γ acts on H20(N), thus on
the space of functions f : H20(N) → Vk by
∀x ∈ H20(N),∀γ ∈ Γ, f |kγ(x) := f(γx)γ.
Theorem 2. There is an isomorphism of Hecke modules
MBk (N)
f : H20(N) → Vk
∣ f |kγ = f, γ ∈ Γ
where the Hecke action on the right hand side is given by
f |kT1(p) =
u∈Θ1(p)
f |ku,
f |kT2(p) =
u∈Θ2(p)
f |ku.
Proof. The canonical map
φ : GB(Z)\GB(Ẑ)/U0(N) → GB(Q)\GB(Q̂)/U0(N)
is an injection. Making use of the fact that the class number in the principal
genus of GB is one (GB(Q̂) = GB(Q)GBZ (Ẑ)), we see that φ is in fact a
8 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
bijection. Since each element f ∈ MBk (N) is determined by its values on a
set of coset representatives of GB(Q)\GB(Q̂)/U0(N), the map φ induces an
isomorphism of complex vector spaces
MBk (N)
f : H20(N) → Vk
∣ f |kγ = f, γ ∈ Γ
f 7−→ f ◦ φ.
We make this into a Hecke module isomorphism by defining the Hecke action
on the right hand side as indicated in the statement of the theorem. �
In the rest of this section, we explain the main steps of the algorithm
provided by Theorem 2.
2.1. The quotient H20(N). Keeping the notations of the previous section,
we recall that N =
p|N p
ep . Let p be a prime dividing N and consider the
rank 4 free
OFp/pep
-module L =
OFp/pep
endowed with the symplectic
pairing 〈 , 〉 given by the matrix
−12 0
where 12 is the identity matrix in M2(OFp/pep ). Let M be a rank 2
OFp/pep
-submodule which is a direct factor in L. We say that M is
isotropic if 〈u, v〉 = 0 for all u, v ∈ M . We recall that GSp4(OFp ) acts
transitively on the set of rank 2, isotropic
OFp/pep
-submodules of L and
that the stabilizer of the submodule generated by e1 = (1, 0, 0, 0)
T and
e2 = (0, 1, 0, 0)
T is U0(p
ep ). The quotient H20(pep ) = GSp4(OFp )/U0(pep )
is the set of rank 2, isotropic
OFp/pep
-submodules of L. Via the reduction
map ÔF → OF /N , the quotient GZ(Ẑ)/U0(N) can be identified with the
product
H20(N) =
H20(pep ).
The cardinality of H20(N) is extremely useful and is determined using the
following lemma.
Lemma 1. Let p be a prime in F and ep ≥ 1 an integer. Then, the cardi-
nality of the set H20(pep ) is given by
#H20(pep ) = N(p)3(ep−1)(N(p) + 1)(N(p)2 + 1).
Proof. For ep = 1, the cardinality of the Lagrange variety over the finite
field Fp = OF /p is given by (N(p) + 1)(N(p)2 + 1). Proceed by induction
on ep. �
We have more to say about elements of H20(pep ) in Subsection 2.5.
COMPUTING HILBERT-SIEGEL MODULAR FORMS 9
2.2. Brandt matrices. Let F = {x1, . . . , xh} be a fundamental domain
for the action of Γ on H20(N) and, for each i, let Γi be the stabilizer of xi.
Then, every element in MBk (N) is completely determined by its values on
F . Thus, there is an isomorphism of complex spaces
MBk (N) →
f 7→ (f(xi)),
where V
is the subspace of Γi-invariants in Vk.
For any x, y ∈ H20(N), we let
Θ1(x, y, p) :=
u ∈ Θ1(p)
∣ ∃γ ∈ Γ, ux = γy
Θ2(x, y, p) :=
u ∈ Θ2(p)
∣ ∃γ ∈ Γ, ux = γy
Proposition 3. The actions of the Hecke operators Ts(p), s = 1, 2, are
given by the Brandt matrices Bs(p) = (bsij(p)), where
bsji(p) : V
k → V
v 7→ v ·
u∈Θs(xi, xj ,p)
γ−1u u
Proof. The proof of Proposition 3 follows the lines of [D1 2005, §3]. �
2.3. Computing the group GB(Z). It is enough to compute the subgroup
Γ consisting of the elements in GB(Z) with similitude factor 1. But it is easy
to see that
u, v ∈ O1B
u, v ∈ O1B
where O1B is the group of norm 1 elements.
2.4. Computing the sets Θ1(p) and Θ2(p). Let us consider the quadratic
form on the vector space V = B2 given by
V → F
(a, b) 7→ ||(a, b)|| := nr(a) + nr(b),
where nr is the reduced norm on B. This determines an inner form
V × V → F
(u, v) 7→ 〈u, v〉.
An element of Θ1(p) (resp. Θ2(p)) is a unitary matrix γ ∈ M2(OB) with
respect to this inner form such that the norm of each row is πp (resp. π
and the rank of the reduced matrix is 1). So we first start by computing
all the vectors u = (a, b) ∈ O2B such that ||u|| = πp (resp. ||u|| = π2p). And
for each such vector u, we compute the vectors v = (c, d) ∈ O2B of the same
10 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
norm such that 〈u, v〉 = 0. The corresponding matrix γ =
belongs
to Θ1(p) (resp. Θ2(p)) when its reduction mod p has the appropriate rank.
We list all these matrices up to equivalence and stop when we reach the
right cardinality.
2.5. The implementation of the algorithm. The implementation of the
algorithm is similar to that of [D1 2005]. However, it is important to note
how we represent elements in H20(N) so that we can retrieve them easily
once stored. As in [D1 2005] we choose to work with the product
H20(N) =
H20(pep ).
Using Plucker’s coordinates, we can view H20(pep ) as a closed subspace of
P5(OFp/pep ). We then represent each element in H20(pep ) by choosing a
point x = (a0 : · · · : a5) = [u ∧ v] ∈ P5(OFp/pep ) such that the submodule
M generated by u and v is a Lagrange submodule, and the first invertible
coordinate is scaled to 1.
Remark 1. In [LP 2002], Lansky and Pollack describe an algorithm which
computes algebraic modular forms on the same inner form of GSp4/Q that
we use. We would like to note that there are some differences between the
two algorithms. Although [LP 2002] also uses the flag variety H20(N) in
order to determine the double coset space GB(Q)\GB(Q̂)/U0(N), it later
returns to the adelic setting in order to compute the Brandt matrices. In
contrast, Theorem 2 and Proposition 3 allow us to avoid that unnecessary
step by describing the Hecke action on the flag variety H20(N) directly. As
a result, we get an algorithm that is more efficient.
3. Numerical examples: F = Q(
5) and B =
−1,−1
In this section, we provide some numerical examples using the quadratic
field F = Q(
5). It is proven in K. Hashimoto and T. Ibukiyama [HI 1980]
that, for the Hamilton quaternion algebra B over F , the class number of
the principal genus of GB is one. We use our algorithm to compute all the
systems of Hecke eigenvalues of Hilbert-Siegel cusp forms of weight 3 and
level N that are defined over real quadratic fields, where N runs over all
prime ideals of norm less than 50. We then determine which of the forms
we obtained are possible lifts of Hilbert cusp forms by comparing the Hecke
eigenvalues for those primes.
3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3. In
Table 1 we list all the systems of eigenvalues of Hilbert-Siegel cusp forms of
weight 3 and level N that are defined over real quadratic fields, where N
runs over all prime ideals in F of norm less than 50. Here are the conventions
we use in the tables.
COMPUTING HILBERT-SIEGEL MODULAR FORMS 11
(1) For a quadratic field K of discriminant D, we let ωD be a generator
of the ring of integers OK of K.
(2) The first row contains the level N , given in the format (Norm(N), α)
for some generator α ∈ F of N , and the dimensions of the relevant
spaces.
(3) The second row lists the Hecke operators that have been computed.
(4) For each eigenform f , the Hecke eigenvalues are given in a row, and
the last entry of that row indicates if the form f is a probable lift.
(5) The levels and the eigenforms are both listed up to Galois conjuga-
tion.
For an eigenform f and a given prime p ∤ N , let a1(p, f) and a2(p, f) be the
eigenvalues of the Hecke operators T1(p) and T2(p), respectively. Then the
Euler factor Lp(f, s) is given (for example, in [AS 2001, §3.4]) by
Lp(f, s) = Qp(q
−s)−1,
where
Qp(x) = 1− a1(p, f)x+ b1(p, f)x2 − a1(p, f)q2k−3x3 + q4k−6x4,
b1(p, f) = a1(p, f)
2 − a2(p, f)− q2k−4,
q = N(p).
3.2. Tables of Hilbert cusp forms of parellel weight 4. In Table 2, we
list all the Hilbert cusp forms of parallel weight 4 and level N that are defined
over real quadratic fields, with N running over all prime ideals of norm less
than 50. (They are computed by using the algorithm in [D1 2005]). We use
this data in order to determine the forms in Table 1 that are possible lifts
from GL2.
3.3. Lifts. There are two types of lifts from GL2 to GSp4. The first one
corresponds to the homomorphism of L-groups determined by the long root
embedding into GSp4, and the second one by the short root embedding.
(See [LP 2002] for more details). Let f be a Hilbert cusp form of parallel
weight k and level N with Hecke eigenvalues a(p, f), where p is a prime not
dividing N . Let φ be the lift of f to GSp4 via the long root, and ψ the one
via the short root. Then the Hecke eigenvalues of φ are given by
a1(p, φ) = a(p, f) N(p)
2 +N(p)2 +N(p)
a2(p, φ) = a(p, f) N(p)
2 (N(p) + 1) +N(p)2 − 1,
and the Hecke eigenvalues of ψ are given by
a1(p, ψ) = a(p, f)
2 − 2 a(p, f) N(p)
a2(p, ψ) = a(p, f)
N(p)4−2k − 3 a(p, f)2 N(p)3−k +N(p)2 − 1.
The second lift ψ is the so-called symmetric cube lifting.
12 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
N = (4, 2) : dimMB
(N) = 2, dimSB
(N) = 1
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 −4 0 20 −36 140 580 yes
N = (5, 2 + ω5) : dimM
(N) = 2, dimSB
(N) = 1
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 20 15 −5 0 40 −420 yes
N = (9, 3) : dimMB
(N) = 3, dimSB
(N) = 2
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 25− 3ω41 40− 15ω41 30 + 6ω41 24 + 36ω41 −9 0 yes
N = (11, 3 + ω5) : dimM
(N) = 3, dimSB
(N) = 2
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 24 35 34 48 88 60 yes
f2 −20 35 −10 4 0 60 no
N = (19, 4 + ω5) : dimM
(N) = 5, dimSB
(N) = 4
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 4 11 −20 28 6 76 no
f2 7 −50 15 −66 73 −90 yes
f3 24 + ω161 35 + 5ω161 36− ω161 60− 6ω161 98− 3ω161 160− 30ω161 yes
N = (29, 5 + ω5) : dimM
(N) = 9, dimSB
(N) = 8
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 −4 11 10 20 30 60 no
f2 8 −45 30 24 50 −320 yes
f3 17 0 9 −102 86 40 yes
N = (31, 5 + 2ω5) : dimM
(N) = 12, dimSB
(N) = 11
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 13 −20 20 −36 76 −60 yes
N = (41, 6 + ω5) : dimM
(N) = 19, dimSB
(N) = 18
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 10 20 −10 29 30 −20 no
f2 −1 1 5 14 −2 −56 no
f3 27 50 40 84 124 420 yes
f4 −12 19 30 65 0 0 no
f5 16− 2ω21 −5− 10ω21 21 + 4ω21 −30 + 24ω21 72− 2ω21 −100− 20ω21 yes
f6 2− 6ω5 11− 2ω5 8 + 4ω5 11− 4ω5 −12 + 54ω5 160 + 40ω5 no
N = (49, 7) : dimMB
(N) = 26, dimSB
(N) = 25
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 5 −60 46 120 40 −420 yes
f2 4 + 4ω65 32 + 3ω65 12− 4ω65 44− 4ω65 −6− 12ω65 145 + 8ω65 no
Table 1. Hilbert-Siegel eigenforms of weight 3
COMPUTING HILBERT-SIEGEL MODULAR FORMS 13
N (4, 2) (5, 2 + ω5) (9, 3) (11, 3 + ω5)
N(p) p a(p, f1) a(p, f1) a(p, f1) a(p, f1)
4 2 −4 0 5− 3ω41 4
5 2 + ω5 −10 −5 6ω41 4
9 3 50 −50 −9 −2
11 3 + 2ω5 −28 32 −18− 6ω41 −10
11 3 + ω5 −28 32 −18− 6ω41 −11
19 4 + 3ω5 60 100 −40 + 24ω41 −94
19 4 + ω5 60 100 −40 + 24ω41 28
N (19, 4 + ω5) (29, 5 + ω5)
N(p) p a(p, f1) a(p, f2) a(p, f1) a(p, f2)
4 2 −13 5− ω161 −12 −3
5 2 + ω5 −15 5 + ω161 0 −21
9 3 −17 5 + 3ω161 −40 −4
11 3 + 2ω5 −6 2 + 8ω161 −68 37
11 3 + ω5 33 7− 7ω161 30 −66
19 4 + 3ω5 −139 −15− 9ω161 −28 −40
19 4 + ω5 19 −19 84 −9
N (31, 5 + 2ω5) (41, 6 + ω5)
N(p) p a(p, f1) a(p, f1) a(p, f2)
4 2 −7 7 −4− 2ω21
5 2 + ω5 −10 10 −9 + 4ω21
9 3 −14 34 −18− 2ω21
11 3 + 2ω5 −20 −60 −19
11 3 + ω5 −28 −2 −24− 4ω21
19 4 + 3ω5 −12 74 4− 50ω21
19 4 + ω5 28 16 −29 + 44ω21
N (49, 7)
N(p) p a(p, f1) a(p, f2)
4 2 −15 −2
5 2 + ω5 16 −10
9 3 −50 −11
11 3 + 2ω5 −8 −7− 28ω13
11 3 + ω5 −8 −35 + 28ω13
19 4 + 3ω5 −110 −26 + 14ω13
19 4 + ω5 −110 −12− 14ω13
Table 2. Hilbert eigenforms of weight 4
Remark 2. So far, our algorithm has been implemented only for congruence
subgroups of Siegel type. We intend to improve the implementation in the
near future so as to include more additional level structures such as the
Klingen type. Indeed, Ramakrishnan and Shahidi [RS 2007] recently showed
the existence of symmetric cube lifts for non-CM elliptic curves E/Q to
GSp4/Q. And their result should hold for other totally real number fields,
with the level structures of the lifts being of Klingen type. Unfortunately,
14 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
those lifts cannot be seen in our current tables. For example, there are
modular elliptic curves over Q(
5) whose conductors have norm 31, 41 and
49, but the corresponding symmetric cubic lifts do not appear in Table 1.
We would like to remedy that in our next implementation.
References
[D1 2005] L. Dembélé, Explicit computations of Hilbert modular forms on Q(
5). Exper-
iment. Math. 14 (2005), no. 4, 457–466.
[D2 2007] L. Dembélé, Quaternionic M -symbols, Brandt matrices and Hilbert modular
forms. Math. Comp. 76, no 258, (2007), 1039-1057. Also available electronically.
[D3 2007] L. Dembélé, On the computation of algebraic modular forms (submitted).
[AS 2001] Mahdi Asgari and Ralf Schmidt, Siegel modular forms and representations,
Manuscripta Math. 104 (2001), 173–200.
[FvdG1 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes
locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes.
I, C. R. Math. Acad. Sci. Paris 338 (2004), no. 5, 381–384.
[FvdG2 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes
locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes.
II, C. R. Math. Acad. Sci. Paris 338 (2004), no. 6, 467–470.
[JL 1970] Hervé Jacquet and Robert Langlands, Automorphic forms on GL(2), Lecture
notes in mathematics 114 and 278, 1970.
[Gr 1999] Benedict H. Gross, Algebraic modular forms. Israel J. Math. 113 (1999), 61–93.
[Gu 2000] P. Gunnells, Symplectic modular symbols, Duke Math. J. 102 (2000), no. 2,
329-350.
[HI 1980] K. Hashimoto and T. Ibukiyama, On the class numbers of positive definite
binary quaternion hermitian forms. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980),
549-601.
[Ib 1984] T. Ibukiyama, On symplectic Euler factors of genus 2. J. Fac. Sci. Univ. Tokyo
30 (1984), 587614.
[Ih 1964] Y. Ihara, On certain Dirichlet series, J. Math. Soc. Japan 16 (1964), 214-225.
[LP 2002] J. Lansky and D. Pollack, Hecke algebras and automorphic forms. Compositio
Math. 130 (2002), no. 1, 21–48.
[RS 2007] Dinakar Ramakrishnan and Freydoon Shahidi, Siegel modular forms of genus
2 attached to elliptic curves (preprint). Available at www.math.arxiv.
[R 2006] N. C. Ryan, Computing the Satake p-parameters of Siegel modular forms. (sub-
mitted).
[Sk 1992] Nils-Peter Skoruppa, Computations of Siegel modular forms of genus two. Math.
Comp. 58 (1992), no. 197, 381–398.
[So 2008] Claus M. Sorensen, Potential level-lowering for GSp(4), arXive:0804.0588v1.
Department of Mathematics, University of Calgary
E-mail address: cunning@math.ucalgary.ca
Institut für Experimentelle Mathematik, Universität Duisburg-Essen
E-mail address: lassina.dembele@uni-duisburg-essen.de
	Introduction
	1. Hilbert-Siegel modular forms and the Jacquet-Langlands correspondence
	1.1. Hilbert-Siegel modular forms
	1.2. The Hecke algebra
	1.3. Algebraic Hilbert-Siegel autormorphic forms
	1.4. The Jacquet-Langlands Correspondence
	2. The Algorithm
	2.1. The quotient H02(N)
	2.2. Brandt matrices
	2.3. Computing the group GB(Z)
	2.4. Computing the sets 1(p) and 2(p)
	2.5. The implementation of the algorithm
	3. Numerical examples: F=Q(5) and B=(-1,-1F)
	3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3
	3.2. Tables of Hilbert cusp forms of parellel weight 4
	3.3. Lifts
	References

  In this paper we present an algorithm for computing Hecke eigensystems of
Hilbert-Siegel cusp forms over real quadratic fields of narrow class number
one. We give some illustrative examples using the quadratic field
$\Q(\sqrt{5})$. In those examples, we identify Hilbert-Siegel eigenforms that
are possible lifts from Hilbert eigenforms.

Introduction
Let F be a real quadratic field of narrow class number one and let B be the
unique (up to isomorphism) quaternion algebra over F which is ramified at
both archimedean places of F and unramified everywhere else. Let GU2(B)
be the unitary similitude group of B⊕2. This is the set of Q-rational points
of an algebraic group GB defined over Q. The group GB is an inner form of
G := ResF/Q(GSp4) such that G
B(R) is compact modulo its centre. (These
notions are reviewed at the beginning of Section 1.)
In this paper we develop an algorithm which computes automorphic forms
on GB in the following sense: given an idealN inOF and an integer k greater
than 2, the algorithm returns the Hecke eigensystems of all automorphic
forms f of level N and parallel weight k. More precisely, given a prime p in
OF , the algorithm returns the Hecke eigenvalues of f at p, and hence the
Euler factor Lp(f, s), for each eigenform f of level N and parallel weight k.
The algorithm is a generalization of the one developed in [D1 2005] to the
genus 2 case. Although we have only described the algorithm in the case of
a real quadratic field in this paper, it should be clear from our presentation
that it can be adapted to any totally real number field of narrow class
number one.
The Jacquet-Langlands Correspondence of the title refers to the conjec-
tural map JL : Π(GB) → Π(G) from automorphic representations of GB to
automorphic representations of G, which is injective, matches L-functions
and enjoys other properties compatible with the principle of functoriality;
Date: October 29, 2018.
1991 Mathematics Subject Classification. Primary: 11F41 (Hilbert and Hilbert-Siegel
modular forms).
Key words and phrases. Hilbert-Siegel modular forms, Jacquet-Langlands Correspon-
dence, Brandt matrices, Satake parameters.
http://arxiv.org/abs/0704.0011v3
2 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
in particular, the image of the Jacquet-Langlands Correspondence is to be
contained in the space of holomorphic automorphic representations. If we
admit this conjecture, then the algorithm above provides a way to produce
examples of cuspidal Hilbert-Siegel modular forms of genus 2 over F and
allows us to compute the L-factors of the corresponding automorphic repre-
sentations for arbitrary finite primes p of F .
In fact, we are also able to use these calculations to provide evidence for
the Jacquet-Langlands Correspondence itself by comparing the Euler factors
we find with those of known Hilbert-Siegel modular forms obtained by lifting.
This we do in the final section of the paper where we observe that some of
the Euler factors we compute match those of lifts of Hilbert modular forms,
for the primes we computed. Although this does not definitively establish
that these Hilbert-Siegel modular forms are indeed lifts, in principle one can
establish equality in this way, using an analogue of the Sturm bound.
The first systematic approach to Siegel modular forms from a computa-
tional viewpoint is due to Skoruppa [Sk 1992] who used Jacobi symbols to
generate spaces of such forms. His algorithm, which has been extensively
exploited by Ryan [R 2006], applies only to the case of full level structure.
More recently, Faber and van der Geer [FvdG1 2004] and [FvdG2 2004]
also produced examples of Siegel modular forms by counting points on hy-
perelliptic curves of genus 2; again their results are available only in the
full level structure case. The most substantial progress toward the com-
putation of Siegel modular forms for proper level structure is by Gunnells
[Gu 2000] who extended the theory of modular symbols to the symplectic
group Sp4/Q. However, this work does not see the cuspidal cohomology,
which is the only part of the cohomology which is relevant to arithmetic
geometric applications. To the best of our knowledge, there are no numer-
ical examples of Hilbert-Siegel modular forms for proper level structure in
the literature, with the exception of those produced from liftings of Hilbert
modular forms.
The outline of the paper is as follows. In Section 1 we recall the basic
properties of Hilbert-Siegel modular forms and algebraic automorphic forms
together with the Jacquet-Langlands Correspondence. In Section 2 we give
a detailed description of our algorithm. Finally, in Section 3 we present
numerical results for the quadratic field Q(
Acknowledgements. During the course of the preparation of this paper,
the second author had helpful email exchanges with several people includ-
ing Alexandru Ghitza, David Helm, Marc-Hubert Nicole, David Pollack,
Jacques Tilouine and Eric Urban. The authors wish to thank them all.
Also, we would like to thank William Stein for allowing us to use the SAGE
computer cluster at the University of Washington. And finally, the sec-
ond author would like to thank the PIMS institute for their postdoctoral
fellowship support, and the University of Calgary for its hospitality.
COMPUTING HILBERT-SIEGEL MODULAR FORMS 3
1. Hilbert-Siegel modular forms and the Jacquet-Langlands
correspondence
Throughout this paper, F denotes a real quadratic field of narrow class
number one. The two archimedean places of F and the real embeddings of
F will both be denoted v0 and v1. For every a ∈ F , we write a0 (resp. a1)
for the image of a under v0 (resp. v1). The ring of integers of F is denoted
by OF . For every prime ideal p in OF , the completion of F and OF at p
will be denoted by Fp and OFp , respectively.
Let B be the unique (up to isomorphism) totally definite quaternion al-
gebra over F which is unramified at all finite primes of F . We fix a maximal
order OB of B. Also, we choose a splitting field K/F of B that is Ga-
lois over Q and such that there exists an isomorphism j : OB ⊗Z OK ∼=
M2(OK)⊕M2(OK), where M2(A) denotes the ring of 2× 2-matrices with
entries from a ring A. For every finite prime p in F , we fix an isomorphism
Bp ∼= M2(Fp) which restricts to an isomorphism from OB, p onto M2(OFp ).
The algebraic group G = ResF/Q(GSp4) is defined as follows. For any
Q-algebra A, the set of A-rational points of G is given by
G(A) =
γ ∈ GL4(A⊗Q F )
t = νG(γ)J2
νG(γ) ∈ (A⊗Q F )×
where
−12 0
This group admits an integral model with A-rational points for every Z-
algebra A given by
GZ(A) =
γ ∈ GL4(A⊗Z OF )
t = νG(γ)J2
νG(γ) ∈ (A⊗Z OF )×
For any Q-algebra A, the conjugation on B extends in a natural way to the
matrix algebra M2(B ⊗Q A).
The algebraic group GB/Q is defined as follows. For any Q-algebra A,
the set of A-rational points of GB is given by
GB(A) =
γ ∈ M2(B ⊗Q A)
γγ̄t = νGB(γ)12
νGB (γ) ∈ (A⊗Q F )×
This group also admits an integral model with A-rational points for every
Z-algebra given by
GBZ (A) =
γ ∈ M2(OB ⊗Z A)
γγ̄t = νGB(γ)12
νGB (γ) ∈ (A⊗Z OF )×
The group GB/Q is an inner form of G/Q such that GB(R) is compact
modulo its center. Combining the isomorphism j (see above) with con-
jugation by a permutation matrix, we obtain an isomorphism GBZ (OK) ∼=
4 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
GZ(OK), which we fix from now on. For every prime ideal p in F , the split-
ting of GB at p amounts to the splitting of the quaternion algebra B at p;
we refer to [D1 2005] for further details.
By the choice of the quaternion algebra B, we have GB(Q̂) ∼= G(Q̂). (We
denote the finite adèles of Q (resp. Z) by Q̂ (resp. Ẑ)).
1.1. Hilbert-Siegel modular forms. We fix an integer k ≥ 3 and, for
simplicity, we restrict ourselves to Hilbert-Siegel modular forms of parallel
weight k. The real embeddings v0 and v1 of F extend to G(Q) = GSp4(F )
in a natural way. We denote by GSp+4 (F ) the subgroup of elements γ with
totally positive similitude factor νG(γ). We recall that the Siegel upper-half
plane of genus 2 is defined by
H2 = {γ ∈ GL2(C)
∣ γt = γ and Im(γ) is positive definite }.
We also recall that GSp+4 (F ) acts on H
(τ0, τ1) :=
(a0τ0 + b0)(c0τ0 + d0)
−1, (a1τ1 + b1)(c1τ1 + d1)
This induces an action on the space of functions f : H22 → C by
, f |kγ(τ) =
νG(γi)
det(ciτi + di)k
f(τ).
Let N be an ideal in OF and set
Γ0(N) =
∈ GSp+4 (OF )
∣ c ≡ 0(N)
A Hilbert-Siegel modular form of level N and parallel weight k is a
holomorphic function f : H22 → C such that
∀γ ∈ Γ0(N), f |kγ = f.
The space of Hilbert-Siegel modular forms of parallel weight k and level N
is denoted Mk(N). Each f ∈Mk(N) admits a Fourier expansion, which by
the Koecher principle takes the form
∀τ ∈ H22, f(τ) =
{Q}∪{0}
2πiTr(Qτ),
where Q ∈ M2(F ) runs over all symmetric totally positive and semi-definite
matrices. A Hilbert-Siegel modular forms f is a cusp form if, for all γ ∈
4 (F ), the constant term in the Fourier expansion of f |kγ is zero. The
space of Hilbert-Siegel cusp forms is denoted Sk(N).
COMPUTING HILBERT-SIEGEL MODULAR FORMS 5
1.2. The Hecke algebra. The space Sk(N) comes equipped with a Hecke
action, which we now recall. Take u ∈ GSp+4 (F ) ∩M4(OF ), and write the
finite disjoint union
Γ0(N)uΓ0(N) =
Γ0(N)ui.
Then the Hecke operator [Γ0(N)uΓ0(N)] on Sk(N) is given by
[Γ0(N)uΓ0(N)]f =
f |kui.
Let p be a prime ideal in OF and let πp be a totally positive generator of
p; let T1(p) and T2(p) be the Hecke operators corresponding to the double
Γ0(N)-cosets of the symplectic similitude matrices
1 0 0 0
0 1 0 0
0 0 πp 0
0 0 0 πp
1 0 0 0
0 πp 0 0
0 0 π2p 0
0 0 0 πp
respectively. (We remind the reader of the symplectic form J2 fixed at
the beginning of Section 1.) The Hecke algebra Tk(N) is the Z-algebra
generated by the operators T1(p) and T2(p), where p runs over all primes
not dividing N .
1.3. Algebraic Hilbert-Siegel autormorphic forms. We only consider
level structure of Siegel type. Namely, we define the compact open subgroup
U0(N) of G(Q̂) by
U0(N) =
GSp4(OFp )×
ep ),
where N =
p|N p
ep and
ep ) :=
∈ GSp4(OFp )
∣ c ≡ 0 mod pep
The weight representation is defined as follows. Let Lk be the repre-
sentation of GSp4(C) of highest weight (k− 3, k− 3). We let Vk = Lk ⊗Lk
and define the complex representation (ρk, Vk) by
ρk : G
B(R) −→ GL(Vk),
where the action on the first factor is via v0, and the action on the second
one is via v1.
The space of algebraic Hilbert-Siegel modular forms of weight k and
level N is given by
MBk (N) :=
f : GB(Q̂)/U0(N) → Vk
∣ ∀γ ∈ GB(Q), f |kγ = f
6 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
where f |kγ(x) = f(γx)γ, for all x ∈ GB(Q̂)/U0(N). When k = 3, we let
IBk (N) :=
f : GB(Q)\GB(Q̂)/U0(N) → C
∣ f is constant
Then, the space of algebraic Hilbert-Siegel cusp forms of weight k and
level N is defined by
SBk (N) :=
MBk (N) if k > 3,
MBk (N)/I
k (N) if k = 3.
The action of the Hecke algebra on SBk (N) is given as follows. For any
u ∈ G(Q̂), write the finite disjoint union
U0(N)uU0(N) =
uiU0(N),
and define
[U0(N)uU0(N)] : S
k (N) → SBk (N)
f 7→ f |k[U0(N)uU0(N)],
f |k[U0(N)uU0(N)](x) =
f(xui), x ∈ G(Q̂).
For any prime p ∤ N , let ̟p be a local uniformizer at p. The local Hecke alge-
bra at p is generated by the Hecke operators T1(p) and T2(p) corresponding
to the double U0(N)-cosets ∆1(p) and ∆2(p) of the matrices
1 0 0 0
0 1 0 0
0 0 ̟p 0
0 0 0 ̟p
1 0 0 0
0 ̟p 0 0
0 0 ̟2
0 0 0 ̟p
respectively. We let TBk (N) be the Hecke algebra generated by T1(p) and
T2(p) for all primes p ∤ N .
1.4. The Jacquet-Langlands Correspondence. The Hecke modules Sk(N)
and SBk (N) are related by the following conjecture known as the Jacquet-
Langlands Correspondence for symplectic similitude groups.
Conjecture 1. The Hecke algebras Tk(N) and T
k (N) are isomorphic and
there is a compatible isomorphism of Hecke modules
Sk(N)
∼−→ SBk (N).
It is common, but perhaps not entirely accurate, to attribute this con-
jecture to Jacquet-Langlands. To the best of our knowledge, the correspon-
dence in this form was first discussed by Ihara [Ih 1964] in the case F = Q. In
[Ib 1984], Ibukiyama provided some numerical evidence. On the other hand,
it is appropriate to refer to Conjecture 1 as the Jacquet-Langlands Corre-
spondence (for GSp(4)) since it is an analogue of the Jacquet-Langlands
COMPUTING HILBERT-SIEGEL MODULAR FORMS 7
Correspondence (for GL(2)) which relates automorphic representations of
the multiplicative group of a quaternion algebra with certain automorphic
representations of GL(2) (see [JL 1970]). Both correspondences are, in turn,
special consequences of the principle of functoriality, as expounded by Lang-
lands. Finally, it appears that Conjecture 1 may soon be a theorem due to
the work of [So 2008] and the forthcoming book by James Arthur on auto-
morphic representations of classical groups.
2. The Algorithm
In this section, we present the algorithm we used in order to compute
the Hecke module of (algebraic) Hilbert-Siegel modular forms. The main
assumption in this section is that the class number of the principal genus
of GB is 1. (We refer to [D3 2007] to see how one can relax this condition
on the class number.) We recall that since B is totally definite, GB satis-
fies Proposition 1.4 in Gross [Gr 1999]. Thus the group GB(R) is compact
modulo its centre, and Γ = GB(Z)/O×F is finite.
For any prime p in F , let Fp = OF /p be the residue field at p and define
the reduction map
M2(OB, p) → M4(Fp)
g 7→ g̃,
where we use the splitting of OB,p that was fixed at the beginning of Sec-
tion 1. Now, choose a totally positive generator πp of p and put
Θ1(p) := Γ\
u ∈ M2(OB)
∣ uūt = πp12and rank(g̃) = 2
Θ2(p) := Γ\
u ∈ M2(OB)
∣ uūt = π2
12 and rank(g̃) = 1
We let H20(N) = G(Ẑ)/U0(N). Then the group Γ acts on H20(N), thus on
the space of functions f : H20(N) → Vk by
∀x ∈ H20(N),∀γ ∈ Γ, f |kγ(x) := f(γx)γ.
Theorem 2. There is an isomorphism of Hecke modules
MBk (N)
f : H20(N) → Vk
∣ f |kγ = f, γ ∈ Γ
where the Hecke action on the right hand side is given by
f |kT1(p) =
u∈Θ1(p)
f |ku,
f |kT2(p) =
u∈Θ2(p)
f |ku.
Proof. The canonical map
φ : GB(Z)\GB(Ẑ)/U0(N) → GB(Q)\GB(Q̂)/U0(N)
is an injection. Making use of the fact that the class number in the principal
genus of GB is one (GB(Q̂) = GB(Q)GBZ (Ẑ)), we see that φ is in fact a
8 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
bijection. Since each element f ∈ MBk (N) is determined by its values on a
set of coset representatives of GB(Q)\GB(Q̂)/U0(N), the map φ induces an
isomorphism of complex vector spaces
MBk (N)
f : H20(N) → Vk
∣ f |kγ = f, γ ∈ Γ
f 7−→ f ◦ φ.
We make this into a Hecke module isomorphism by defining the Hecke action
on the right hand side as indicated in the statement of the theorem. �
In the rest of this section, we explain the main steps of the algorithm
provided by Theorem 2.
2.1. The quotient H20(N). Keeping the notations of the previous section,
we recall that N =
p|N p
ep . Let p be a prime dividing N and consider the
rank 4 free
OFp/pep
-module L =
OFp/pep
endowed with the symplectic
pairing 〈 , 〉 given by the matrix
−12 0
where 12 is the identity matrix in M2(OFp/pep ). Let M be a rank 2
OFp/pep
-submodule which is a direct factor in L. We say that M is
isotropic if 〈u, v〉 = 0 for all u, v ∈ M . We recall that GSp4(OFp ) acts
transitively on the set of rank 2, isotropic
OFp/pep
-submodules of L and
that the stabilizer of the submodule generated by e1 = (1, 0, 0, 0)
T and
e2 = (0, 1, 0, 0)
T is U0(p
ep ). The quotient H20(pep ) = GSp4(OFp )/U0(pep )
is the set of rank 2, isotropic
OFp/pep
-submodules of L. Via the reduction
map ÔF → OF /N , the quotient GZ(Ẑ)/U0(N) can be identified with the
product
H20(N) =
H20(pep ).
The cardinality of H20(N) is extremely useful and is determined using the
following lemma.
Lemma 1. Let p be a prime in F and ep ≥ 1 an integer. Then, the cardi-
nality of the set H20(pep ) is given by
#H20(pep ) = N(p)3(ep−1)(N(p) + 1)(N(p)2 + 1).
Proof. For ep = 1, the cardinality of the Lagrange variety over the finite
field Fp = OF /p is given by (N(p) + 1)(N(p)2 + 1). Proceed by induction
on ep. �
We have more to say about elements of H20(pep ) in Subsection 2.5.
COMPUTING HILBERT-SIEGEL MODULAR FORMS 9
2.2. Brandt matrices. Let F = {x1, . . . , xh} be a fundamental domain
for the action of Γ on H20(N) and, for each i, let Γi be the stabilizer of xi.
Then, every element in MBk (N) is completely determined by its values on
F . Thus, there is an isomorphism of complex spaces
MBk (N) →
f 7→ (f(xi)),
where V
is the subspace of Γi-invariants in Vk.
For any x, y ∈ H20(N), we let
Θ1(x, y, p) :=
u ∈ Θ1(p)
∣ ∃γ ∈ Γ, ux = γy
Θ2(x, y, p) :=
u ∈ Θ2(p)
∣ ∃γ ∈ Γ, ux = γy
Proposition 3. The actions of the Hecke operators Ts(p), s = 1, 2, are
given by the Brandt matrices Bs(p) = (bsij(p)), where
bsji(p) : V
k → V
v 7→ v ·
u∈Θs(xi, xj ,p)
γ−1u u
Proof. The proof of Proposition 3 follows the lines of [D1 2005, §3]. �
2.3. Computing the group GB(Z). It is enough to compute the subgroup
Γ consisting of the elements in GB(Z) with similitude factor 1. But it is easy
to see that
u, v ∈ O1B
u, v ∈ O1B
where O1B is the group of norm 1 elements.
2.4. Computing the sets Θ1(p) and Θ2(p). Let us consider the quadratic
form on the vector space V = B2 given by
V → F
(a, b) 7→ ||(a, b)|| := nr(a) + nr(b),
where nr is the reduced norm on B. This determines an inner form
V × V → F
(u, v) 7→ 〈u, v〉.
An element of Θ1(p) (resp. Θ2(p)) is a unitary matrix γ ∈ M2(OB) with
respect to this inner form such that the norm of each row is πp (resp. π
and the rank of the reduced matrix is 1). So we first start by computing
all the vectors u = (a, b) ∈ O2B such that ||u|| = πp (resp. ||u|| = π2p). And
for each such vector u, we compute the vectors v = (c, d) ∈ O2B of the same
10 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
norm such that 〈u, v〉 = 0. The corresponding matrix γ =
belongs
to Θ1(p) (resp. Θ2(p)) when its reduction mod p has the appropriate rank.
We list all these matrices up to equivalence and stop when we reach the
right cardinality.
2.5. The implementation of the algorithm. The implementation of the
algorithm is similar to that of [D1 2005]. However, it is important to note
how we represent elements in H20(N) so that we can retrieve them easily
once stored. As in [D1 2005] we choose to work with the product
H20(N) =
H20(pep ).
Using Plucker’s coordinates, we can view H20(pep ) as a closed subspace of
P5(OFp/pep ). We then represent each element in H20(pep ) by choosing a
point x = (a0 : · · · : a5) = [u ∧ v] ∈ P5(OFp/pep ) such that the submodule
M generated by u and v is a Lagrange submodule, and the first invertible
coordinate is scaled to 1.
Remark 1. In [LP 2002], Lansky and Pollack describe an algorithm which
computes algebraic modular forms on the same inner form of GSp4/Q that
we use. We would like to note that there are some differences between the
two algorithms. Although [LP 2002] also uses the flag variety H20(N) in
order to determine the double coset space GB(Q)\GB(Q̂)/U0(N), it later
returns to the adelic setting in order to compute the Brandt matrices. In
contrast, Theorem 2 and Proposition 3 allow us to avoid that unnecessary
step by describing the Hecke action on the flag variety H20(N) directly. As
a result, we get an algorithm that is more efficient.
3. Numerical examples: F = Q(
5) and B =
−1,−1
In this section, we provide some numerical examples using the quadratic
field F = Q(
5). It is proven in K. Hashimoto and T. Ibukiyama [HI 1980]
that, for the Hamilton quaternion algebra B over F , the class number of
the principal genus of GB is one. We use our algorithm to compute all the
systems of Hecke eigenvalues of Hilbert-Siegel cusp forms of weight 3 and
level N that are defined over real quadratic fields, where N runs over all
prime ideals of norm less than 50. We then determine which of the forms
we obtained are possible lifts of Hilbert cusp forms by comparing the Hecke
eigenvalues for those primes.
3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3. In
Table 1 we list all the systems of eigenvalues of Hilbert-Siegel cusp forms of
weight 3 and level N that are defined over real quadratic fields, where N
runs over all prime ideals in F of norm less than 50. Here are the conventions
we use in the tables.
COMPUTING HILBERT-SIEGEL MODULAR FORMS 11
(1) For a quadratic field K of discriminant D, we let ωD be a generator
of the ring of integers OK of K.
(2) The first row contains the level N , given in the format (Norm(N), α)
for some generator α ∈ F of N , and the dimensions of the relevant
spaces.
(3) The second row lists the Hecke operators that have been computed.
(4) For each eigenform f , the Hecke eigenvalues are given in a row, and
the last entry of that row indicates if the form f is a probable lift.
(5) The levels and the eigenforms are both listed up to Galois conjuga-
tion.
For an eigenform f and a given prime p ∤ N , let a1(p, f) and a2(p, f) be the
eigenvalues of the Hecke operators T1(p) and T2(p), respectively. Then the
Euler factor Lp(f, s) is given (for example, in [AS 2001, §3.4]) by
Lp(f, s) = Qp(q
−s)−1,
where
Qp(x) = 1− a1(p, f)x+ b1(p, f)x2 − a1(p, f)q2k−3x3 + q4k−6x4,
b1(p, f) = a1(p, f)
2 − a2(p, f)− q2k−4,
q = N(p).
3.2. Tables of Hilbert cusp forms of parellel weight 4. In Table 2, we
list all the Hilbert cusp forms of parallel weight 4 and level N that are defined
over real quadratic fields, with N running over all prime ideals of norm less
than 50. (They are computed by using the algorithm in [D1 2005]). We use
this data in order to determine the forms in Table 1 that are possible lifts
from GL2.
3.3. Lifts. There are two types of lifts from GL2 to GSp4. The first one
corresponds to the homomorphism of L-groups determined by the long root
embedding into GSp4, and the second one by the short root embedding.
(See [LP 2002] for more details). Let f be a Hilbert cusp form of parallel
weight k and level N with Hecke eigenvalues a(p, f), where p is a prime not
dividing N . Let φ be the lift of f to GSp4 via the long root, and ψ the one
via the short root. Then the Hecke eigenvalues of φ are given by
a1(p, φ) = a(p, f) N(p)
2 +N(p)2 +N(p)
a2(p, φ) = a(p, f) N(p)
2 (N(p) + 1) +N(p)2 − 1,
and the Hecke eigenvalues of ψ are given by
a1(p, ψ) = a(p, f)
2 − 2 a(p, f) N(p)
a2(p, ψ) = a(p, f)
N(p)4−2k − 3 a(p, f)2 N(p)3−k +N(p)2 − 1.
The second lift ψ is the so-called symmetric cube lifting.
12 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
N = (4, 2) : dimMB
(N) = 2, dimSB
(N) = 1
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 −4 0 20 −36 140 580 yes
N = (5, 2 + ω5) : dimM
(N) = 2, dimSB
(N) = 1
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 20 15 −5 0 40 −420 yes
N = (9, 3) : dimMB
(N) = 3, dimSB
(N) = 2
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 25− 3ω41 40− 15ω41 30 + 6ω41 24 + 36ω41 −9 0 yes
N = (11, 3 + ω5) : dimM
(N) = 3, dimSB
(N) = 2
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 24 35 34 48 88 60 yes
f2 −20 35 −10 4 0 60 no
N = (19, 4 + ω5) : dimM
(N) = 5, dimSB
(N) = 4
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 4 11 −20 28 6 76 no
f2 7 −50 15 −66 73 −90 yes
f3 24 + ω161 35 + 5ω161 36− ω161 60− 6ω161 98− 3ω161 160− 30ω161 yes
N = (29, 5 + ω5) : dimM
(N) = 9, dimSB
(N) = 8
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 −4 11 10 20 30 60 no
f2 8 −45 30 24 50 −320 yes
f3 17 0 9 −102 86 40 yes
N = (31, 5 + 2ω5) : dimM
(N) = 12, dimSB
(N) = 11
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 13 −20 20 −36 76 −60 yes
N = (41, 6 + ω5) : dimM
(N) = 19, dimSB
(N) = 18
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 10 20 −10 29 30 −20 no
f2 −1 1 5 14 −2 −56 no
f3 27 50 40 84 124 420 yes
f4 −12 19 30 65 0 0 no
f5 16− 2ω21 −5− 10ω21 21 + 4ω21 −30 + 24ω21 72− 2ω21 −100− 20ω21 yes
f6 2− 6ω5 11− 2ω5 8 + 4ω5 11− 4ω5 −12 + 54ω5 160 + 40ω5 no
N = (49, 7) : dimMB
(N) = 26, dimSB
(N) = 25
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 5 −60 46 120 40 −420 yes
f2 4 + 4ω65 32 + 3ω65 12− 4ω65 44− 4ω65 −6− 12ω65 145 + 8ω65 no
Table 1. Hilbert-Siegel eigenforms of weight 3
COMPUTING HILBERT-SIEGEL MODULAR FORMS 13
N (4, 2) (5, 2 + ω5) (9, 3) (11, 3 + ω5)
N(p) p a(p, f1) a(p, f1) a(p, f1) a(p, f1)
4 2 −4 0 5− 3ω41 4
5 2 + ω5 −10 −5 6ω41 4
9 3 50 −50 −9 −2
11 3 + 2ω5 −28 32 −18− 6ω41 −10
11 3 + ω5 −28 32 −18− 6ω41 −11
19 4 + 3ω5 60 100 −40 + 24ω41 −94
19 4 + ω5 60 100 −40 + 24ω41 28
N (19, 4 + ω5) (29, 5 + ω5)
N(p) p a(p, f1) a(p, f2) a(p, f1) a(p, f2)
4 2 −13 5− ω161 −12 −3
5 2 + ω5 −15 5 + ω161 0 −21
9 3 −17 5 + 3ω161 −40 −4
11 3 + 2ω5 −6 2 + 8ω161 −68 37
11 3 + ω5 33 7− 7ω161 30 −66
19 4 + 3ω5 −139 −15− 9ω161 −28 −40
19 4 + ω5 19 −19 84 −9
N (31, 5 + 2ω5) (41, 6 + ω5)
N(p) p a(p, f1) a(p, f1) a(p, f2)
4 2 −7 7 −4− 2ω21
5 2 + ω5 −10 10 −9 + 4ω21
9 3 −14 34 −18− 2ω21
11 3 + 2ω5 −20 −60 −19
11 3 + ω5 −28 −2 −24− 4ω21
19 4 + 3ω5 −12 74 4− 50ω21
19 4 + ω5 28 16 −29 + 44ω21
N (49, 7)
N(p) p a(p, f1) a(p, f2)
4 2 −15 −2
5 2 + ω5 16 −10
9 3 −50 −11
11 3 + 2ω5 −8 −7− 28ω13
11 3 + ω5 −8 −35 + 28ω13
19 4 + 3ω5 −110 −26 + 14ω13
19 4 + ω5 −110 −12− 14ω13
Table 2. Hilbert eigenforms of weight 4
Remark 2. So far, our algorithm has been implemented only for congruence
subgroups of Siegel type. We intend to improve the implementation in the
near future so as to include more additional level structures such as the
Klingen type. Indeed, Ramakrishnan and Shahidi [RS 2007] recently showed
the existence of symmetric cube lifts for non-CM elliptic curves E/Q to
GSp4/Q. And their result should hold for other totally real number fields,
with the level structures of the lifts being of Klingen type. Unfortunately,
14 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
those lifts cannot be seen in our current tables. For example, there are
modular elliptic curves over Q(
5) whose conductors have norm 31, 41 and
49, but the corresponding symmetric cubic lifts do not appear in Table 1.
We would like to remedy that in our next implementation.
References
[D1 2005] L. Dembélé, Explicit computations of Hilbert modular forms on Q(
5). Exper-
iment. Math. 14 (2005), no. 4, 457–466.
[D2 2007] L. Dembélé, Quaternionic M -symbols, Brandt matrices and Hilbert modular
forms. Math. Comp. 76, no 258, (2007), 1039-1057. Also available electronically.
[D3 2007] L. Dembélé, On the computation of algebraic modular forms (submitted).
[AS 2001] Mahdi Asgari and Ralf Schmidt, Siegel modular forms and representations,
Manuscripta Math. 104 (2001), 173–200.
[FvdG1 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes
locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes.
I, C. R. Math. Acad. Sci. Paris 338 (2004), no. 5, 381–384.
[FvdG2 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes
locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes.
II, C. R. Math. Acad. Sci. Paris 338 (2004), no. 6, 467–470.
[JL 1970] Hervé Jacquet and Robert Langlands, Automorphic forms on GL(2), Lecture
notes in mathematics 114 and 278, 1970.
[Gr 1999] Benedict H. Gross, Algebraic modular forms. Israel J. Math. 113 (1999), 61–93.
[Gu 2000] P. Gunnells, Symplectic modular symbols, Duke Math. J. 102 (2000), no. 2,
329-350.
[HI 1980] K. Hashimoto and T. Ibukiyama, On the class numbers of positive definite
binary quaternion hermitian forms. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980),
549-601.
[Ib 1984] T. Ibukiyama, On symplectic Euler factors of genus 2. J. Fac. Sci. Univ. Tokyo
30 (1984), 587614.
[Ih 1964] Y. Ihara, On certain Dirichlet series, J. Math. Soc. Japan 16 (1964), 214-225.
[LP 2002] J. Lansky and D. Pollack, Hecke algebras and automorphic forms. Compositio
Math. 130 (2002), no. 1, 21–48.
[RS 2007] Dinakar Ramakrishnan and Freydoon Shahidi, Siegel modular forms of genus
2 attached to elliptic curves (preprint). Available at www.math.arxiv.
[R 2006] N. C. Ryan, Computing the Satake p-parameters of Siegel modular forms. (sub-
mitted).
[Sk 1992] Nils-Peter Skoruppa, Computations of Siegel modular forms of genus two. Math.
Comp. 58 (1992), no. 197, 381–398.
[So 2008] Claus M. Sorensen, Potential level-lowering for GSp(4), arXive:0804.0588v1.
Department of Mathematics, University of Calgary
E-mail address: cunning@math.ucalgary.ca
Institut für Experimentelle Mathematik, Universität Duisburg-Essen
E-mail address: lassina.dembele@uni-duisburg-essen.de
	Introduction
	1. Hilbert-Siegel modular forms and the Jacquet-Langlands correspondence
	1.1. Hilbert-Siegel modular forms
	1.2. The Hecke algebra
	1.3. Algebraic Hilbert-Siegel autormorphic forms
	1.4. The Jacquet-Langlands Correspondence
	2. The Algorithm
	2.1. The quotient H02(N)
	2.2. Brandt matrices
	2.3. Computing the group GB(Z)
	2.4. Computing the sets 1(p) and 2(p)
	2.5. The implementation of the algorithm
	3. Numerical examples: F=Q(5) and B=(-1,-1F)
	3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3
	3.2. Tables of Hilbert cusp forms of parellel weight 4
	3.3. Lifts
	References
