Almost every vector valued modular form is an oldform

0.4 MB·

by  Fabian Werner

Almost every vector valued modular form is an
4 oldform
1
0
2 Fabian Werner

n
a [email protected]
J

7
2 January 28, 2014

]
T
N Abstract
. In this article we show that ’most’ of the vector valued modular forms
h
w.r.t. the Weil representation on the groups rings C[D] of discriminant
t
a formsDareoldforms. Theprecisemeaningofoldformisthattheformcan
m
be represented as a sum of lifts of vectorvalued modular forms on group
[ ringsofquotientsH⊥/H forisotropicsubgroupsH ofD. Inthiscontext,
e
’most’ means that all forms are oldforms if there is a part Z/p Z inside
2
a p-part of D that is repeated several times (i.e. ≥ 4,5,7,9 depending
v
on p and e). Wewill proceed by giving an oldform detection mechanism.
4
This criterion also gives rise to an efficient algorithm for computing the
1
2 decomposition ofcuspformsintotheirspacesofold-andnewforms when
6 only given the Fouriercoefficients of a basis of the space of cusp forms.
.
1
0 1 Overview
4
1
: Let D=(D,Q) be a discriminant form. To simplify the exposition, we assume
v
throughout that the signature of D is even. All results carry over naturally to
i
X thecaseofanoddsignatureandthemetaplecticcover. Recently,liftsforvector
r valued modular forms for quotients H⊥/H – where H is an isotopic subgroup
a
– of the form
G= Gaea 7→G↑ := G e
H γ+H γ
a
∈XDH γX∈H⊥
have gained attention. Note that in the body of the paper, this map will be
called↑initinordertodistinguishitfromitspurelyalgebraicversion(seeSection
H
3). This map is expected to replace the lifting process for dividing levels in the
scalar valued case, thus giving rise to an oldform/newform theory. Following
the scalar valued ideas, one defines the space of oldforms as
S (H⊥/H)↑
k H
H isotropic
X
1
where S (D) is the space of entirely holomorphic vector valued modular cusp
k
forms of weight k for the discriminant form D.
The main purpose of this paper is to show the following:
If N ∈ N is fixed and D is a discriminant form of level N with |D| ≥ N9,
then every vector valued modular form for D is an oldform. This bound (N9)
is absolutely not optimal.
ThisisstatedasCor. 27insection7. Inotherwords,themeaningofthisresult
isthatforeveryfixedlevelN weonlyhavetostudyfinitely manyvectorvalued
modular forms for discriminant forms of this level. In fact, the number N9 is
absolutely not optimal, it suffices if a certain p-part of the discriminant form is
repeated often enough, see Thm 26.
We achieve this by giving a purely algebraic characterizationfor detecting old-
forms which is interesting in its own right. More precisely, we show in Thm 10
that for an arbitrary selection of isotropic subgroups H ,...,H ,
1 n
F is an oldform with respect to the H ,...,H ⇐⇒ ker(↓)⊂ker(F)
1 n
where ↓ is a certain ’algebraic part’ of the ’converse’ map of ↑ and F is the
C-linearmapsendinge toitscomponentF asamodularformforΓ(N). This
γ γ
resultcanbeunderstoodasageneralizationoftheworkofBruinier,citebruinier-
converse Thm. 3.6.
Moreover, we present the following: given a basis of vector valued modular
formsforsomeconcreteweightanddiscriminantformuptosomeSturmbound
(which have been created using a computer algebra system for example) this
characterization allows us to compute bases for the spaces of oldforms (with
respect to any selection of isotropic subgroups) and, if we restrict ourselves
to cusp forms, we can compute its orthogonal complement, i.e. the space of
newforms with a little trick as well, see Thm. 14. This is great for doing
concrete experiments with vector valued modular forms, especially in view of
the fact that M. Raum has recently given an algorith that computes bases of
vector valued modular forms, see [11]. Secondly, more abstractly, the same
strategy as in the proof of the theorem above allows us to solve the converse
problem, i.e. the question of whether a vector valued modular form for the
smaller discriminant form H⊥/H is a ’down’ lift of some form above.
The paper is organized as follows:
2
Section Contents
Sec. 2 Basic terminology, lattices, discriminant forms, Weil representa-
tions, scalar valued and vector valued modular forms, up/down
maps
Sec. 3 Separationof up/downmaps into algebraicandtrivial part, alge-
braic parts are homomorphisms of Weil representations
Sec. 4 Characterization: F is old iff. ker(↓)⊂ker(F)
Sec. 5 An algorithm for splitting vector valued cusp forms into old and
new spaces
Sec. 6 Technical lemmas needed for the proof of the main theorem
Sec. 7 Proof of the main theorem, the version announced above is then
a corollary
2 Prerequisites
The groupGL+(R)={M ∈Z2×2|det(M)>0} acts from the left on the upper
2
half plane H:={τ ∈C|Im(τ)>0} by
aτ +b a b
M.τ = , M =
cτ +d c d
(cid:18) (cid:19)
We can continue this action to H∪R∪{∞} by putting M.∞ := a/c and if
cτ+d=0 thenwe putM.τ :=∞. We alsoput (M :τ):=cτ+d. This induces
a right action on functions from H to C by
f| M :=(M :τ)−kf(Mτ), k∈Z
k
Mostly, k will be fixed throughout, so we will drop it from the notation. If Γ is
a group, a group homomorphism χ : Γ → C is called a character. To simplify
the exposition, let us assume that Γ is one of the well studied subgroups,
a b
Γ (N):= M = ∈SL (Z):c≡0 mod N
0 c d 2
(cid:26) (cid:18) (cid:19) (cid:27)
a b
Γ (N):= M = ∈SL (Z):a≡d≡1 mod N, c≡0 mod N
1 c d 2
(cid:26) (cid:18) (cid:19) (cid:27)
Γ(N):={M ∈SL (Z):M ≡Id mod N}
2
ofSL (Z). We willonlyneedthese subgroupsinthebody ofthe paperanyhow.
2
Definition 1. Let Γ be one of the subgroups as above. Let χ : Γ → C× be
a character with the property that Γ(N) ⊂ ker(χ). An entirely holomorphic
3
modular form of weight k ∈ Z for Γ with character χ is a function f : H → C
such that
1. f is holomorphic.
2. f|γ =χ(γ)f for all γ ∈Γ.
3. for every M ∈SL (Z) (not merely all M ∈Γ!), f|M(τ) is bounded when
2
τ →∞.
The C-vector space of all these functions will be denoted by M (Γ,χ).
k
In case that f ∈ M (Γ,χ), one can show ([4] pp. 1-5, [15], Thm. 2.4.7 or any
k
other book on modular forms) that f possesses a Fourier expansion
∞
f(τ)= a qn/N, q =exp(2πiτ)=e(τ)
n
n=0
X
or, more general, for every M ∈SL (Z),
2
∞
f| (τ)= a(M)qn/N
M n
n=0
X
f is called cusp form if a(M) = 0 for all M ∈SL (Z). The subspace of all such
0 2
functions will be denoted by S (Γ,χ).
k
We want to describe one possible generalization of the theory of scalar valued
modular forms, namely vector valued modular forms. These are functions that
’behave well’ under slashing with matrices in SL (Z) but now they map H to
2
certain finite dimensional C-vector space. Before we are going to describe the
structure we need some terminology.
Let throughout R be a commutative ring and let X,Z be R-modules. If R is
an integral domain then let F denote its field of fractions. A Z-valued bilinear
form is a map
b:X ×X →Z
such that b(·,x) : X → Z and b(x,·) : X → Z are R-linear for all x ∈ X. If
Z =R then we call b integral. b is called non-degenerate (resp. unimodular) if
the maps x7→ b(·,x) and x 7→b(x,·) (from X to Hom (X →Z)) are injective
R
(resp. bijective). We saythatbis symmetric ifb(x,y)=b(y,x)for allx,y ∈X.
We say that b is even, if b(x,x) ∈ 2R for all x ∈ X. If b is symmetric and
E,W ⊂X aresubmodules,then we write E⊥W if b(x,y)=0 forallx∈E,y ∈
W. IfE⊥W andforeverys∈E+W,theelementse,wins=e+wareunique,
then we write EkW.
A Z-valued quadratic form is a map Q : X → Z with the properties that
Q(ax)=a2Q(x) for all x∈X,a∈R and
b :X ×X →Z, (x,y)7→Q(x+y)−Q(x)−Q(y)
Q
4
is a bilinear map. Q is called integral if Z = R. Q is called non-degenerate
(respectively unimodular) if b has the respective property.
Q
A Z-valued R-lattice is a touple L=(L,b) consisting of a freely, finitely gener-
atedR-moduleL(i.e. thereisafinitesetv ,...,v suchthatL=Rv ⊕...⊕Rv
1 n 1 n
meaning that every v ∈ L can be written as a unique R-linear combination of
the v ) together with a Z-valued bilinear form b. L is called integral, even,
i
non-degenerate or unimodular if b has the respective property.
Together with every integral lattice over an integral domain R comes its F-
vector space V = L⊗F and the F-valued bilinear form b = b⊗id . For a
F F
lattice L we define L′ to be the touple consisting of
L′ :={v ∈V :b(v,l)∈R ∀l ∈L}
together with the F-valued bilinear form bF|L′×L′ and call this the dual lattice
to L.
A discriminant form is a touple D=(D,Q) consisting of a finite abelian group
D and a so-called finite quadratic form, that is, a non-degenerate quadratic
form Q : D → Q/Z such i.e. for the associated bilinear form (γ,δ) = Q(γ +
δ)−Q(γ)−Q(δ) we have D⊥ = {0}. Two discriminant forms (D,Q),(D˜,Q˜)
are called isomorphic if there exists a Z-module isomorphism φ : D → D˜ such
that Q˜(φ(γ))=Q(γ) for all γ ∈D.
Mostly, we are rather sloppy with the notation and just write L for L and D
for D because the bilinear, resp. quadratic form will be fixed or clear from the
context.
One of the key-features of discriminant forms is the following:
Theorem 2. Every discriminant form D=(D,Q) possesses a so-called Jordan
splitting, i.e. onefindsabasis inthesenseoffinitelygeneratedabelian groupsof
D suchthat thematrix consistingof thebilinear pairings (moduloZ) is diagonal
on the odd p-parts and almost diagonal on the 2-adic part. More precisely: D
is the orthogonal sum over components C of the form
1. C ∼= Zpe for some odd prime p and C is generated by a single element γ
with (γ,γ) = a where a ∈ Z,gcd(a,p) = 1 and Q(γ) = 2−1a +Z where
pe pe
the inversion of 2 takes place in Z .
pe
2. C ∼= Z2e is generated by a single element γ with (γ,γ) = 2ae where a ∈
Z,gcd(a,2)=1 and Q(γ)= a+v2e +Z where v is either 0 or 1.
2e+1
3. C ∼=Z2e×Z2e isgeneratedbytwoelementsγ,δ suchthattheGrammatrix
of pairings of γ and δ is given by
x 1
2−e
1 x
(cid:18) (cid:19)
where x is either 0 or 2. If x=0 then Q(γ)=Q(δ)=0+Z. We say that
this is a block of type (A). If x = 2 then Q(γ)= Q(δ) = 1 +Z. We say
2e
that this is a block of type (B).
5
Proof. A proof can be found in [18].
Examples of discriminantforms canbe obtained by using even, non-degenerate
Z-lattices L = (L,b): As L is even, we can define an integral quadratic form
Q : L → Z,Q(x) := b(x,x)/2. Its associated bilinear form b is nothing else
Q
than b, so b(x,y) = Q(x+y)−Q(x)−Q(y) is integral. Hence, L ⊂ L′ and
it makes sense to consider D := L′/L. Then Q(x+L) := Q(x)+Z gives a
discriminant form. One can show that in fact, all discriminant forms arise in
suchaway. One uses the followingstrategy: usingThm. 2, we obtaina Jordan
splitting of D. Then one only needs to show the existence of a lattice for the
Jordan constituents and this problem can be solved in a surprisingly easy way,
see [13], Thm 6, mainly p. 297.
Let L be an even lattice. For τ ∈ C, let e(τ) := exp(2πiτ). Milgrams formula
([9] Appendix 4) shows that
Q(γ)= |D|e(s/8)
γ∈XL′/L p
wheresisthesignature(overR)ofthelatticeL. Forthisreason,foragivendis-
criminantformD thereisanumbers∈Z/8Zsuchthatallevennon-degenerate
integral Z-lattices L having D as their discriminant form (up to isomorphy)
have a signature congruent to s modulo 8. This element is called the signature
sign(D) of D.
If the signature of D is even, then there is a unitary representation ρ of
SL (Z)onC[D],theC-vectorspaceofdimension|D|withcanonicalorthonormal
2
basis (e ) . This representation works as follows: SL (Z) is generated by
γ γ∈D 2
S = 0−1 and T =(11). Their C-linear actions are given by
1 0 01
(cid:0) (cid:1) ρ(T)e =e(Q(γ))e
γ γ
ρ(S)e =c e(−(γ,β))e
γ D β
β∈D
X
where
e(−sign(D)/8)
c :=
D
|D|
Ofcourse,no one came up with suchforpmulae outof nowhere,they are conrete
instances of a more general construction due to A. Weil. One can see some
traces of this process: The action of S is essentially a Fourier transform. The
constructionin full generalitycan be found in [14]. A down-to-earthproof that
justmakesuseofMilgramsFormula(whichinturnisprovedinadown-to-earth
way in [9] Appendix 4) is written down in [17].
If the signature is not even, then still, there is a representation but one has to
pass to a degree 2 metaplectic cover of SL (Z). We are going to skip this case
2
for the sake of readability but all the results carry over naturally.
6
LetDbeadiscriminantformofevensignatureandρtheWeilrepresentation
on C[D]. A holomorphic vector valued modular form of weigth k ∈ Z is a
holomorphic function F :H→C[D] satisfying
aτ +b a b
F =(cτ +d)kρ F(τ)
cτ +d c d
(cid:18) (cid:19) (cid:18) (cid:19)
for all τ ∈ H, a b ∈ SL (Z), in short: F|M = ρ(M)F, and is holomorphic at
c d 2
∞ meaning that, for example, F(τ) stays bounded as τ → ∞. The set of all
(cid:0) (cid:1)
such functions will be denoted by M (D).
k
ThelevelN ofadiscriminantformD isthesmallestnaturalnumbermsuch
that mQ(γ)=0+Z for all γ ∈D. It is widely believed to be provenfor a long
time thatρ(M)=IdC[D] for allM ∈Γ(N). However,tothe bestofthe authors
knowledge, the only proof that was officially, completely written down is due
to S. Zemel ([19], Thm. 3.2) in 2011. An alternative down-to-earth proof is
due to N.-P. Skoruppa. Unfortunately his book about the Weil representation
is not published yet. As F – or F|M, which is just a linear combination of
the components of F – stays bounded when τ →∞, the same is true for every
component. Hence,everycomponentofavectorvaluedmodularformisascalar
valuedmodularformforΓ(N). Weusethisassertionwithoutmentioningitany
further. F is called a cusp form iff. every component is a cusp form. The
subspace of cusp forms will be written as S (D).
k
We describe the Petersson scalar product: The measure
1
ν(M)= dxdy, M ⊂H Lebesgue-measurable
y2
ZM
is GL+(R)-invariant (see [8], Kap. IV, §3). Let A be an arbitrary fundamental
2
domainforΓ,thatis,a’nice’systemofrepresentativesforΓ\Hwiththeproperty
that ν(∂A) = 0 where ∂A is the topological boundary of A. Different authors
givedifferent(wrong!,see[5])definitionsof’nice’andforgetabouttheadditional
condition. However, for the three subgroups Γ (N),Γ (N),Γ(N), every of the
0 1
definitionsfloatingaroundincurrentliterature(forexample: [10],§1.6,[8]Kap.
II §3) together with the condition ν(∂A)=0 will do.
Let f,g ∈ S (Γ) with Γ being a ’nice’ subgroup of SL (Z), say one of the
k 2
examples given above.
The map
1
hf,gi:= f(τ)g(τ)ykdxdy/y2
[SL (Z):Γ]
2 ZA
is convergent ([8] Kap. IV, §3, [4] §5.4, etc.), is independent of the chosen
fundamental domain ([8], Kap. IV, §3, pp. 231-232) and turns S (Γ) into a
k
Hilbert space. In complete analogy we define the Petersson scalar product for
vector valued cusp forms F,G to be
hF,Gi:= F G ykdxdy/y2
γ γ
ZAγ∈D
X
7
where here, A is a fundamental domain for SL (Z)\H.
2
We are going to describe a part of the theory of (scalar valued) modular forms
called ’old/newform-theory’: Let f ∈S (Γ (A)) for some A|B. Then there are
k 0
multiple ways of interpreting f as an element in S (Γ (B)). Generally, one can
k 0
consider f 7→ f(cτ) where c|A; c = 1 corresponding to the trivial inclusion
B
S (Γ (A)) ⊂ S (Γ (B)). The span of the images of all these maps is called
k 0 k 0
the subspace of oldforms. The reason why we restrict ourselves to cusp forms
is that we want to take the orthogonal complement w.r.t. the Petersson scalar
product of the space of oldforms and call this the space of newforms. This
spacehassomeextremelyimportantproperties(eigenbasisforHeckeoperators,
Euler products,connections to elliptic curves,etc), see for example [10], §4.6 ff.
Hence, it is important to ask whether there is a similar construction for vector
valued modular forms.
LetD=(D,Q)beadiscriminantform. Anelementγ ofDiscalledisotropic
if Q(γ) = 0+Z. A subgroup H of D is called isotropic if all elements of H
are isotropic. If H is an isotropic subgroup we put D := H⊥/H. Then,
H
D := (D ,Q ) with Q (γ +H) := Q(γ) becomes a discriminant form and
H H H H
satisfies sign(D )=sign(D) and|D |=|D|/|H|2. (The proof of this assertion
H H
is left to the reader). When isotropic subgroups H ,...,H are given we just
1 n
write D in place of D and D in place of H⊥/H .
i Hi i i i
Recently, operators of the form
↑init:M (D )→M (D), Gaea 7→G↑init:= G e (1)
H k H k H γ+H γ
a
∈XDH γX∈H⊥
have gained attention. Abstractly, these operators are expected to replace the
lifting process for dividing levels in the scalar valued case, hence giving rise to
a vector valued oldform/newform theory. They have been used for example, to
study in which cases certain orthogonal modular forms arise as Borcherds lifts
(see [1]) and under which conditions a vector valued modular form is induced
by a scalar valued one (see [12]). There is also a ’converse’map:
↓init:M (D)→M (D ), F e 7→F↓init:= F ea (2)
H k k H γ γ H a  a γ
γX∈D ∈XDH Xγ∈
 
(Remarkthatitisnotclearthattheseoperatorsreallymapvectorvaluedmod-
ular forms to vector valued modular forms again; we will prove it in the next
section). We write them with a superscript ’init’ for ’initial’ in order not to
confuse them with their ’algebraic’parts, see Sec. 3.
Following the ideas in the scalar valued case we define old- and newforms:
Take isotropic subgroups H ,...,H of D. We define the space of vector valued
1 k
oldforms w.r.t. H ,...,H to be
1 k
S (D)old,H1,...,Hk :=S (D )↑init +...+S (D )↑init
k k 1 H1 k k Hk
Analogously, the space of newforms is
S (D)new,H1,...,Hk := S (D)old,H1,...,Hk ⊥
k k
(cid:0) (cid:1)
8
where the orthogonalcomplement is taken with respect to the Petersson scalar
product for vector valued modular forms.
3 Separarion of the up and down maps
In this section we will separate the up and down maps from the introduction
into two parts. In order not to confuse them we will call the up/down maps on
modularforms↑func and↓func ifindoubt. Itturnsoutthatthey canbe written
H H
as ↑func=↑ ⊗id for some C-linear, purely algebraic map ↑ (and similarly with
H H
↓). These maps are of crucial importance for the characterizationof oldforms.
Generallyspeaking,givenvectorspacesV,V ,...,V andvectorspacehomomor-
1 n
phisms α : V → V then we denote by α := (α ,...,α ) the homomorphism
i i 1 n
from V to V ⊕...⊕V given by α(v) := (α (v),...,α (v)). Conversly, given
1 n 1 n
vector spaces V,V ,...,V and vector space homomorphisms β : V → V then
1 n i i
wedenotebyβ :=β +...+β thehomomorphismfromV ⊕...⊕V toV given
1 n 1 n
by β(v ,...,v ):=β (v )+...+β (v ).
1 n 1 1 n n
It is easy to see that
Remark 3. Ifφ,φ ,...,φ arerepresentationsofsome groupGonV,V ,...,V .
1 n 1 n
We endow V ⊕...⊕V with the representationφ ⊕...⊕φ .
1 n 1 n
(a) If all α are homomorphisms of representations then so is α=(α ,...,α )
i 1 n
(b) If all β are homomorphisms of representationsthen so is β =β +...+β .
i 1 n
Definition 4. Let D = (D,Q) be a discriminant form and H be an arbitrary
isotropic subgroup. We let π : H⊥ →D denote the projection π(γ)= γ+H
H
and we put ↓ :C[D]→C[D ] to be the C–linear map
H H
↓ c e := c ea
H γ γ γ
(cid:18)γX∈D (cid:19) a∈XDH(cid:18)γ∈Xπ−1(a) (cid:19)
Further we define a C–linear map ↑ :C[D ]→C[D] as
H H
↑ caea := c e
H γ+H γ
a
(cid:18) ∈XDH (cid:19) γX∈H⊥
WhenisotropicsubgroupsH ,...,H aregivenwejustwriteD inplaceofD ,
1 n i Hi
↓ in place of ↓ and similarly with the up arrow maps and we let
i Hi
↓ :=(↓ ,...,↓ )
H1,...,Hn 1 n
↑ :=↑ +...+↓
H1,...,Hn 1 n
and dropthe H from the notation as they will be clear from the context. If no
i
isotropicsubgroupsaregiven,thenwejustwrite↑,↓asassociated(inthe sense
above) to all isotropic subgroups in D.
9
Lemma 5. Let D = (D,Q) be a discriminant form. Let ρ : SL (Z) → C[D]
2
denote the Weil representation of D.
(a) Let H be an arbitrary isotropic subgroup and let η be the Weil rep. of D .
H
Then the maps ↓ ,↑ are homomorphisms of the Weil representations, i.e.
H H
ρ(M)
C[D] //C[D]
OO OO
↓H ↑H ↑H ↓H
(cid:15)(cid:15) η(M) (cid:15)(cid:15)
C[D ] // C[D ]
H H
commutes for every M ∈SL (Z).
2
(b) Let H ,...,H be arbitrary isotropic subgroups of D and let ρ be the Weil
1 n i
representation of D for i = 1,...,n. Then ↓ and ↑ are homomorphisms of
i
representations, i.e.
ρ(M)
C[D] // C[D]
OO OO
↓ ↑ ↑ ↓
C[D ]⊕..(cid:15)(cid:15).⊕C[D ](ρ1⊕...⊕ρn)(M)//C[D ]⊕..(cid:15)(cid:15).⊕C[D ]
H1 Hn H1 Hn
commutes for every M ∈SL (Z).
2
Proof. (a): We need to show that for all x∈C[D] and all M ∈SL (Z),
2
η(M)↓ (x)= ↓ (ρ(M)x)
H H
Sinceallmaps↓ ,↑ ,ρ(M),η(M)areC-linear,itsufficestoshowtheassertion
H H
for x=e . Since SL (Z) is generated by S,T, and both, ρ,η are left actions, it
γ 2
suffices to show the assertion for M =S,M =T.
On M =T:
↓ (ρ(T)e )= ↓ (e(Q(γ))e )=e(Q(γ))↓ (e )=e(Q(γ))e
H γ H γ H γ γ+H
=e(Q (γ+H))e =η(T)e =η(T)↓ (e )
H γ+H γ+H H γ
On M =S: we write
e = c e with c =1 (3)
γ δ δ δ γ=δ
δ∈D
X
10