A relation of cusp forms and Maass forms on product of hyperbolic Riemann orbisurfaces of finite volume

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by  Anilatmaja Aryasomayajula

A relation of cusp forms and Maass forms on product of
hyperbolic Riemann orbisurfaces of finite volume
Anilatmaja Aryasomayajula
Abstract
4 In [2], J. Jorgenson and J. Kramer proved a certain key identity which relates the two nat-
1 ural metrics, namely the hyperbolic metric and the canonical metric defined on a compact
0 hyperbolicRiemannsurface. Inthisarticle,weextendthisidentitytoproductofnoncompact
2 hyperbolic Riemann orbisurfaces of finite volume, which can be realized as a quotient space
of theaction of a Fuchsian subgroup of first kind on thehyperbolic upperhalf plane.
n
a Mathematics Subject Classification (2010): 30F30, 30F35, 30F45.
J

8
2 1 Introduction

]
T For i = 1,2, let X be a noncompact hyperbolic Riemann orbisurface of finite volume vol (X )
i hyp i
N with genus g ≥ 1, and can be realized as the quotient space Γ \H, where Γ ⊂ PSL (R) is a
i i i 2
. Fuchsian subgroup of the first kind acting on the hyperbolic upper half-plane H, via fractional
h
linear transformations. Let E and P denote the finite sets of elliptic fixed points and cusps of Γ ,
t i i i
a respectively. PutX =X ∪P . Then,X admitsthestructureofaRiemannsurface. Nowconsider
m i i i i
the complex surface X = X ×X , which admits the structure of a noncompact K¨ahler-orbifold
1 2
[ of dimension two. Put X =X ×X .
1 2

1 For i = 1,2, let µi denote the (1,1)-form associated to hyperbolic metric, which is the natural
v hyp
6 metriconXi,andofconstantnegativecurvatureminusone. Nowµ1hyp+µ2hyp isthenaturalmetric
2 on X, and let µvol denote the volume form associated to µ1 +µ2 .
1 hyp hyp hyp
7
For i = 1,2, the Riemann surface X is embedded in its Jacobian variety Jac(X ) via the Abel-
. i i
1 Jacobimap. Then, the pull backof the flat Euclideanmetric by the Abel-Jacobimap is calledthe
0 canonical metric, and the (1,1)-form associated to it is denoted by µi . We denote its restriction
4 to X by µi . Now, µ1 +µ2 defines a metric on X, which correcsapnonds to the flat Euclidean
1 i can can can
metric, and let µvol denote the volume form associated to µ1 +µ2 .
: can can can
v
i For i = 1,2, let ∆i denote the hyperbolic Laplacian acting on smooth functions on X . The
X hyp i
hyperbolic heatkernelKi (t;z ,w ) onR ×X ×X is the unique solutionof the heat equation
r hyp i i >0 i i
a
∂
i i
∆ + K (t;z ,w )=0,
(cid:18) hyp ∂t(cid:19) hyp i i
with the normalization condition
lim Ki (t;z ,w )f(z )µi (z )=f(w ),
hyp i i i hyp i i
t→0ZXi
for any fixed w ∈X and any smoothfunction f onX . When z =w , for brevity of notation,we
i i i i
denote the hyperbolic heat kernel by Ki (t;z ).
hyp i
Main result With notation as above, for z =(z ,z )∈(X \E )×(X \E ), we have the relation
1 2 1 1 2 2
1
of differential forms
1 1 1 1 1 1 1
g g µvol(z)= + · + µvol (z)+ + ×
1 2 can (cid:18)4π vol (X )(cid:19) (cid:18)4π vol (X )(cid:19) hyp 2(cid:18)4π vol (X )(cid:19)
hyp 1 hyp 2 hyp 1
∞ 1 1 1 ∞
∆2 K2 (t;z )dt µvol (z)+ + · ∆1 K1 (t;z )dt µvol (z)+
(cid:18)Z0 hyp hyp 2 (cid:19) hyp 2(cid:18)4π volhyp(X2)(cid:19) (cid:18)Z0 hyp hyp 1 (cid:19) hyp
1 ∞ ∞
∆1 K1 (t;z )dt · ∆2 K2 (t;z )dt µvol (z).
4(cid:18)Z hyp hyp 1 (cid:19) (cid:18)Z hyp hyp 2 (cid:19) hyp
0 0
The above relation, which we call the key-identity, relates the two natural metrics defined on a
K¨ahler-orbifold of dimension two. The key-identity is proved for compact hyperbolic Riemann
surfaces, by J. Jorgenson and J. Kramer in [2]. The same authors extended the key-identity to
noncompact hyperbolic Riemann surfaces of finite volume in [3]. Following diferent methods, the
key-identity is extended to noncompact hyperbolic Riemann orbisurfaces in [1].
Arithmetic significance The key-identity is the most significant technical result of [2], which
transforms a problem in Arakelov theory into that of hyperbolic geometry. The key-identity has
enabled J. Jorgenson and J. Kramer to derive optimal bounds for the canonical Green’s func-
tion defined on a compact hyperbolic Riemann surface X in terms of invariants coming from the
hyperbolic geometry of X.
Using the key-identity one can relate the holomorphic world of cusp forms with the C∞ world of
Ma¨ss forms, via the spectral expansion of the hyperbolic heat kernel in terms of Ma¨ss forms. In
fact, J. Jorgenson and J. Kramer have derived a Rankin-Selberg L-function relation relating the
Fourier coefficients of cusp forms with those of Ma¨ss forms in [3].
Our main result is the first instance of an extension of the key-identity to higher dimensions,
relating the cusp forms of the group Γ ×Γ with the Ma¨ss forms defined on X and X via the
1 2 1 2
spectral expansions of the hyperbolic heat kernels K1 (t;z ) and K2 (t;z ), respectively.
hyp 1 hyp 2
Acknowledgements This article was realized during the author’s graduate studies, which was com-
pleted under the supervision of J. Kramer at Humboldt Universit¨at zu Berlin. The author would like
to express his gratitude to J. Jorgenson and J. Kramer for introducing him to the exciting area of heat
kernelsandautomorphicforms, andforgenerously discussingmanynewscientificideas, which resultedin
thecompletion of this project.
The author would also like to extend his gratitude to N. Walji, who through many interesting scientific
discussions has conveyedtheimportance of thekey-identity.
2 Background material
For i=1,2, let Γ ⊂PSL (R) be a Fuchsian subgroup of the first kind acting by fractional linear
i 2
transformations on the upper half-plane H. Let X be the quotient space Γ \H, and let g denote
i i i
the genus of X . The quotient space X admits the structure of a Riemann orbisurface.
i i
LetE andP bethefinitesetsofellipticfixedpointsandcuspsofX ,respectively;putS =E ∪P .
i i i i i i
For ei ∈Ei, let mei denote the order of ei; for pi ∈Pi, put mpi =∞; for zi ∈Xi\Ei, put mzi =1.
Let X denote X =X ∪P .
i i i i
Locally, away from the elliptic fixed points and the cusps, we identify X with its universal cover
i
H, and hence, denote the points on X \S by the same letter as the points on H.
i i
The quotient space X admits the structure of a compact Riemann surface. We refer the reader
i
to Section 1.8 in [4], for the details regarding the structure of X as a compact Riemann surface.
i
Let X denote the product of the Riemann orbisurfacesX ×X . Then, X admits the structure of
1 2
a complex K¨ahler-orbifold of dimension two. The boundary of X is given by ∂X = X ×P ∪
1 2
X ×P , and the compactification of X is given by X =X ×X . (cid:0) (cid:1)
2 1 1 2
(cid:0) (cid:1)
2
Hyperbolic metric For i = 1,2, we denote the (1,1)-form corresponding to the hyperbolic
metric of X , which is compatible with the complex structure on X and has constant negative
i i
curvature equal to minus one, by µi (z ). Locally, for z ∈X \E , it is given by
hyp i i i i
i dz ∧dz
µi (z )= · i i.
hyp i 2 Im(z )2
i
From the above formula, it follows that the hyperbolic metric µi (z ) is singular at the elliptic
hyp i
fixed points and at the cusps.
Let vol (X ) be the volume of X with respect to the hyperbolic metric µi (z ). It is given by
hyp i i hyp i
the formula
1
vol (X )=2π 2g −2+|P |+ 1− .
hyp i i i
(cid:18) eXi∈Ei(cid:18) mei(cid:19)(cid:19)
We denote the (1,1)-form corresponding to the hyperbolic metric of X, which is compatible with
the complex structure on X, by µ (z) and the corresponding volume form by µvol (z). Locally,
hyp hyp
for z =(z ,z )∈(X \E )×(X \E ), it is given by
1 2 1 1 2 2
i dz ∧dz i dz ∧dz
µ (z)=µ1 (z )+µ2 (z )= · 1 1 + · 2 2,
hyp hyp 1 hyp 2 2 Im(z )2 2 Im(z )2
1 2
and the corresponding volume form is given by
dz ∧dz ∧dz ∧dz
µvol (z)=µ1 (z )∧µ2 (z )=− 1 1 2 2.
hyp hyp i hyp 2 2Im(z )2Im(z )2
1 2
Canonical metric For i = 1,2, let S (Γ ) denote the C-vector space of cusp forms of weight
2 i
2 with respect to Γ equipped with the Petersson inner-product. Let {fi,...,fi } denote an
i 1 gi
orthonormal basis of S (Γ ) with respect to the Petersson inner-product. Then, the (1,1)-form
2 i
µi (z ) corresponding to the canonical metric of X is given by
can i i
µi (z )= i gi fi(z ) 2dz ∧dz .
can i 2g j i i i
i Xj=1(cid:12) (cid:12)
(cid:12) (cid:12)
The canonical metric µi (z) remains smooth at the elliptic fixed points and at the cusps, and
can
measures the volume of X to be one.
LetΩ1 denotethecotangentbundleofholomorphicdifferentialformsofdegreeoneonX . Recall
Xi i
that for each fi ∈ S (Γ ), fi(z )dz defines a holomorphic differential form of degree one on X ,
2 i i i i
and every holomorphicdifferential form of degree one on X comes from a weight2 cusp form. So
i
{fi,...,fi } the orthonormalbasis of S (Γ ) with respect to the Peterssoninner-product gives us
1 gi 2 i
anorthonormalbasis{fidz ,...,fi dz }ofH0(X ,Ω1 )endowedwiththe L2-innerproductgiven
1 i gi i i Xi
by
i
hαi,βii= αi(z )βi(z ),
i i
2Z
X
where αi,βi ∈Ω1 .
Xi
LetΩ2 denotethe spaceofholomorphicdifferentialformsofdegree2,andlet{ω ,...,ω }denote
X 1 n
an orthonormal basis of H0(X,Ω2 ) endowed with the L2-inner product given by
X
1
hα,βi=− α(z)β(z),
4Z
X
where n denotes the dimension of H0(X,Ω2 ) as a vector space overC, and α,β ∈Ω2 . Then, the
X X
canonical volume form on X is defined as
n
1
µvol(z)=− ω (z)∧ω (z).
can 4n j j
Xj=1
The canonical volume form µvol(z) measures the volume of X to be one.
can
3
Hyperbolic Laplacian For i = 1,2, the hyperbolic Laplacian acting on smooth functions de-
fined on X is given by
i
∂2 ∂2
∆i =−y2 + ,
hyp i(cid:18)∂x2 ∂y2(cid:19)
i i
and the hyperbolic Laplacian acting on smooth functions defined on X is given by
∂2 ∂2 ∂2 ∂2
∆ =∆1 +∆2 =−y2 + −y2 + .
hyp hyp hyp 1(cid:18)∂x2 ∂y2(cid:19) 2(cid:18)∂x2 ∂y2(cid:19)
1 1 2 2
Hyperbolic heat kernels Fort∈R>0 andz,w∈H,let KH(t;z,w)denote the hyperbolicheat
kernel on R ×H×H.
>0
For i=1,2, t∈R and z ,w ∈X , the hyperbolic heat kernel Ki (t;z ,w ) on R ×X ×X
>0 i i i hyp i i >0 i i
is defined as
Khiyp(t;zi,wi)= KH(t;zi,γiwi). (1)
γXi∈Γi
For z ,w ∈X , the hyperbolic heat kernel Ki (t;z ,w ) satisfies the differential equation
i i i hyp i i
∂
∆i + Ki (t;z,w)=0, (2)
(cid:18) hyp,z ∂t(cid:19) hyp
Furthermore, for a fixed w ∈ X , and any smooth function fi on X , the hyperbolic heat kernel
i i i
Ki (t;z ,w ) satisfies the equation
hyp i i
lim Ki (t;z ,w )fi(z)µi (z )=fi(w ). (3)
hyp i i hyp i i
t→0ZXi
To simplify notation, we write Ki (t;z ) instead of Ki (t;z ,z ), when z =w .
hyp i hyp i i i i
Key-identity on X For i=1,2, z ∈X \E , we have the relation of differential forms
i i i i
1 1 1 ∞
g µi (z )= + µi (z )+ ∆i Ki (t;z )dt µi (z ). (4)
i can i (cid:18)4π volhyp(Xi)(cid:19) hyp i 2(cid:18)Z0 hyp hyp i (cid:19) hyp i
This relation has been established as Theorem 3.4 in [2], when X is compact. The proof given in
i
[2] applies to our case where X does admit elliptic fixed points and cusps, as long as z ∈X \E .
i i i
In [1], the above identity is extended to elliptic fixed points and cusps at the level of currents.
3 Key-identity on X
Lemma1. ThedimensionofH0(X,Ω2 )asavectorspaceoverCisg g ,andforz =(z ,z )∈X,
1 2 1 2
X
the canonical volume form is given by
1 g1 g2
µvol(z)=µ1 (z )∧µ2 (z )=− |f1(z )|2·|f2(z )|2dz ∧dz ∧dz ∧dz . (5)
can can 1 can 2 4g g j 1 k 2 1 1 2 2
1 2 Xj=1kX=1
Proof. From Ku¨nneth theorem of algebraic geometry, we have
H0 X,Ω2 =H0 X ,Ω1 ⊗H0 X ,Ω1 .
X 1 X1 2 X2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
So from the isomorphism S2(Γi)∼=H0(Xi,Ω1 ), it follows that the set
Xi
f1f2
j k 1≤j≤g1
(cid:8) (cid:9)1≤k≤g2
4
forms an orthonormal basis of H0(X,Ω2 ), which implies that
X
1 g1 g2
µvol(z)=− |f1(z )|2·|f2(z )|2dz ∧dz ∧dz ∧dz .
can 4g g j 1 k 2 1 1 2 2
1 2 Xj=1Xk=1
Furthermore, a direct calculation, shows that µ1 (z )∧µ2 (z ) = µvol(z), which completes the
can 1 can 2 can
proof of the lemma.
Theorem 2. For z =(z ,z )∈(X \E )×(X \E ), we have the relation of differential forms
1 2 1 1 2 2
1 1 1 1 1 1 1
g g µvol(z)= + · + µvol (z)+ + ×
1 2 can (cid:18)4π vol (X )(cid:19) (cid:18)4π vol (X )(cid:19) hyp 2(cid:18)4π vol (X )(cid:19)
hyp 1 hyp 2 hyp 1
∞ 1 1 1 ∞
∆2 K2 (t;z )dt µvol (z)+ + · ∆1 K1 (t;z )dt µvol (z)+
(cid:18)Z0 hyp hyp 2 (cid:19) hyp 2(cid:18)4π volhyp(X2)(cid:19) (cid:18)Z0 hyp hyp 1 (cid:19) hyp
1 ∞ ∞
∆1 K1 (t;z )dt · ∆2 K2 (t;z )dt µvol (z).
4(cid:18)Z hyp hyp 1 (cid:19) (cid:18)Z hyp hyp 2 (cid:19) hyp
0 0
Proof. The proof of the theorem follows from combining equations (4) and (5).
References
[1] Anilatmaja Aryasomayajula,Extension of a key identity, arXiv:1310.4336.
[2] J. Jorgenson and J. Kramer, Bounds on canonical Green’s functions, Compositio Math. 142
(2006), 679–700.
[3] J. Jorgenson and J. Kramer, A relation involving Rankin-Selberg L-functions of cusp forms
and Maass forms, In: B. Kr¨otz, O. Offen, E. Sayag (eds.), Representation Theory, Complex
Analysis, and Integral Geometry, 9–40, Birkh¨auser-Verlag,2012.
[4] T. Miyake, Modular Forms, Springer-Verlag,Berlin, 2006.
Department of Mathematics,
Universityof Hyderabad,
Prof. C. R.Rao Road, Gachibowli,
Hyderabad,500046, India
email: [email protected]
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