Hyper-Algebras of Vector-Valued Modular Forms
We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ℚ, acting on these hyper-algebras. These definitions bridge the classical and represent...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
2018
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| Цитувати: | Hyper-Algebras of Vector-Valued Modular Forms / M. Raum // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859593822365810688 |
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| author_facet | Raum, M. |
| citation_txt | Hyper-Algebras of Vector-Valued Modular Forms / M. Raum // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ℚ, acting on these hyper-algebras. These definitions bridge the classical and representation-theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 108, 17 pages
Hyper-Algebras of Vector-Valued Modular Forms
Martin RAUM
Chalmers tekniska högskola och Göteborgs Universitet,
Institutionen för Matematiska vetenskaper, SE-412 96 Göteborg, Sweden
E-mail: martin@raum-brothers.eu
URL: http://raum-brothers.eu/martin/
Received May 07, 2018, in final form September 30, 2018; Published online October 04, 2018
https://doi.org/10.3842/SIGMA.2018.108
Abstract. We define graded hyper-algebras of vector-valued Siegel modular forms, which
allow us to study tensor products of the latter. We also define vector-valued Hecke operators
for Siegel modular forms at all places of Q, acting on these hyper-algebras. These definitions
bridge the classical and representation theoretic approach to Siegel modular forms. Com-
bining both the product structure and the action of Hecke operators, we prove in the case of
elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from
products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of
cuspidal automorphic representations into the tensor product of global principal series.
Key words: Siegel modular forms; vector-valued Hecke operators; automorphic representa-
tions
2010 Mathematics Subject Classification: 11F40; 11F60; 11F70
1 Introduction
Products of scalar-valued modular forms can be used to construct new ones, and in special
situations they reveal deep arithmetic information. For example, Rankin showed in [14] that
the Petersson scalar product of an elliptic cusp form f and the product EkEl of two Eisenstein
series, can be expressed in terms of special L-values attached to f . This very relation reappeared
in [8], where Kohnen and Zagier define a Q-structure on elliptic modular forms that is different
from the one originating in Fourier expansions. The product of scalar-valued modular forms
naturally leads to the definition of the graded algebra M(•) of modular forms.
Vector-valued modular forms in the elliptic case can be associated with any representation ρ
of the modular group SL2(Z). As opposed to scalar-valued ones, their product structure has
mostly been neglected. Instead, one considers graded modules M(•⊗ρ) over M(•). For example,
Marks and Mason proved in [11] that M(•⊗ρ) is free over M(•) and that its rank relates directly
to the eigenvalues of ρ
(
−I(2)) for I(2) the 2× 2 identity matrix.
It is possible to define tensor products of vector-valued modular forms by means of the tensor
product of smooth functions from the Poincaré upper half plane H = {τ ∈ C : Im τ > 0} to the
representation spaces V (ρ) and V (ρ′) of ρ and ρ′
⊗ : C∞
(
H→ V (ρ)
)
× C∞
(
H→ V (ρ′)
)
−→ C∞
(
H→ V (ρ⊗ ρ′)
)
,
(f ⊗ g)(τ) 7−→ f(τ)⊗ g(τ).
Tensor products of vector-valued modular forms have seldom been studied. In [23], we employed
them to express elliptic cusp forms of any level as tensor products of at most two Eisenstein
series. Tensor products can be conveniently subsumed under the concept of hyper-algebras.
This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko
Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.html
mailto:martin@raum-brothers.eu
http://raum-brothers.eu/martin/
https://doi.org/10.3842/SIGMA.2018.108
http://www.emis.de/journals/SIGMA/modular-forms.html
2 M. Raum
Our description of tensor products of and differential operators on almost holomorphic Siegel
modular forms that we gave in [6], for example, employed hyper-algebras to classify almost
holomorphic Siegel modular forms. In this work, we review the concept of hyper-algebras in the
context of modular forms.
Hyper-algebras mimic algebras but they have multivalued multiplication. In particular, they
are natural analogues of hyper-groups [21]. Note that we suppress Wall’s assumption of dimen-
sionality of hyper-groups, which does not hold for examples that we treat. Hyper-groups arise
naturally in the context of double cosets in group theory. The definition of a group G includes
the product a · b ∈ G of a, b ∈ G. Given a sufficiently nice subgroup H – that is, if (G,H)
is a Gelfand pair – it is standard in modular forms to define a (commutative) Hecke algebra.
Specifically, Hecke algebras assign meaning to the product of double cosets HaH · HbH. For
instance, if G = GSpg(Q) and K = GSpg(Z), then the Hecke algebra with integral coefficients is
free as a Z-module and has basis HcH for certain diagonal c ∈ G. Every product of double cosets
can be written as a sum
∑
c∈GmcHcH, where 0 ≤ mc ∈ Z and mc 6= 0 for only finitely many c.
The corresponding notion of hyper-groups, to the author’s knowledge, has not yet appeared in
the context of modular forms, but is standard in other areas of mathematics. Rephrasing the
product in Hecke algebras, we arrive at a notion of multivalued multiplication of double cosets.
One can decompose HaH =
⋃
c∈L(H,a)Hc and HbH =
⋃
c∈L(H,b)Hc as a disjoint union of left
cosets with coset representatives L(H, a),L(H, b) ⊆ G. By definition of the multiplication in
Hecke algebras, the multiset {{H(da · db) : da ∈ L(H, a), db ∈ L(H, b)}} has a decomposition into
the disjoint union of {{Hdc : dc ∈ L(H, c)}} where each L(H, c) occurs with multiplicity mc that
occurred before. A more detailed, explicit explanation can be found in [9]. Summarizing, we
obtain a multiplication of double cosets that takes values in multisets of double cosets:
H\G/H ×H\G/H −→ Multiset(H\G/H).
In this way, we obtain a commutative hyper-group. Note that every group can be naturally
viewed as a hyper-group. Vice versa, a hyper-group whose multiplication takes values in multi-
sets of size 1 is a group.
It is possible to extend the definition of hyper-groups to hyper-algebras. Addition is axioma-
tized as in the case of algebras. Multiplication takes values in the set of all subspaces and
compatibility relations that mimic those for algebras are imposed on it. We give a precise
definition of hyper-algebras in Section 3. Given an algebra A we obtain a hyper-algebra by
assigning to two elements the module spanned by their product. In general, it is not possible to
recover the algebra from this, because the hyper-algebra product a · b associated with a, b ∈ A
and (ra) · (r′b) is the same for all units r, r′ in the base ring R.
Siegel modular forms are assigned to a weight σ and a type ρ, which are representations of
GLg(C) and Spg(Z), respectively. Given Siegel modular forms f and g of weights and types σf ,
ρf , σg, and ρg, then their tensor product f ⊗ g has weight σf ⊗ σg and type ρf ⊗ ρg. Even if
the weights and types of f and g are irreducible, their tensor products can be reducible. On the
other hand, to avoid redundancies, graded modules of Siegel modular forms are preferably build
from irreducible weights and types. As a consequence, tensor products do not yield an algebra
structure on such modules.
Since weights and types behave similarly with respect to the construction that we discuss
now, we focus on the former. To remedy the lack of algebra structures on the graded module of
Siegel modular forms, we have previously suggested to attach to f and g the span
f ⊗ g = span
{
φ ◦ (f ⊗ g) : φ ∈ Hom(σf ⊗ σg, σ)
}
⊂
⊕
σ
M(g)(σ),
where σ runs through a fixed set of representatives of isomorphism classes of irreducible weights,
and M(g)(σ) denotes the corresponding space of Siegel modular forms. In [6], we identified the
Hyper-Algebras of Vector-Valued Modular Forms 3
resulting structure as a hyper-algebra. It is a special case of hyper-algebras of Siegel modular
forms, defined in Section 3.1.
In [6], we also defined the concept of differential hyper-algebras to subsume the action of
covariant differential operators. In Section 3.4, we revisit this definition and reinterpret it in
terms of a Hecke operator T∞ at the infinite place of Q. Returning to the case of general weights
and types, we also study vector-valued Hecke operators Tv at all places v of Q. If v = p is a prime,
then Tp generalizes classical Hecke operators for Siegel modular forms. It extends the vector-
valued Hecke operators for elliptic modular forms that we have already defined in [23]. The
polynomial algebra T generated by formal elements Tv acts by means of these Hecke operators
on the hyper-algebra of Siegel modular forms. Specifically, at the infinite place, the Hecke
operator acts as a hyper-derivation. At the finite places, Hecke operators respect the hyper-
product structure. A precise description can be found in Section 3.6.
Section 4 contains a more detailed study of tensor products and the action of T on modular
forms is studied in more detail in the genus 1 case. Let S(1)(k) be the space of genus 1 cusp
forms of weight k and level 1. Write E(1)(l) for the level 1 Eisenstein series of weight l. The
following is a slight modification of the main result in [23].
Theorem 1.1. Let l, l′ ≥ 4 be even integers and ρ a finite-dimensional representation of SL2(Z)
whose kernel is a congruence subgroup. Then for every k ≥ l + l′, we have
S(1)(k ⊗ ρ) ⊂ T
(
TE(1)(l)⊗TE(1)(l′)
)
. (1.1)
The right-hand side of (1.1) can be efficiently computed and its left-hand side is genuinely
interesting. We argue that the proof can serve as a blueprint to a whole family of analogous
statements. Thus Theorem 1.1 sets the path towards a general method of computing Siegel
modular forms. Section 4 contains a specific conjecture in the case of genus 2.
Next, we discuss a connection between Theorem 1.1 and automorphic representations for
PGL2. To this end, we restrict to representations ρ whose kernel is a congruence subgroup.
Vector-valued Hecke operators defined in this paper can be related to adelic automorphic rep-
resentations. Covariant differential operators correspond to the action of an appropriate Lie
algebra on Harish-Chandra modules. Hecke operators at finite places arise directly from a rep-
resentation of a group over Qp. It is natural to ask how much Theorem 1.1 relates to results
from automorphic representation theory. We make such a relation precise, and determine some
constituents in the tensor product of principal series. Specifically, we consider automorphic rep-
resentations for PGL2. We let $(k− 1,1) be the principal series that is unramified at the finite
places and has Harish-Chandra parameter k− 1 at infinity. Let $∞(k− 1) be the discrete series
with Harish-Chandra parameter k − 1.
Theorem 1.2. Given even integers l, l′ ≥ 4, then
$ ↪−→ $(l − 1,1)⊗$(l′ − 1,1)
for every cuspidal automorphic representation $ = $∞(k − 1) ⊗ $f with k ≥ l + l′ and $f
a representation of PSL2(Af).
When phrased in this language, similarity to the Gross–Prasad conjectures for the inclusion
O1,2
∼= SL2 ↪→ SL2 × SL2
∼= O2,2 becomes apparent.
In Section 2, we collect preliminaries on Siegel modular forms. In Section 3, we define hyper-
algebras of Siegel modular forms. In Section 4, we show that products of certain Eisenstein
series yield cusp forms. In Section 5, we give an interpretation of our results in terms of adelic
automorphic representations.
4 M. Raum
2 Modular forms
2.1 The classical setup
The Siegel upper half space
{
τ ∈ Matg(C) : tτ = τ, Im τ > 0
}
of genus g is denoted by H(g). It
carries an action of the real symplectic group
Spg(R) =
{
γ ∈ Mat2g(R) : tγJ (g)γ = J (g)
}
, J (g) =
(
0 −I(g)
I(g) 0
)
,
where I(g) is the g×g identity matrix. This action is explicitly given by γτ = (aτ+b)(cτ+d)−1,
where we employ the block decomposition of γ =
(
a b
c d
)
. The subgroup Γ(g) = Spg(Z) of
symplectic transformation matrices with integral entries is called the Siegel modular group of
genus g.
Generally, a complex representation σ of GLg(C) is called a weight. For the purpose of this
paper, we restrict to representations that factor through GLg(C)/
{
±I(g)
}
. A finite-dimensional
representation ρ of Spg(Z) is called a type. Throughout this note, we focus on types whose
kernel is a congruence subgroup that contains −I(g). The representation spaces of σ and ρ are
denoted by V (σ) and V (ρ). A weight and a type together determine a slash action(
f
∣∣
σ,ρ
γ
)
(τ) = σ(cτ + d)−1ρ(γ)−1f(γτ)
on functions f : H(g) → V (σ)⊗ V (ρ).
The space of genus g Siegel modular forms of weight σ and type ρ is defined as the space of
holomorphic functions f : H(g) → V (σ) ⊗ V (ρ) such that f |σ,ργ = f for all γ ∈ Γ(g) and which
satisfy f(x+ iy) = O(1) as y →∞, if g = 1. We write
M(g)(σ ⊗ ρ) = E(g)(σ ⊗ ρ)⊕ S(g)(σ ⊗ ρ)
for this space, the space of Eisenstein series, and the space of cusp forms, respectively.
2.2 The symplectic group and its Lie algebra
We write G = PGSpg for the Q-split algebraic group of projective symplectic similitudes.
Throughout we work with the model
PGSpg(Z) =
{
γ ∈ Mat2g(Z) : tγJ (g)γ = s(γ)J (g), s(γ) ∈ Z
}/{
sI(g) : s ∈ Z
}
.
Compact subgroups K
(g)
∞ and K
(g)
p of G(R) and G(Qp) are Ug(R)/
{
±I(g)
}
and PGSpg(Zp),
where the former is embedded into G(R) by ai + b 7→
(
a −b
b a
)
. Weights correspond to complex
representations of K
(g)
∞ by means of the restriction along Ug(R)/
{
±I(g)
}
→ GLg(C)/
{
±I(g)
}
.
We let Rep
(
K
(g)
∞
)
be a fixed set of representatives of isomorphism classes of finite-dimensional,
complex representations of K
(g)
∞ . For all g, the representation detk : Ug(R) → C, k 7→ det(k)
with k ∈ Z, 2 | gk yields an irreducible representation of K
(g)
∞ via the above isomorphism of K
(g)
∞
and Ug(R)/
{
±I(g)
}
. In the context of spaces of modular forms, we will occasionally write k
instead of detk to match more closely the classical notation. For example,
M(g)(k ⊗ ρ) = M(g)
(
detk ⊗ ρ
)
.
We denote the l-th symmetric power of the standard representation by syml. The dual of
a representation is indicated by a superscript ∨: σ∨ is the dual of σ.
We let Rep
(
K
(g)
p
)
be a fixed set of representatives of isomorphism classes of finite-dimensional,
complex representations of K
(g)
p . To simplify notation, we let Rep
(
K
(g)
f
)
be the set of restricted
Hyper-Algebras of Vector-Valued Modular Forms 5
tensor products of the representations in Rep
(
K
(g)
p
)
. That is, it consists of exterior tensor
products of representations of K
(g)
p which are trivial for all but finitely many p. There is
a correspondence between Rep
(
K
(g)
f
)
and finite-dimensional, complex representations of Γ(g)
with a congruence subgroup and the matrix −I2g in their kernel.
Subsets of irreducible representations are denoted by
IrrRep
(
K(g)
∞
)
⊂ Rep
(
K(g)
∞
)
and IrrRep
(
K
(g)
f
)
⊂ Rep
(
K
(g)
f
)
.
2.3 Covariant differential operators
We call a differential operator D covariant from
∣∣
σ,ρ
to
∣∣
σ′,ρ
if for all g ∈ G(R) and all smooth
functions f : H(g) → V (σ)⊗ V (ρ) , we have
D
(
f
∣∣
σ,ρ
g
)
= D(f)
∣∣
σ′,ρ
g.
If D is covariant for the trivial type then it yields a differential operator that is covariant for all
types.
A theorem of Helgason [3], allows us to classify order 1 differential operators. For each σ
there is a lowering operator L = Lσ that is covariant from σ to Lσ = sym2∨σ and a raising
operator R = Rσ that is covariant from σ to Rσ = sym2σ. We only fix a normalization in the
case g = 1, setting
L = −2iy2∂τ and R = 2i∂τ + ky−1.
In [6], we gave explicit expressions for all g. The normalization employed there, however, does
not coincide with the one we choose here.
Covariant differential operators allow us to define almost holomorphic Siegel modular forms.
A function f : H(g) → V (σ) ⊗ V (ρ) that vanishes under the (d + 1)-th tensor power of the
lowering operator, i.e., Ldf = 0, that is invariant of weight σ and type ρ, i.e., f
∣∣
σ,ρ
γ = f for all
γ ∈ Γ(g), and that satisfies f(τ) = O(1) as y → ∞, if g = 1, is called an almost holomorphic
Siegel modular form of weight σ, type ρ, and depth d. The space of such functions is denoted by
[d]
M
(g)(σ ⊗ ρ).
2.4 Hecke operators
We extend the definition of vector-valued Hecke operators in [23] for elliptic modular forms to
Siegel modular forms. Many of the proofs in op. cit. apply to the general case word by word.
For this reason, we skip several arguments in this subsection.
Hecke operators on representations
For a positive integer M , we let
∆M =
{(
a b
0 d
)
∈ GSpg(Q) : tad = MI(g), a, b, d ∈ Matg(Z), d upper triangular,
∀ i < j : 0 ≤ di,j < dj,j , ∀ i : , 0 ≤ bi,j < dj,j
}
,
where GSpg is the group of symplectic similitude transformations. We have a right action
of Spg(Z) on ∆M defined by (m, γ) 7→ mγ with γ′mγ = mγ for some γ′ ∈ Spg(Z). This, in
particular, defines a cocycle Im(γ) = γ′; That is, we have Im(γ1γ2) = Im(γ1)Imγ1(γ2).
6 M. Raum
We denote the natural basis of C[∆M ] by em, m ∈ ∆M . To every type ρ we associated the
type TMρ defined by
V (TMρ) := V (ρ)⊗ C[∆M ] and (TMρ)(γ)(v ⊗ em) := ρ
(
I−1
m
(
γ−1
))
(v)⊗ emγ−1 .
The cocycle property of Im(γ) implies that it is a representation of Spg(Z). Given a scalar
product 〈 · , · 〉ρ on V (ρ), we obtain one on V (TMρ) by
〈v ⊗ emv , w ⊗ emw〉 =
{
〈v, w〉ρ, if mv = mw,
0, otherwise.
Hecke operators on representations are compatible with homomorphisms between types and
with tensor products. Specifically, the following are homomorphism of Spg(Z)-representations:
TMφ : TMρ −→ TMρ
′, v ⊗ em 7−→ φ(v)⊗ em,
1 ↪−→ TM1, c 7−→ c
∑
m∈∆M
em,
(TMρ)⊗ (TMσ) −� TM (ρ⊗ σ),
(v ⊗ em)⊗ (w ⊗ em′) 7−→
{
(v ⊗ w)⊗ em, if m = m′,
0, otherwise,
(2.1)
where φ ∈ Hom(ρ, ρ′). If ψ : ρ → ρ′′ is a further homomorphism, then TM (ψ ◦ φ) = (TMψ) ◦
(TMφ). For later reference, we denote the morphism in (2.1) by πM,ρ,ρ′ .
Hecke operators on modular forms
For m =
(
a b
0 d
)
∈ GSpg(R) of similitude M , and for f : H→ V (σ), we define(
f
∣∣
σ
m
)
(τ) = σ
(
d/
√
M
)−1
f
(
(aτ + b)d−1
)
.
Fixing a type ρ and a positive integer M , we define the vector-valued Hecke operator TM
acting on f ∈ M(g)
(
σ ⊗ ρ
)
by(
TMf
)
(τ) =
∑
m∈∆M
(
f
∣∣
σ
m
)
(τ)⊗ em ∈ M(g)
(
σ ⊗ TMρ
)
.
The above homomorphism πM,ρ,ρ′ is compatible with Hecke operators. Specifically, we have
πM,ρ,ρ′
(
(TMf)⊗ (TMg)
)
= TM
(
f ⊗ g
)
.
3 Hyper-algebras
We start with the formal definition of hyper-groups, which is the blueprint to our Definition 3.2
of hyper-algebras. Given a set S, let Multiset(S) ∼= {f : S → Z≥0} be the set of all multisubsets
of S. Multisets are throughout denoted by double curly brackets {{· · · }}. The union of multisets
corresponds to the sum of functions S → Z≥0. As a shorthand notation, for the binary operator
appearing in the next definition, we set a · {{bi}}i =
⋃
bi
a · bi and similarly {{ai}}i · b =
⋃
ai
ai · b.
Definition 3.1. A pair (G, · ) of a set G and a binary operator · : G × G → Multiset(G) is
called a hyper-group if the following axioms are satisfied:
(i) (Finite image) For any a, b ∈ G, the multiset a · b is finite.
Hyper-Algebras of Vector-Valued Modular Forms 7
(ii) (Associativity) For any a, b, c ∈ G, we have (a · b) · c = a · (b · c), that is,⋃
h∈a · b
h · c =
⋃
h∈b · c
a ·h.
(iii) (Identity) There is an element e ∈ G such that for every a ∈ G we have a ∈ a · e and
a ∈ e · a.
(iv) (Inverse) For every element a ∈ G there is element a−1 ∈ G such that e ∈ a · a−1 and
e ∈ a−1 · a.
Extending this notion to hyper-algebras is straightforward. However, the reader should be
warned that the word “hyper-algebra” is used in the context of algebraic groups [18], too, and
these two notions should not be confused.
Given a commutative ring R (with identity), and an R-module M , we let SubModR(M) be the
set of all R-submodules of M . In analogy with the definition of multisets, we extend the binary
operator · in the following definition to submodules by a ·N =
⋃
b∈N a · b and N · b =
⋃
a∈N a · b
for N ∈ SubModR(M).
Definition 3.2. Let R be a ring. A triple (A,+, · ) with (A,+) an R-module and binary
operator
· : A×A −→ SubModR(A)
is called a hyper-algebra (with identity) if
(i) (Linearity) Given a, b ∈ A and r ∈ R, we have (ra) · b = r
(
a · b
)
= a · (rb).
(ii) (Identity) There is e ∈ A such that we have e · a = a · e = spanR a for all a ∈ A.
(iii) (Associativity) Given a, b, c ∈ A, we have a · (b · c) = (a · b) · c.
(iv) (Distributivity) for a, b, c ∈ A, we have a · (b+ c) ⊆ a · b+ a · c.
Concepts like commutativity, grading, and derivation extend to hyper-algebras. Fixing
a hyper-algebra A, we call it commutative if for all a, b ∈ A, we have ab = ba. We say that it
is graded by a hyper-group G if A =
⊕
g Ag as an R-module and AgAg′ ⊆
⋃
h∈gg′ Ah. A hyper-
derivation d on A is an R-module endomorphism such that d(a · b) ⊆ (da) · b+ a · (db).
Note that for a hyper-algebra A, we can define a (left) hyper-module M as an R-module
with binary operator · : A × M → SubModR(M) satisfying the analogue of the axioms in
Definition 3.2.
3.1 Hyper-algebras of modular forms
Recall the various sets of representations defined in Section 2.2. The hyper-algebra of holomor-
phic Siegel modular forms as a vector space is
M(g)(• ⊗ •) =
⊕
σ,ρ
M(g)(σ ⊗ ρ), σ ∈ IrrRep
(
K(g)
∞
)
, ρ ∈ IrrRep
(
K
(g)
f
)
with product fg = f ⊗ g ⊂ M(g)(• ⊗ •) defined as
span
{
(φσ ⊗ φρ) ◦ (f ⊗ g) : σ ∈ IrrRep
(
K(g)
∞
)
, φσ : σf ⊗ σg → σ,
ρ ∈ IrrRep
(
K
(g)
f
)
, φρ : ρf ⊗ ρg → ρ
}
. (3.1)
8 M. Raum
Note that both IrrRep
(
K
(g)
∞
)
and IrrRep
(
K
(g)
f
)
are hyper-groups, and the hyper-algebra of Siegel
modular forms is graded by both. Our notation for submodules, for example M(g)(• ⊗ ρ), has
the obvious meaning.
We will also work with the hyper-algebras of almost holomorphic Siegel modular forms
[•]
M
(g)(• ⊗ •) =
⋃
0≤d
⊕
σ,ρ
[d]
M
(g)(σ ⊗ ρ), σ ∈ IrrRep
(
K(g)
∞
)
, ρ ∈ IrrRep
(
K
(g)
f
)
with product as in (3.1).
3.2 Computing the hyper-algebra product in Sage
In the case of elliptic modular forms, all irreducible weights are 1-dimensional. Types can,
however, be arbitrary dimensional. We adopt an example from [23], which illustrates how to
compute products in M(g)(• ⊗ •) of modular forms with nontrivial type. For convenience, we
give Sage code [16]1, which the reader can modify to perform his or her own computation – the
code is not optimized for performance, but for clarity.
We consider the representation ρ3, which we realize as a matrix representation by
ρ3(T ) =
1 0 0
0 0 1
−1 −1 −1
, ρ3(S) =
0 1 0
1 0 0
−1 −1 −1
.
Its kernel contains the congruence subgroup Γ(3). The space of Eisenstein series of weight 12
and type ρ3 has dimension one. A basis element E12,ρ3 can be obtained from vector-valued
Hecke operators that are discussed in Section 2.4. We let ζ be a third root of unity. The image
of T3 on the level 1 Eisenstein series of weight 12 is
K.< zeta> = Cyc l o t om i cF i e l d (3 )
R.<q3> = K [ [ ] ]
E12 = E i s e n s t e i nFo rms ( 1 , 1 2 ) . b a s i s ( ) [ 0 ]
E12T3 = lambda n : v e c t o r ( [
3∗∗ 6 ∗ E12 . qexp ( n//3+1). subs ( q=q3 ∗∗9 ) . add b igoh (3∗n )
, 3∗∗−6 ∗ E12 . qexp (3∗n ) . subs ( q=q3 )
, 3∗∗−6 ∗ E12 . qexp (3∗n ) . subs ( q=ze t a ∗q3 )
, 3∗∗−6 ∗ E12 . qexp (3∗n ) . subs ( q=ze t a ∗∗2∗q3 )
] )
Fourier expansions are computed in terms of q3 = e(τ/3). The components of an Eisenstein
series of type ρ3 can be found by applying a homomorphism from T31 to ρ3, which can be
computed by the method that we describe below when decomposing ρ3ρ3,
E12rho3 = lambda n : mat r i x (3 , [ 1 ,−1/3 ,−1/3 ,−1/3
, −1/3 ,1 ,−1/3 ,−1/3
, −1/3 ,−1/3 ,1 ,−1/3
] ) ∗ E12T3 (n )
To compute E12,ρ3 ⊗E12,ρ3 , we have to decompose the 9-dimensional representation ρ3 ⊗ ρ3.
Since the kernel of ρ3 has finite index in SL2(Z), character theory for finite groups is one way
to achieve this. We use a more general method that applies to all representation: By exhibiting
the trivial representation in (ρ3 ⊗ ρ3)∨ ⊗ ρ for various representations ρ, we compute the space
of homomorphisms Hom(ρ3⊗ρ3, ρ). For a systematic decomposition of ρ3⊗ρ3, one could apply
the MeatAxe algorithm [4] in conjunction with a multimodular approach.
1Computations make implicit use of the libraries [2, 12] and possibly further ones that are less obvious from
the source code of Sage.
Hyper-Algebras of Vector-Valued Modular Forms 9
We start by defining the representation matrices of the trivial representation 1, ρ3, and
ρ3 ⊗ ρ3,
s t r i v = t t r i v = i d e n t i t y m a t r i x (QQ, 1)
s3 = mat r i x (QQ, 3 , [ 0 , 1 , 0 , 1 ,0 ,0 , −1 ,−1 ,−1])
t3 = mat r i x (QQ, 3 , [ 1 , 0 , 0 , 0 ,0 ,1 , −1 ,−1 ,−1])
s33 = s3 . t e n s o r p r o d u c t ( s3 )
t33 = t3 . t e n s o r p r o d u c t ( t3 )
We compute homomorphisms between representations by employing the isomorphism of vector
spaces Hom(ρ, ρ′) ∼=
(
ρ∨ ⊗ ρ′
)
(1). Representation matrices of the dual representation ρ∨ are
given by the transpose inverses of those of ρ. We can immediately compute homomorphisms
from ρ3ρ3 to 1 and ρ3,
dua l = lambda m: m. t r a n s p o s e ( ) . i n v e r s e ( )
hom = lambda s1 , t1 , s2 , t2 : \
( dua l ( s1 ) . t e n s o r p r o d u c t ( s2 )−1). r i g h t k e r n e l ( ) \
. i n t e r s e c t i o n ( ( dua l ( t1 ) . t e n s o r p r o d u c t ( t2 )−1). r i g h t k e r n e l ( ) )
hom t r i v = hom( s33 , t33 , s t r i v , t t r i v )
hom rho3 = hom( s33 , t33 , s3 , t3 )
There is one copy of 1 and two copies of ρ3 in ρ3ρ3. We are facing the problem of decomposing
their complement. In our case, it turns out that it consists of two inequivalent one-dimensional
representations with representation matrices
ρζ(S) =
(
1
)
, ρζ(T ) =
(
ζ
)
and ρζ2(S) =
(
1
)
, ρζ2(T ) =
(
ζ2
)
.
In Sage, we implement them by means of
K.< zeta> = Cyc l o t om i cF i e l d (3 )
s z e t a = i d e n t i t y m a t r i x (K, 1 )
t z e t a = mat r i x (K, 1 , [ z e t a ] )
s z e t a 2 = i d e n t i t y m a t r i x (K, 1 )
t z e t a 2 = mat r i x (K, 1 , [ z e t a ∗∗2 ] )
First, we determine the kernel of the homomorphism that we have determined so far
bm = reduce ( lambda s , l : s . i n t e r s e c t i o n ( mat r i x ( [ l ] ) . r i g h t k e r n e l ( ) )
, [ hom t r i v . b a s i s ( ) [ 0 ] ]
+ [ b [ i x : : 3 ] f o r b i n hom rho3 . b a s i s ( ) f o r i x i n range ( 3 ) ]
, VectorSpace (QQ, 9 ) ) \
. b a s i s m a t r i x ( ) . t r a n s p o s e ( )
s r e s t = bm. s o l v e r i g h t ( s33 ∗bm)
t r e s t = bm. s o l v e r i g h t ( t33 ∗bm)
Second, we observe that S acts trivially on that kernel, so that it suffices to decompose the
action of T into eigenspaces. In the present case they are defined over a third order cyclotomic
extension of the rationals. In fact, they are isomorphic to ρζ and ρζ2 given above
hom zeta = hom( s33 , t33 , s z e ta , t z e t a )
hom zeta2 = hom( s33 , t33 , s ze ta2 , t z e t a 2 )
We construct matrices from the homomorphism spaces that we previously determined
p h i z e t a = mat r i x ( hom zeta . b a s i s ( ) [ 0 ] )
p h i z e t a 2 = mat r i x ( hom zeta2 . b a s i s ( ) [ 0 ] )
p h i r h o 3 1 = mat r i x ( [ hom rho3 . b a s i s ( ) [ 0 ] [ i x : : 3 ] f o r i x i n range ( 3 ) ] )
p h i r h o 3 2 = mat r i x ( [ hom rho3 . b a s i s ( ) [ 1 ] [ i x : : 3 ] f o r i x i n range ( 3 ) ] )
10 M. Raum
Assembling the results that we have computed via Sage, we find that with respect to the given
bases, we have homomorphisms
φ1 : ρ3 ⊗ ρ3 −→ 1,
(
1 1
2
1
2
1
2 1 1
2
1
2
1
2 1
)
,
φζ : ρ3 ⊗ ρ3 −→ ρζ ,
(
1 ζ + 1 −ζ ζ + 1 ζ −1 −ζ −1 −ζ − 1
)
,
φζ2 : ρ3 ⊗ ρ3 −→ ρζ2 ,
(
1 −ζ ζ + 1 −ζ −ζ − 1 −1 ζ + 1 −1 ζ
)
,
φρ3,1 : ρ3 ⊗ ρ3 −→ ρ3,
1 0 −1 −1 −1 −2 0 1 −1
−1 0 1 −1 1 0 −2 −1 −1
−1 −2 −1 1 −1 0 0 −1 1
,
φρ3,2 : ρ3 ⊗ ρ3 −→ ρ3,
0 1 −1 −1 0 −3 1 3 0
0 1 3 −1 0 1 −3 −1 0
0 −3 −1 3 0 1 1 −1 0
.
Summarizing, we find that E12,ρ3 ⊗E12,ρ3 is supported on 1, ρζ , ρζ2 , and ρ3. The correspon-
ding subspaces are spanned by elements whose Fourier coefficients are too large to display them
all. We confine ourselves to the trivial type:
564856947200
1594323
− 1894333004462080000
84584326707
q − 1261863434802833408000
28194775569
q2 +O
(
q3
)
∈ E12,ρ3 ⊗E12,ρ3 ∩M
(
24⊗ 1
)
.
Beyond this, we illustrate how to compute the remaining Fourier expansions with Sage. The
tensor square of E12,ρ3 with precision at least n is given by the following function Esq(n)
Esq = lambda n : v e c t o r (R , [ c1∗ c2 f o r c1 i n E12rho3 ( n )
f o r c2 i n E12rho3 ( n ) ] )
p r i n t p h i t r i v ∗ Esq (3 )
p r i n t p h i z e t a ∗ Esq (3 )
p r i n t ph i z e t a 2 ∗ Esq (3 )
p r i n t ( p h i r h o 3 1 ∗ Esq (3 ) , p h i r h o 3 2 ∗ Esq ( 3 ) )
3.3 From weights to isomorphism classes of irreducible weights
We have defined weights as complex, finite-dimensional representations of K
(g)
∞ without any
further restriction. If we wished to construct a graded algebra of modular forms whose grading
includes all these weights, this would not be possible. All representations of K
(g)
∞ together do
not constitute a set, but rather they are objects in a category Rep
(
K
(g)
∞
)
. Given this fact,
one might be inclined to pass from weights to isomorphism classes of weights – the skeleton of
Rep
(
K
(g)
∞
)
. Alternatively, we can and will focus on a fixed set Rep
(
K
(g)
∞
)
of representatives of
isomorphism class. For every σ ∈ Rep
(
K
(g)
∞
)
the space of modular forms M(g)(σ) of weight σ is
a vector space.
Let us consider to what extent we can define a multiplication. Given f ∈ M(g)(σf ) and
g ∈ M(g)(σg) with σf , σg ∈ Rep
(
K
(g)
∞
)
, we find that
fg = f ⊗ g ∈ M(g)(σf ⊗ σg)
for their tensor product. Its weight, in general, is not in Rep
(
K
(g)
∞
)
. Denote by σfσg the
corresponding representative. A priori, fg lies in C∞(H(g) → V (σfσg)). We have to obtain from
it an element in C∞
(
H(g) → V (σfσg)
)
. The most naive way is to choose an endomorphism φ in
Hyper-Algebras of Vector-Valued Modular Forms 11
Hom(σfσg, σfσg) and compose it with fg. The resulting function φ ◦ fg depends on φ. More
precisely, it depends on f and g up to automorphisms of σfσg.
Except in the case g = 1, there seems to be no natural choice of endomorphisms for all tensor
products. For instance, symlsyml′ contains syml+l′ in a natural way by realizing syml and syml′
as a representation on polynomials of degree l and l′ in g variables. But, for example if g = 2,
there is no natural inclusion of syml+l′−2 into symlsyml′ for l, l′ ≥ 2. The situation becomes even
more difficult for g ≥ 3, because then irreducible representations can occur with multiplicities
greater than 1 in tensor products of irreducible representations.
Since there is no natural choice of a single homomorphism from σfσg to σfσg, we resort to
taking the span over all of them. The product of f and g in this setting is defined as the space
span
{
φ ◦ fg : φ ∈ Hom
(
σfσg, σfσg
)}
⊂
⊕
σ∈Rep
(
K
(g)
∞
)M(g)(σ).
This product yields a hyper-algebra of modular forms whose weights run through all isomorphism
classes of representations. Since at this point, we have already lost the algebra structure and
arrived at hyper-algebras, there is no additional harm in considering only irreducible weights.
This motivates our definition in Section 3.1.
3.4 Covariant differential operators on hyper-algebras
As in the case of tensor products the image of lowering and raising operators has, in general,
reducible weight. For example, in the case g = 2 and if l ≥ 2, the weight R detk syml =
detk sym2syml allows for a decomposition into irreducible weights detk−2 syml−2, detk−1 syml,
and detk syml+2. It is therefore natural to define a vector space valued action of covariant
differential operators.
Let C[L,R] be the polynomial algebra in two formal variables L and R. Notation overlaps
with the one for lowering and raising operators, but it will be clear from the context to what we
refer when writing Lf or Rf . The action of L and R on [•]
M
(g)(• ⊗ •) is given by
Lf = span
{
φ ◦ Lσf f : σ ∈ IrrRep
(
K(g)
∞
)
, φ : σ → Lσf
}
and
Rf = span
{
φ ◦ Rσf f : σ ∈ IrrRep
(
K(g)
∞
)
, φ : σ → Rσf
}
.
Viewing the differential operators L and R jointly as a vector-valued Hecke operator at the
infinite place acting on almost holomorphic Siegel modular forms, we set for f ∈ [•]
M
(g)(• ⊗ •)
T∞ = C[T∞], T∞f = Lf + Rf ⊂ [•]
M
(g)(• ⊗ •).
One readily verifies by employing the defining formulas of lowering and raising operators in [6]
that T∞ acts as a hyper-derivation.
3.5 Hecke actions on hyper-algebras
Based on the vector-valued Hecke operators that we have introduced in Section 2.4, we get an
additional hyper-module structure on (almost) holomorphic Siegel modular forms. Let
Tf = C[Tp : p prime]
be the polynomial ring in infinitely many formal variables Tp. As in the case of covariant
differential operators, notation coincides with the one for actual Hecke operators, but this should
not lead to confusion. The action of Tp on [•]
M
(g)(• ⊗ •) is defined by
Tpf = span
{
φ ◦ Tpf : ρ ∈ IrrRep
(
K
(g)
f
)
, φ : Tpρf → ρ
}
⊂ [•]
M
(g)(• ⊗ •).
This equips [•]
M
(g)(• ⊗ •) with the structure of a Tf -hyper-module.
12 M. Raum
3.6 The formal Hecke algebra
We combine both formal Hecke algebras into one
T = T∞TAf
= C[T∞,Tp : p prime].
Recall that T∞ ⊂ T acts on almost holomorphic Siegel modular forms by hyper-derivations,
and Tf acts by endomorphisms.
4 Essential surjectivity of tensor products of modular forms
The hyper-algebra structure and the action of the formal Hecke algebra make it rather easy to
formulate essential surjectivity for products of modular forms. The blueprint for such results
are the ones in [6, 23]. The first of them, phrased in the language that we have developed reads
as follows. For all k ≥ 8, all l, k − l ≥ 4, and all congruence representations ρ, we have
M(1)(k ⊗ •) = E(1)(k ⊗ •) + TfE
(1)(l)⊗TfE
(1)(k − l). (4.1)
It is now straightforward to ask for analogues.
4.1 A conjecture in the case of genus 2
We start with a conjecture, that we will not prove in this paper. There is experimental evidence
in [15] that
M(2)(k ⊗ 1) = SK(2)(k ⊗ 1) +
∑
4≤l≤k−4
SK(2)(l ⊗ 1) · SK(2)(k − l ⊗ 1), (4.2)
where SK denotes the space of (holomorphic) Saito–Kurokawa lifts. In light of this and of
formula (4.1) an extension to all ρ seems possible. Could it be that for sufficiently large k, l,
and k − l, and for any congruence type ρ of genus 2 Siegel modular forms, we have
M(2)(k ⊗ ρ) = SK(2)(k ⊗ ρ) + TfSK(2)(l ⊗ 1)⊗TfSK(2)(k − l ⊗ 1) ? (4.3)
Note that we have restricted to scalar weights in the above. To cover vector-valued weights, we
have to first study the effect of covariant differential operators on tensor products.
On inspection of the proofs in [23], we observe that applying vector-valued Hecke operators on
the right-hand side of (4.3) reduces the proof to a certain unfolding of Petersson scalar products
and a nonvanishing of twisted L-values. In the case of Siegel modular forms, unfolding to twisted
spinor L-series was studied in [10] following older ideas from [7]. A sufficient nonvanishing result
for the purpose of [23] follows from Waldspurger’s treatment of half-integral weight modular
forms [8, 20]. Since the untwisted analogue in the case of genus 2 Siegel modular forms has now
been established [1], there is hope to prove the following, stronger statement:
M(2)(k ⊗ ρ) = SK(2)(k ⊗ ρ) + TfE
(2)(l ⊗ 1)⊗TfSK(2)(k − l ⊗ 1) ?
4.2 Hecke operators at all places in genus 1
Making use of not only Hecke operators at the finite places, but of the full formal Hecke algebra T,
one can strengthen surjectivity results such as the one in (4.1).
Theorem 4.1. Let l, l′ ≥ 4 be even integers, and ρ a congruence type. Then for every even
k ≥ l + l′ we have
S(1)(k ⊗ ρ) ⊂ T∞
(
TE(1)(l)⊗TE(1)(l′)
)
. (4.4)
Hyper-Algebras of Vector-Valued Modular Forms 13
Proof. To ease notation, we suppress the superscript (1) for Eisenstein series throughout this
proof. The notation in this proof is adopted from [23]: We let 1|D| be the square of the Kronecker
character εD for a negative fundamental discriminant D. Note hat 1|D| is a trivial non-primitive
Dirichlet character. The modular form fεD is the twist of f by εD. The induction of modular
forms for Γ0(N) to vector-valued modular forms for SL2(Z) is denoted by Ind. The induction
of a Dirichlet character χ of modulus N from Γ0(N) to SL2(Z) is denoted by ρχ.
Without loss of generality, we can assume that l ≤ l′. In complete analogy with [23], we note
that the right-hand side of (4.4) is a Tf -module. It therefore suffices to show the following: Any
scalar multiple of a newform f for the congruence subgroup Γ0(N) vanishes, if for t = (l+l′−k)/
2 ∈ Z≥0 and all D, the (holomorphic) modular form fεD is orthogonal to the almost holomorphic
modular form
Ind
(
El,1|D|
)
⊗ Rt
(
El′,1|D|,∞
)
.
We define the following two vector-valued Eisenstein series:
El,1|D|(τ) =
∞∑
n=1
σl−1,1|D|(n)qn,
El,1|D|,∞,s =
∑
γ∈Γ∞\SL2(Z)
eΓ0(N) ⊗ eΓ0(N)y
s
∣∣∣
k,ρ1|D|⊗ρ1|D|
γ.
Real-analytic vector-valued Eisenstein series have, for example, appeared in [19].
We establish the described vanishing condition by relating the Petersson scalar product to
special L-values, as in [5, 8, 23]. Combining regularization and unfolding as is described in detail
in [23], we find that〈
IndfεD , Ind
(
El,1|D|
)
⊗ Rt
(
El′,1|D|,∞,s
)〉
=
∫
Γ(1)\H(1)
π
(
f · El,1|D|(τ) · eΓ0(N) ⊗ eΓ0(N)R
tys
)dxdy
y2−k , (4.5)
where π is the projection adjoint to the inclusion
1 −→ ρ|D| ⊗
(
ρ|D| ⊗ ρN |D|2
)
⊗ ρN |D|2 , 1 7−→
∑
γ : Γ0(|D|)\SL2(Z)
γ′ : Γ0(|D|N)\SL2(Z)
eγ ⊗
(
eγ ⊗ eγ′
)
⊗ eγ′ .
We will next evaluate the integral, employ analytic continuation, and then insert s = 0, to
obtain a product of special values of Dirichlet series.
A direct verification shows that lim
s→0
Rt
l′y
s = (l′)↑t y
−t, where (a)↑n = a(a+ 1) · · · (a+ n− 1) is
the upper factorial. The above scalar product (4.5) therefore equals
(l′)↑t
( ∞∑
n=0
∫ ∞
0
c(fεD ;n)e(niy)σl−1,1|D|(n)e(niy)y−t+s
dy
y2−k
)
s=0
= (l′)↑t
(
Γ(k − 1− t)
(4π)k−1−t
∞∑
n=0
σl(n)c(fεD ;n)
nk−1−t+s
)
s=0
.
By the extension of Rankin’s result [14] in [23], we find that it does converge absolutely, if l > l′
or allows for a suitable analytic continuation of l = l′. It can be expressed in terms of special
L-values as follows
L
(
fεD , k − 1− t
)
L
(
fεD × 1|D|, k − l − t
)
L
(
1|D|1|D|1, k − 1− 2t− l)
) .
14 M. Raum
The denominator equals L
(
1|D|, l
′ − 1)
)
by the relation k = 2t + l + l′. The first factor in the
numerator is a special value of an Euler product, since t = (k − l − l′)/2 < k
2 − 2 and thus
k − 1− t > k
2 + 1. In order to inspect the second factor in the numerator, note that l + t ≤ k
2 ,
since we have assumed that l ≤ l′. If l + t < k
2 then k − t− l ≥ k
2 + 1, so that L(fεD , k − t− l)
is the special value of a convergent Euler product. Otherwise, we infer that the central value
L
(
fεD ,
k
2
)
vanishes for all negative fundamental discriminants D. Using Waldspurger’s and
Kohnen–Zagier’s results [8, 20], we can argue as in [23] to finish the proof. More precisely,
the vanishing of L
(
fεD ,
k
2
)
implies the vanishing of the Shintani lift of f , because f is a scalar
multiple of a newform. Since the Shintani lift is injective, we find that f = 0 as desired. �
Theorem 4.1 does not cover the case of small weights k. The next statement clarifies that
k < l + l′ cannot appear on the right-hand side of (4.4).
Proposition 4.2. Suppose that k < l1 + l2 for positive even l1, l2 ≥ 4 and positive even k. Then
any weight k cusp form f is orthogonal with respect to the Petersson scalar product to Rt1g1·Rt2g2
for g1 and g2 modular forms of weight l1 and l2, and t1, t2 ∈ Z≥0 such that k = l1 +2t1 + l2 +2t2.
Proof. The almost holomorphic modular form Rt1g1 ·Rt2g2 has depth t1 + t2. That is, it allows
for a decomposition
Rt1g1 · Rt2g2 =
t1+t2∑
t=0
Rtht
for holomorphic modular forms ht of weight k − 2t. Since d > t1 + t2, each term in this sum is
orthogonal to f by Shimura’s orthogonality relations [17] – also confer [13]. �
5 Adelic automorphic representations
Automorphic representation theory in most modern settings focuses on adelic representations.
That is, one investigates the right regular representation of G(A) on L2(G(Q)\G(A)), where
G = PGSpg. It is an important aspect of the theory that one can split any automorphic
representation into a restricted tensor product of local components. The local theory is over
p-adic fields and the theory over the infinite places.
In the classical theory of modular forms these two aspects of automorphic representation
theory are reflected by Hecke operators and covariant differential operators. To support our
claim that hyper-algebras of modular forms together with the action of T are mitigating between
the classical language and the representation theoretic one, we show how to pass from classical
modular forms to the attached local representations and how some of their aspects can be
interpreted in terms of hyper-algebras. In particular, we rephrase our result in Section 4 in
terms of tensor products of automorphic representations.
5.1 Harish-Chandra modules
It is formally correct to work with the group G = PGSpg in this subsection. However, as is
common, we will instead work with G = Spg, i.e., G(R) = Spg(R) for which all aspects that we
discuss here are the same. Further, we adopt notation from [24] to shorten this exposition.
Given any almost holomorphic Siegel modular form f , we can associate to it an irreducible
Harish-Chandra module. Recall that a Harish-Chandra module is an admissible (g,K)-module.
A (g,K)-module is a representation of K = K
(g)
∞ that is simultaneously a g-module with g = spg,
and for which these two structures are compatible. A detailed definition can be found in [22,
Hyper-Algebras of Vector-Valued Modular Forms 15
Section 3.3.1]. A (g,K)-module M is called admissible if it is a unitarizable K-representation
and if HomK(σ,M) is finite-dimensional for every finite-dimensional K-representation σ.
Starting with f , we produce a function A∞(f) on G, A∞ stands for adelization at the infinite
place, which here is isomorphic to R:
A∞(f)(g) =
(
f
∣∣
σ,ρ
g
)(
iI(g)
)
.
It takes values in V (σ)⊗V (ρ), and when contracting with V (σ)∨, we obtain a space of functions
that take values in V (ρ) and which is a K-module isomorphic to σ∨ ⊗ V (ρ). We denote the
contraction by A∞(f) · V (σ)∨, and the (g,K)-module generated by it will be denoted by
A∞(f) = (g,K)
(
A∞(f) · V (σ)∨
)
.
Let k ⊂ g be the Lie algebra of K. It was established in [24] that application of covariant
differential operators and the action of the k-complement m ⊆ g commute with passing back
and forth between modular forms and K-types in Harish-Chandra modules. Concretely, we
established commutativity of the following diagram, featuring the hyper-derivations L and R:
f A∞(f)
Rf m+A∞(f),
A∞
A∞
R m+
f A∞(f)
Lf m−A∞(f).
A∞
A∞
L m−
The product of vector-valued modular forms then contains information about the tensor
product of Harish-Chandra modules. And thus it yields a lower bound on smooth functions
in the tensor product of G(R)-representations. Given a Harish-Chandra module we can obtain
from it a representation of G(R). In our case, we denote this representation by G(R)A∞.
Corollary 5.1. Given two almost holomorphic Siegel modular forms f and g, we have the
following inclusion of (g,K)-modules:
A∞
(
T∞f ⊗T∞g
)
⊆ A∞(f)⊗A∞(g).
By passing back to representations of Spg(R), we find that
G(R)A∞
(
T∞f ⊗T∞g
)
⊆ G(R)A∞(f)⊗G(R)A∞(g).
Proof. This follows from the commutative diagrams above, except that we have to verify that
the hyper-product T∞f ⊗T∞g yields an admissible (g,K)-module. This becomes clear when
inspecting the action of the center of k. �
5.2 The finite places
As opposed to the infinite place it is important to insist on the group G = PGSpg when treating
the finite places. Recall that as a maximal compact group over Qp, we choose K
(g)
p = G(Zp).
Also recall that any type in this paper is a congruence type, which means that it gives rise to
a representation over Af .
To approximate at p, we need the relation G(Qp) = G(Q)K
(g)
p . Consider a vector-valued
almost holomorphic Siegel modular form f of type ρ = ⊗′ρp. Given gk ∈ G(Qp) we set
Ap(f)(gk) = ρ−1
p (k)
(
f
∣∣
σ,ρ
g
)(
iI(g)
)
.
16 M. Raum
As a first remark note that f is constant on ker ρp ⊆ K(g)
p . The contraction of Ap(f) with V (ρp)
∨
will be denoted by Ap(f). It is a space of functions on G(Qp) taking values in V (σ)⊗V (⊗′p 6=p′ρp′).
As a K
(g)
p -module it has isomorphism type ρ∨p ⊗
(
V (σ)⊗ V (⊗′p 6=p′ρp′)
)
.
The commutative diagram corresponding to the one in Section 5.1 is
f Ap(f)
Tpf ∆pAp(f),
Ap
Ap
Tp ∆p
where ∆p is the vector space with basis consisting of determinant p matrices in K
(g)
p . In analogy
with the infinite case, we write G(Qp)Ap(f) for the representation over Qp that corresponds
to f .
Corollary 5.2. Given Siegel modular forms f and g, we have the following inclusion of G(Qp)-
representations:
G(Qp)Ap
(
Tpf ⊗Tpg
)
⊆ G(Qp)Ap(f)⊗G(Qp)Ap(g).
5.3 Global tensor products
We now transfer the statements of Corollaries 5.1 and 5.2 to the global setting.
Theorem 5.3. Given almost holomorphic Siegel modular forms f and g, with automorphic
representations $(f) and $(g) associated with them. Suppose that for a newform h we have
Ind(h) ∈ Tf ⊗Tg.
Then the associated automorphic representation $(h) is contained in the tensor product of those
associated with f and g:
$(h) ↪−→ $(f)⊗$(g).
Proof. Using strong approximation, we obtain a map from holomorphic Siegel modular forms f
to functions
A(f) = A∞(f)⊗
(⊗
p
Ap(f)
)
on G(A), which generate an automorphic representation $(f) = $∞(f) ⊗ ⊗′p$p(f) associated
with f . Note that the Harish-Chandra module attached to a representation of G(R) consists of
the subspace of smooth vectors. In particular, it is a subspace of V ($∞(f)). The commutative
diagrams in Sections 5.1 and 5.2 show that A∞(T∞f) ⊂ V ($∞) and Ap(Tpf) ⊂ V ($p). �
As a consequence of Theorem 5.3, we can reinterpret (4.1) and (4.4) in terms of tensor
products of global representations. This provides a proof of Theorem 1.2.
Acknowledgements
The author was partially supported by Vetenskapsr̊adet Grant 2015-04139. The author is grate-
ful to the referees for various helpful comments.
Hyper-Algebras of Vector-Valued Modular Forms 17
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1 Introduction
2 Modular forms
2.1 The classical setup
2.2 The symplectic group and its Lie algebra
2.3 Covariant differential operators
2.4 Hecke operators
3 Hyper-algebras
3.1 Hyper-algebras of modular forms
3.2 Computing the hyper-algebra product in Sage
3.3 From weights to isomorphism classes of irreducible weights
3.4 Covariant differential operators on hyper-algebras
3.5 Hecke actions on hyper-algebras
3.6 The formal Hecke algebra
4 Essential surjectivity of tensor products of modular forms
4.1 A conjecture in the case of genus 2
4.2 Hecke operators at all places in genus 1
5 Adelic automorphic representations
5.1 Harish-Chandra modules
5.2 The finite places
5.3 Global tensor products
References
|
| id | nasplib_isofts_kiev_ua-123456789-209849 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T12:29:56Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Raum, M. 2025-11-27T14:53:25Z 2018 Hyper-Algebras of Vector-Valued Modular Forms / M. Raum // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 11F40; 11F60; 11F70 arXiv: 1810.02048 https://nasplib.isofts.kiev.ua/handle/123456789/209849 https://doi.org/10.3842/SIGMA.2018.108 We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ℚ, acting on these hyper-algebras. These definitions bridge the classical and representation-theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series. The author was partially supported by Vetenskapsrådet Grant 2015-04139. The author is grateful to the referees for various helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hyper-Algebras of Vector-Valued Modular Forms Article published earlier |
| spellingShingle | Hyper-Algebras of Vector-Valued Modular Forms Raum, M. |
| title | Hyper-Algebras of Vector-Valued Modular Forms |
| title_full | Hyper-Algebras of Vector-Valued Modular Forms |
| title_fullStr | Hyper-Algebras of Vector-Valued Modular Forms |
| title_full_unstemmed | Hyper-Algebras of Vector-Valued Modular Forms |
| title_short | Hyper-Algebras of Vector-Valued Modular Forms |
| title_sort | hyper-algebras of vector-valued modular forms |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209849 |
| work_keys_str_mv | AT raumm hyperalgebrasofvectorvaluedmodularforms |