REGULARIZED PETERSSON INNER PRODUCTS FOR MEROMORPHIC MODULAR FORMS (Automorphic Forms, Automorphic L-Functions and Related Topics)

REGULARIZED PETERSSON INNER PRODUCTS FOR MEROMORPHIC MODULAR FORMS

BENKANE

ABSTRACT. We_minvestigatethe_historyof innerproductswithin the_theoryof mod‐

ularforms. We first_givethe_historyofthe_applicationsof Petersson’s_originaldef‐ inition for the inner product ofS_{2k} and then recall _Zagier’s extension to a non‐ degenerate

(but

not_{necessarilypositivedefinite)} innerproductonallholomorphic modular forms. We then recall the_historyof the so‐called _{‘‘regularization”} of the inner_producttoextend itto_{weakly holomorphic}modular forms_{originally by}Pe‐

terssonand then later_{independently}rediscovered_{by Harvey‐Moore}andBorcherds, as wellas its applications totheta lifts _by Borcherds, Bruinier‐Funke, and _many morerecentauthors. This has been_recentlyextendedtoawell‐defined inner_prod‐

uctonallweakly holomorphicmodular formsby Bringmann,Diamantis,and Ehlen. Finally,weconsider innerproductsonmeromorphicmodularforms which havepoles intheupperhalf‐plane. Petersson also definedaregularizationinthiscase_bycut‐ tingoutsmall_{neighborhoods}around each_poleoccurringinthefundamentaldomain; Bringmann,vonPippich, and theauthorhaverecentlyconstructedanextension of this _{regularization, which,} when combined with the_{regularization} of_Bringmann,

Diamantis, and_{Ehlen, yields}an innerproductthat is well‐defined and finiteonall meromorphicmodular forms.

1. INTRODUCTION

The Petersson inner

product

has a

long history

within the

theory

of

automorphic

forms. This

_expository

paperservesas abrief

sojourn through

that

history.

Petersson

[14]

provided

a well‐defined and finite

(see

Section

2)

Hermitian inner

product

on

the space

S_{2k}

of

weight

2k \in 2\mathrm{N} cusp forms on

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z}) (Petersson

considered his

inner

_product

on modular forms for much more

general

Fuchsian _groups, but for

simplicity

of the

_exposition,

we restrict ourselves to

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})).

Roughly speaking,

the

idea of Petersson’s inner

_product

is to construct a function which is invariant under

the action of

_{\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})}

and then

_integrate

over an

arbitrary

fundamental domain for

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H}

, where \mathbb{H} is the

complex

upper

half‐plane.

For

_f,

_g \in

_{S_{2k}}

, we denote Petersson’s inner

product by

\langle f, g\rangle

. The inner

product

has anumber of

applications.

Firstly,

theinner

product

is

non‐degenerate

(and

even

positive‐definite)

on

S_{2k}

,

yielding

an

orthogonal splitting;

this

splitting

may be ex‐

plicitly

realized

_by

_decomposing

intothe

_{(onedimensional)}

simultaneous

_eigenspaces

under the Heckeoperators.

Secondly,

Petersson used his inner

_product

toestablish the

Date:_August23,2016.

2010 Mathematics_{Subject Classification.} IIFII, IIF12, IIF30.

Keywordsand_{phrases. meromorphic}modularforms, inner_products.

The research of the author wassupported by _{grant project} numbers 27300314, 17302515, and 17316416of the Research Grants Council.

well‐known Petersson coefficient formula

_(see

Section 2.3 and

_particularly

Theo‐

rem

2.1).

The coefficient formula

gives

a_{way to}relate the coefficients of_cusp forms

with the inner

_product

of the the cusp forms

against

certain

distinguished

elements

called the Poincaréseries. Poincaréseriesare

generalizations

of the well‐known Eisen‐

steinsenes

E_{2k}(z) :=M=(^{ab})\displaystyle \in$\Gamma$_{\infty}\backslash \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})\sum_{cd}(cz+d)^{-2k}

,

(1.1)

where

_{$\Gamma$_{\infty}}

:=

\{\pm T^{n} : n\in \mathbb{Z}\}

with T:=

(_{01}^{11}

).

Petersson’s coefficient formula uses a

techmique

called

_{“unfolding”,}

where the sum in

(1.1)

is used to extend the

integral

over

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H}

to an

integral

over

$\Gamma$_{\infty}\backslash \mathbb{H}

. Thefundamentaldomainfor

$\Gamma$_{\infty}\backslash \mathbb{H}

is very

simple,

allowing

oneto

_explicitly

_compute

_{integral by plugging}

in Fourier

_expansions.

In

_doing

_so,Peterssonobtains the Fourier coefficients of the modular forms

_{by replacing}

the summand

_(cz+d)^{-2k}

with another

_appropriate

function.

It is natural to ask whether one can extend the inner

product

to include inner

products

with the Eisenstein series

E_{2k}

definedin

_(1.1).

Petersson’s

_original

definition suffices whenone takes the inner

product

of

E_{2k}

with a _cusp

form,

and reveals that

E_{2k}

is

_orthogonal

toallcusp forms.

However,

theinner

product diverges

when

trying

to_computethe Petersson norm

\Vert f\Vert^{2}:=(f,f\}

(1.2)

for

_{f=E_{2k}}

.

Zagier

[21]

later

managed

to extend theinner

product

tothis case and

proved

thatthe Peterssoninner

_product

on

holomorphic

modular forms is indeednon‐

degenerate,

but in

_general

it is not

_{positive‐definite}

_(in

_particular,

thenormof

E_{2k}

is

either

_positive

or

negative, depending

onthe

parity

of k

).

We nextconsiderthe inner

_product

on formsinthe_space

M_{2k}^{1}

of

weight

2k

weakly

holomorphic

modular

_forms

_(i.e.,

_meromorphic

modular forms all of whose

poles

are contained at

_cusps).

_{Unfortunately,}

the naive definition

_{usually diverges,}

even be‐

tween a_cuspform anda

weakly holomorphic

modular form. There is howeveratrick

which allows one to consider inner

products

on this _space, which _{appears to} have

been first realized

_by

Petersson

_[15]

and then later rediscovered

_{by Harvey‐Moore}

_[11]

and Borcherds

_[2].

One

_{“regularizes”}

the

_integral

over

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H} (see

Section

3).

Pe‐

tersson’s

_original

_attempt to doso involved

taking

the

Cauchy principal

vaìue of the

integral by

integrating

over a_part

\overline{J_{T}\prime}

(T\in \mathbb{R})

of the fundamental domain bounded

away from the cusp of

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H}

such that the limit of

\overline{J^{-}}_{T}

as T\rightarrow \infty becomes an

entirefundamental domain for

_{\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H}}

.

Essentially,

this isthesame as

choosing

an

ordering

on the

integral

overthe fundamental domain. Borcherds

[2],

Bruinier‐ $\Gamma$ur&e

[6],

andnumerous other authors haveused this

regularized

inner

product

to compute

thetalifts between modular formson

orthogonal

_groups.

Finally,

we

study

the inner

product

on

meromorphic

modular forms with

poles

in the upper

half‐plane.

The naive inner

product again diverges,

and one

requires

a

regularization.

Petersson

[15]

defined the

Cauchy principal

value in this case

by

cutting

out small

_{neighborhoods}

around each

_pole

and

_shrinking

thevolumeof these

neighborhoods

tozerointhe limit. Hisdefinition extended theinner

product

to many

cases, but it still

diverges

in many cases; in

particular,

the Petersson inner

product

fornon‐cusp forms

always diverges

with Petersson’s

regularization.

InSection

4,

we

Pippich,

and the author

_[5]

which _maybe combined with

_Bringmann,

_Diamantis,

and Ehlen’s

_[3]

_{regularization}

to

_yield

awell‐defined and finiteinmer

_product

on the_space

&k

of all

_meromorphic

modular forms. One

_application

of thenew

regularization

isa

formula

_relating

the

_higher

Green’s functions evaluated at

_CM‐points

with the inner

product

betweencertain

_distingu

shed

_weight

2k

_meromorphic

modular forms

_{f_{Q}(Q\mathrm{a}}

positive‐definite integral binary quadratic

form)

which

_generalize

thecusp forms

f_{k,D}

(

D>0a

discriminant)

whichfirst occurredin

Zagier’s

_paper

[20]

andwerelater used

by

Kohnen and

_Zagier

_[12]

to construct a kernel function for the Shimura

[18]

and

Shintani

_[19]

lifts between

_integral

and

_{half‐integral weight}

modular forms.

2. PETERSSON INNER PRODUCTS

2.1.

_Holomorphic

modular forms and their

_{generalizations.}

Define the

_weight

2k slashaction

_{|_{2k}}

withamatrix

M=(_{\mathrm{c}d}^{ab}

) \in \mathrm{S}\mathrm{L}_{2}(\mathbb{Z})

by

f|_{2k}M(z)

:=(cz+d)^{-2k}f

(Mz),

where Macts on\mathbb{H} via fractional linear transformations. A

weight

2k

(holomorphic)

modular

_form

_{(on \mathrm{S}\mathrm{L}_{2}(\mathbb{Z}))}

isafunction

f

:\mathbb{H}\rightarrow \mathbb{C} for which the

following

hold.

(1)

For all

_{M\in \mathrm{S}\mathrm{L}_{2}(\mathbb{Z})}

, wehave

f|_{2k}M=f

.

(2.1)

(2)

The function

_f

is

_holomorphic

on \mathbb{H}.

(3)

The function

_f

hasaFourier

expansion

of the_type

f(z)=\displaystyle \sum_{n\geq 0}a_{f}(n)e^{2 $\pi$ inz}

.

(2.2)

If

_{a_{f}(0)=0}

, thenwecall

f

a cusp

form.

More

_generally,

ifwe

replace

condition

(2)

with

meromorphicity

(resp.

holomor‐

phicity)

and condition

₍₃₎

with Fourier

_expansions

_(2.2)

with the weaker restriction

n\gg-\infty, thenweobtain the definition for

meromorphic

modular forms

(resp.

weakly

holomorphic

modular

_forms).

Later in the paper, wewill even

replace

condition

(2)

with the_property that

_f

is real

_analytic

and annihilated

_by

acertain differential _op‐

erator

_{\triangle_{2k}}

called the

_weight

2k

_{hyperbolic Laplacian}

_{(see (3.4));}

in this case, the

coefficients

_{a_{f}(n)}

in

_(2.2)

are

replaced

with coefficients

a_{f}(y;n)

which _may

depend

on the

imaginary

_{part y} of z and there is not restriction on n

(i.e.,

n \in \mathbb{Z}

_).

_Doing

so

(replacing (2)

with annihilation

by

$\Delta$_{2k}

)

_\mathrm{y}.elds the definition of a

special

class of

non‐holomorphic

modular

_forms

knownasharmonic Maass

forms.

Analogously

tothe

change

in condition

₍₂₎

from

_holomorphic

modular forms to

_meromorphic

modular

forms,

for

_{non‐holomorphic}

modular formswe_may also allow

(not

necessarily

mero‐

morphic)

singularities

intheupper

half‐plane

orat cusps. This final class of formsare

called

_polar

harmonic Maass

_forms.

In allof the above

_{generalizations,}

theone_propertywhich has remained

unchanged

is

_(2.1).

Thisisthemainessenceof the definition. Of_course,thereare

generalizations

wherethe condition

_{M\in \mathrm{S}\mathrm{L}_{2}(\mathbb{Z})}

isrestrictedto M\in $\Gamma$ forsome

subgroup

$\Gamma$\subseteq \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})

and one can

slightly

_augment

the definition of the slash _operator

|_{2k}

(for

example,

allowing

a

character)

orallow

k\in \mathbb{Q},

k\in \mathbb{R}or evenk\in \mathbb{C},but

essentially

these

changes

2.2. Definition of the inner

_{product. Considering}

the variables z and7as inde‐

pendent variables,

note that for a

weight

2k modular form

f(z)

, the function

\overline{f(z)}

satisfies

_weight

2k

_modularity

as afunction of7.

_Furthermore,

_writing

_{z=x+iy\in \mathbb{H},}

the function

_y^{2k}

satisfiessimultaneous

_weight

-2k

_modularity

inbothzand2because

{\rm Im}(Mz)={\rm Im}(\displaystyle \frac{az+b}{cz+d}) =\frac{{\rm Im}((az+b)(c\overline{z}+d))}{|cz+d|^{2}}=\frac{y}{|\mathrm{c}z+d|^{2}},

whereweused thefactthat ad‐bc=1.

Petersson

_[14]

then realized

_that,

forfunctions

_f

andg

satisfying

(2.1) (i.e.,

satisfying

modularity)

for

_{\mathfrak{N}M\in \mathrm{S}\mathrm{L}_{2}(\mathbb{Z})}

, the function

f(z)\overline{g(z)}y^{2k}

is

_{\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})}

‐invariant.

_Moreover,

the metric

\displaystyle \frac{dxdy}{y^{2}}

isalso

_{\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})}

‐invariant. Hencethe

_integral

\displaystyle \{f, g\rangle :=\int_{\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})\backslash \mathbb{H}}f(z)\overline{g(z)}y^{2k}\frac{dxdy}{y^{2}}

(2.3)

is well‐defined whenever it converges

absolutely. Using

bounds for cusp forms

(in

particular, they exponentially decay

as _{y\rightarrow\infty}

),

one canshow that the

integral

(2.3)

converges

absolutely

for

f,g

\in

S_{2k}

. This

exponential decay

also suffices to show

convergencewhen

taking

the inner

product

between

f\in S_{2K}

and the Eisensteinseries

E_{2k}

definedin

_(1.1).

2.3. Petersson coefficient formula. The Petersson coefficient formulauses an ex‐

plicit

evaluation of theinner

_product

to_computethe Fouriercoefficients

_(in

the_expan‐

sion

_(2.2))

of modularforms. To describethis

_result,

we

require

the classical Poincaré

series

_{(see [16, 17])}

P_{2k,m}(z):=\displaystyle \sum_{M\in$\Gamma$_{\infty}\backslash \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})}$\varphi$_{m}|_{2k}M(z)

,

(2.4)

where

_{k\in \mathrm{N}_{\geq 2}}

and for m\in \mathbb{Z}

$\varphi$_{m}(z):=e^{2 $\pi$ irnz}.

These _converge

_locally

and

_{absolutely uniformly.}

For m =0, the Poincaré series is

precisely

the Eisenstein series

_(1.1),

while for m > 0 we have

P_{2k,m}

\in

_{S_{2k}}

and for m<0 wehave

P_{2k,m}\in M_{2k}^{!}.

Theorem 2.1

_(Petersson

coefficient

_formula).

_{If f\in S_{2k}}

and m\in \mathrm{N}, then

\displaystyle \{f, P_{2k,rn}\rangle=\frac{(2k-2)!}{(4 $\pi$ m)^{2k-1}}a_{f}(m)

.

Sketch

_{of proof. Plugging}

inthe definition

_(2.4)

of the Poincaré series

_{P_{2k,m}}

and choos‐

ing

a

fundamental

domain \mathcal{F}for

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H}(\mathrm{a}

“nice” connected set of

_{representatives}

z\in \mathbb{H}of the orbits of

_{\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H}}

under fractional linear

_{transformations),}

we

unfold

the

_integral

on the left‐hand side

by rewriting

(formally,

but this isvalid because of

\displaystyle \int_{\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})\backslash \mathbb{H}}f(z)\sum_{M\in$\Gamma$_{\infty}\backslash \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})}(c^{\frac{r}{z}}+d)^{2k}y^{2k}\frac{dxdy}{y^{2}}\overline{$\varphi$_{n}(Mz)}

=\displaystyle \sum_{M\in$\Gamma$_{\infty}\backslash \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})}\int_{F}f(Mz)$\varphi$_{n\mathrm{z}}(Mz){\rm Im}(Mz)^{2k}\frac{dxdy}{y^{2}}

=\displaystyle \sum_{M\in$\Gamma$_{\infty}\backslash \mathrm{S}\mathrm{L}_{2}(\mathbb{Z})}\int_{MF}f(z)$\varphi$_{m}(z)y^{2k}\frac{dxdy}{y^{2}}=\int_{$\Gamma$_{\infty}\backslash \mathbb{H}}f(z)\overline{$\varphi$_{m}(z)}y^{2k}\frac{dxdy}{y^{2}}

.

(2.5)

Since the fundamental domain for

_{$\Gamma$_{\infty}\backslash \mathbb{H}}

is very

simple,

this

_unfolding

_argument

results

inthe double

_integral

\displaystyle \int_{0}^{\infty}\int_{0}^{1}f(z)\overline{$\varphi$_{m}(z)}y^{2k}\frac{dxdy}{y^{2}}.

The

_integral

over x

essentially picks

off the mth coefficient and then

explicitly

com‐

puting

the

_integral

over_y

yields

the claim. \square

2.4.

_Orthogonal

_splitting.

The inner

_product

on

S_{2k}

is

positive‐definite. Hence,

by

the Gram‐Schmidtprocess, one canconstruct anorthonormalbasis. A

particular

choiceof the basis elementsturns out to beverynatural.

Thereare certain operators

T_{n}

known as the Hecke

operators

and defined for each

n\in \mathrm{N}

_by

_(these

are normalized

differently

in different books and_papers for various

purposes, but the normalization isnot

important

for the discussion at

hand)

f|{}_{2k}T_{n}:=\displaystyle \sum_{M\in \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})\backslash \mathcal{M}_{n}}f|_{2k}M,

where

_{$\Lambda$ t_{n}}

denotes theset of2\times 2

_integral

matrices with determinant n. The Hecke

operators

commuteandareHermitian with_respecttothe Peterssoninner

product. By

the

_{Spectral Theorem,}

one_maytherefore

diagonalize

toobtain simultaneous

eigenfunc‐

tionsunder

\mathrm{a}\mathrm{U}T_{n}

. Thesesimultaneous

eigenfunctions

areknownasHecke

eigenforms.

The Hecke

_{eigenforms f}

\in

_{S_{2k}}

are often normalized to have

a_{f}(1)

= 1

, but another

natural normalizationtotakeis

_{\Vert f\Vert^{2}=1}

,wherethe Peterssonnorm

\Vert\cdot\Vert^{2}

wasdefined

in

_(1.2).

The Heckeoperators

satisfy

what is knownas

multiplicity

_one, whichmeans

that the

_eigenspaces

of simultaneous

_{eigenfunctions}

under all Hecke operatorsare all

one‐dimensional

_(indeed,

_{they satisfy}

amuch_strongercondition knownasstrongmul‐

tiplicity

one).

Hence,

fortwo distinctHecke

_{eigenforms f,g\in S_{2k}}

,there exists n\in \mathrm{N}

for which the

_eigenvalues

_{$\lambda$_{f}(n)}

and

_{$\lambda$_{g}(n)}

differ.

_However,

since the Hecke _operators

are

Hermitian,

wehave

$\lambda$_{f}(n)\langle f,g\rangle=\langle$\lambda$_{f}(n)f,g\rangle=\{f|_{2k}T_{n},g\rangle=(f,g|_{2k}T_{n}\rangle=\{f, $\lambda$_{g}(n)g\rangle=$\lambda$_{g}(n)\langle f,g\rangle.

Since

_{$\lambda$_{f}(n)}

_\neq

_{$\lambda$_{g}(n)}

, this leads to a contradiction if

\langle f,g}

\neq

0. We thus conclude

that

_f

and_gare

orthogonal

toeach other. Hence the

splitting

of

S_{2k}

into

eigenspaces

precisely yields

the

_{orthogonal splitting,}

with the orthonormal basis

_{given by}

theHecke

eigenforms

normalizedsuch that

_{\Vert f\Vert^{2}=1.}

We notethat the othernormalization

_{a_{f}(1)=1}

isalso natural. Under this normal‐

ization

_(and

_{appropriately normalizing}

the Hecke

_operators),

thecoefficients

_{a_{f}(n)}

and

the

eigenvalues

$\lambda$_{f}(n)

coincide. This realization

_{“de‐mystifies”}

thecoefficients of the Hecke

_eigenforms

and

_plays

an

_important

rolein

understanding

Fourier

expansions.

3. INNER PRODUCTS FOR WEAKLY HOLOMORPHIC MODULAR FORMS

3.1. The

_{regularization}

of

_{Petersson, Harvey‐Moore,}

and Borcherds and its

extension. For

_f,

_{g\in M_{2k}^{!}}

,the

integral

(2.3)

generally diverges.

Peterssonestablished

a

Cauchy principal

value for the

integral

as a

partial

solutiontothis

problem. Firstly,

one chooses a

specific

fundamental domain for

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H}

. We choose the standard

fundamental

domain

_(for

_simplicity,

we take the closed fundamental

domain;

this is

easier to write

_down,

but

_technically

there are

points

on the

boundary

which are

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})

‐equivalent; however,

sincewewill

ultimately integrate

overitand the

boundary

isameasure zero_set, this is irrelevant forour

consideration)

\displaystyle \mathcal{F}:=\{z\in \mathbb{H}:|z|\geq 1, -\frac{1}{2}\leq x\leq\frac{1}{2}\}.

Instead of

_integrating

over\overline{Jr}in

(2.3),

we

integrate

over a

cut‐off fundamental

domain

whose closure does not include the cusp onthe

boundary

of the chosen fundamental

domain. In our _case,the _cuspis i\infty and the cut‐off fundamental domainis

given

by

\displaystyle \mathcal{F}_{T}:=\{z\in \mathbb{H}:|z|\geq 1, y\leq T, -\frac{1}{2}\leq x\leq\frac{1}{2}\}.

For

_{f,g\in M_{2k}^{!}}

, Peterssonthen defined the

regularized

inner

product

(see

[15])

\displaystyle \{f,g\rangle := $\tau$\rightarrow\infty \mathrm{h}\mathrm{m}\'{I}_{\mathcal{F}_{T}}f(z)\overline{g(z)}y^{2k}\frac{dxdy}{y^{2}}

.

(3.1)

The

_key

to the above

_{regularization}

is that it

_{essentially gives}

an

ordering

to the

integrals

over x and_y.

This constructionwasfurther

independently

rediscovered and extended

by Harvey‐

Moore

_[11]

and Borcherds

_[2]

_by

_multiplying

the

_{integrand by}

_y^{S}

for some s \in \mathbb{C}

with

_{{\rm Re}(s)}

\gg 0 and then

_taking

the constant term of the Laurent

_expansion

of the

meromorphic

continuation

_(in

_s)

at s=0.

Onecan usethe

regularized

inner

product

toshow that for m<0 the Poincaréseries

P_{2k,m}

, defined in

(2.4),

is

orthogonal

to cuspforms. This wasshown

by

Peterssonina

muchmore

general setting

in

[15,

Satz

4].

The

_{regularization}

of Petersson

_{/\mathrm{H}\mathrm{a}\mathrm{r}\mathrm{v}\mathrm{e}\mathrm{y}‐Moore/Borcherds}

doesnot

_always

_converge,

however. In

_particular,

Petersson found a _necessary and sufficient condition for his

regularization

(3.1)\mathrm{t}\mathrm{o}

converge

(see

[15,

Satz

1])

and Peterssonnormsonce

again

_posea

problem,

as

they

did for the Eisensteinseries. This

problem

has been

recently

resolved

by Bringmann, Diamantis,

and Ehlen

_[3],

who wereable to extend the

_{regularization}

in a_way so that theinner

product

\langle f, g\rangle

is well‐defined and finite for all

f,

g\in M_{2k}^{!}.

We donot

_give

_anyof thetechnicaldetails

_here,

but the reader is

_encouraged

tolook

at

_[3,

Section

_3,

and in

_particular

Theorem

_3.2].

3.2. Thetalifts. Theinner

_product

has been used

_by

manyauthors

(for

example,

in

[2]

and

_[6])

to obtain theta lifts from modular forms ofone

type

tomodular forms of

another

_type.

To

_give

a

rough idea,

onedefines atwo‐variable theta function

$\Theta$(z, $\tau$)

whichismodularinboth variables

_(one

calls this function the theta

_kernel),

but which satisfies adifferent kind of

modularity

in each variable

(for

example,

_supposethat it

function of $\tau$

).

Taking

the inner

product

in one variable

_against

another function

f

satisfying

thesame

_type

of

modularity

then

_yields

a newfunctioninthe othervariable

satisfying

the other

_type

of

_modularity.

In other

_words,

in the

_{example above,}

if

_f

satisfies

_weight

2k

_modularity,

then

$\Phi$(f)( $\tau$):=\langle $\Theta$(\cdot, $\tau$) , f\rangle

satisfies

_weight

_k+1/2

_modularity.

This

_yields

atheta

lift

$\Phi$from

weight

2kmodular

forms to

_weight

_k+1/2

modular forms. The

_example

illustrated aboveis Shintani’s

construction

_[19]

of his lift from

_{integral weight}

to

_{half‐integral weight}

modular forms

and the lift in the

_opposite

direction can be shown to be one of Shimura’s lifts

[18]

from

_{half‐integral weight}

to

_{integral weight}

_(see

_[13]

and

_[12]

fortwoalternative

_options

for the theta

_kernel).

Note:

_although

we do not define

_{half‐integral weight}

modular forms

_here,

one _may

simply

thinkof theseas

generalizations

of modular forms where

the slash _operator is

_{slightly augmented}

to resolve the issue that the square root is

multi‐valued and then

_modularity

is

_again

defined

_by

_(2.1).

Lifts from

_“simpler”

spaceswith

special properties

often

yield

strangeor

exceptional

modular forms whichcanbe usedtounderstand or narrowdown

conjectures

that are

often

_precisely

fakeonthe

_image

or

pre‐image

of such lifts. For

example,

the \mathrm{S}}\mathrm{u}mura

lift

_generally

sends cusp forms to cusp

forms,

but there is an

exceptional

class of

forms known as_unary theta functions in

weight

3/2

whichare_cusp forms but whose

image

underthe Shimura hft isanEisenstein series. These_unarytheta functions are

also

_{counter‐examples}

tothe

_{Ramanujan‐Petersson}

_conjecture,

which statesthat the coefficients of

_weight

_{$\kappa$\displaystyle \in\frac{1}{2}\mathbb{Z}}

cusp forms

f satisfy

|a_{f}(n)|\ll f, $\epsilon$ n^{\frac{ $\kappa$-1}{2}+ $\epsilon$}.

The coefficients of the unarythetafunctionsgrow like

n^{1/2}

,

contradicting

the

conjec‐

ture in this wide of

_{generality. However,}

for

_{integral weight}

cusp forms

f

\in

_{S_{2k},}

the

_conjecture

is a celebrated result of

Deligne

[7]

and it is

conjectured

that the

Ramanujan‐Petersson conjecture

holds in

_{half‐integral weight}

as

long

as

f

is

orthog‐

onalto unarythetafunctions.

3.3.

_Computation

of the inner

_{product by}

the Brunier‐Funke

_pairing.

For

f,g\in M_{2k}^{!}

, wenext describe awaytocompute theinner

product

between thesetwo

forms. There is anatural function G

satisfying weight

2-2k associated with_g. The

inner

_product

between

_f

and_{g is}then

_given

_by

a

pairing

between thefunction G and

f given by

\displaystyle \{f, G\} :=\sum_{n\in \mathrm{Z}}a_{f}(-n)a_{G}^{+}(n)

,

(3.2)

where

_{a_{G}^{+}(n)}

is the nth coefficient of the

_holomorphic

_part of the Fourier

_expansion

(which

has the same

shape

as

(2.2)).

In

particular,

wehave

\{f,g\}=\{f, G\}

.

(3.3)

The

_pairing

is useful for

_computing

inner

_products

because

_{only finitely}

many terms

in

_(3.2)

are non‐zero.

Roughly speaking,

the

_pairing

is shown

_by

_using

Stokes Theorem to evaluate the

integral

instead of the

_unfolding

method described in Section 2.3. When

_applying

Stokes

_Theorem,

a

pre‐image

G of_{g under theoperator}

$\xi$_{2-2k}

:=2iy^{2-2k}\overline{\frac{\partial}{z}}

naturally

appears. Sincegis

weakly holomorphic,

wehave

$\Delta$_{2-2k}(G)=-$\xi$_{2k}(g)=0,

where

$\Delta$_{2-2k}:=-$\xi$_{2k}0$\xi$_{2-2k}

(3.4)

isthe

_weight

2-2k

_{hyperbolic Laplacian. Therefore,}

the

pre‐image

Giswhatisknown

as a

weight

2-2k harmonic Maass

form

(i.e.,

it satisfies

weight

2-2k

modularity,

it

isannihilated

_by

$\Delta$_{2-2k}

, andit grows at mostlinear

exponentially

towards the

cusps).

The

_pairing

was first introduced

by

Bruinier and Funke in

[6].

Its connection to

inner

_products

defined as

regularized integrals

wasthen realized inanumberofcases

by

many authors andone_may

interpret

the recentresults in

[3]

as _\mathrm{g}

.ving

an

analytic

interpretation

viaa

regularized

integral

for the Bruinier‐Funke

pairing

inthe

general

casefor_any

arbitrary

f,g\in M_{2k}^{1}.

4. INNER PRODUCTS FOR MEROMORPHIC MODULAR FORMS

We would now like to define aninner

product

on

arbitrary meromorphic

modular

forms

_{f,g\in S_{2k}}

.

However,

an

arbitrary meromorphic

modular form

f\in \mathcal{S}_{2k}

maybe

decomposed

into two

_pieces,

oneof which

only

has

poles

atthe_cusps

₍

\mathrm{i}.\mathrm{e}.,it is in

M_{2k}^{1}

)

andoneof which

only

has

poles

inthe_upper

half‐plane

(vanishing

towards all

cusps);

we can forms of the second type

weight

2k

meromorphic

cusp

forms

and denote the

subspace

of such forms

_by

\mathrm{S}_{2k}

. It thus

essentially

sufficesto consider inner

products

between forms

_f,g

\in

_{\mathrm{S}_{2k}}

_{(technically,}

we also have to take inner

_products

between forms

_{f\in M_{2k}^{!}}

and

_{g\in \mathrm{S}_{2k}}

, but

hybrid approaches

for the

regularizations

will work infull

_generality

andwe

ignore

thedetails

here).

4.1.

_{Regularization}

of Petersson. The idea that Petersson usedto

_generalize

_(2.3)

is verysimilar to the idea used in the

_{regulaxization(3.1).}

Instead of

_cutting

off the

fundamental domain away from i\infty, one cuts out small

neighborhoods

around each

pole

3 of

f

or_g and then shrinks the

hyperbolic

volume of the

neighborhoods

to zero

inalimit. In

particular,

for

3\in \mathbb{H}

define the ball

\mathcal{B}_{ $\epsilon$}(3):=\{z\in \mathbb{H}:r_{f}(z)< $\epsilon$\},

where

_{r_{t}(z) :=|X_{t}(z)|}

with

X_{s}(z):=\displaystyle \frac{z- $\delta$}{z-\overline{3}}.

The functions

_{r_{3}(z)}

are

naturally

connectedtothe

hyperbolic

distance

d(z,3)

between z and

3=\mathfrak{z}_{1}+i_{f2}

in\mathbb{H} via the formula

r_{t}(z)=\displaystyle \tanh(\frac{d(z,f)}{2})

;

recall that the

_hyperbolic

distance maybe

expressed through

(see

p. 131 of

[1])

\cosh

_(d(

z,

ồ)

)=1+\displaystyle \frac{|_{Z-f}|^{2}}{2y32}

.

(4.1)

Let

_[z1],

..._,

[z_{r}]

\in

\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H}

be the distinct

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})

‐equivalence

classes of all of

interior of

_{$\Gamma$_{z\ell}\mathcal{F}^{*}}

, where

$\Gamma$_{\`{i}}

is the stabilizer of $\delta$ in

\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})

. Petersson’s

regularized

inner

_product

isthendefined

_by

\langle f,g\rangle:=$\epsilon$_{1},\ldots,$\epsilon$_{r}\rightarrow 0+1\dot{\mathrm{m}}\`{I}_{\mathcal{F}^{*}\backslash (\bigcup_{l=1}^{r}B_{ $\epsilon$ p}(z\ell))^{f(z)\overline{g(z)}y^{2k}\frac{dxdy}{y^{2}}}}

.

(4.2)

A _necessary and sufficient condition for the convergence of the

regularization

(4.2)

is

_given

in

_[15,

Satz

_1].

_Furthermore,

certain Poincaré series related to the

_elliptic

expansions

(Petersson

proved

an

elliptic

coefficient formula as

well;

cf.

[15,

Satz

9])

with

_poles

intheupper

half‐plane

were also shownto be

_orthogonal

to cusp formsin

[15,

Satz

_7].

Once

_again,

Petersson’s _necessary and sufficient condition

_implies

that

his

_{regularization diverges}

in

_particular

when

_evaluating

Peterssonnormsfor elements

of

_{\mathrm{S}_{2k}}

which are not cuspforms.

4.2. Anew

regularization.

Since Petersson’s

regularization

stillsometimes

diverges,

one

_requires

a further

regularization;

we recall the construction from

[5].

Roughly

speaking,

the

_integrand

in

_(2.3)

is

_{multiplied by}

an

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})

‐invariant function

H_{s}( $\tau$)

whichremovesthe

poles

of the

integrand

whenever

{\rm Re}(s)

is

sufficiently large.

We then

take theconstant termof the Laurent

_expansion

around s=0tobeour

regularization.

To bemore

precise,

let

[z1],

\cdots

,

[z_{r}]

\in \mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\backslash \mathbb{H}

be the distinct

\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})

‐equivalence

classesof all of the

_poles

of

_f

andgand define

(f,g\displaystyle \rangle:=\mathrm{C}\mathrm{T}_{s\triangleleft}-(\int_{\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})\backslash \mathrm{r}\mathrm{i}}f(z)H_{s}(z)\overline{g(z)}y^{2k}\frac{dxdy}{y^{2}})

,

(4.3)

where

H_{s}(z)=H_{s $\epsilon$ zz_{r}}1,\displaystyle \ldots,r,1,\ldots,(z):=\prod_{\ell=1}^{r}h_{s\ell,z_{l}}(z)

.

Here

h_{$\epsilon$_{\ell},z_{l}}(z):=r_{z_{\ell}}^{28\ell}(Mz)

,

with

_{M\in \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})}

chosen such that Mz\in \mathcal{F}^{*}. Moreover

CT,

=0 denotes theconstant

term in the Laurent

_expansion

around s_{1} = _{s_{2}} =\cdots = _{s_{r}} =0 of the

meromorphic

continuation

_{(if existent).}

In thesame sense that the results in

[3]

_may be viewed as an

analytic

definition

for a

regularized

inner

product satislYing

the Bruinier‐IMnke

pairing

for

arbitrary

f,

g \in

M_{2k}^{1}

, the above

regularization

may be viewed as an

analytic

definition for a

regularized integral giving

a similar

pairing

for all

f,g

\in

_{\mathrm{S}_{2k}}

.

However,

instead of

defining

the

_pairing

viathe Fourier

_expansions,

the

_pairing

is defined via the

_elliptic

expansions

of

_f

and a

weight

2-2k

polar

harmonic Maass

form

(i.e.,

a harmonic

Maass form with

_{singularities}

in the upper

half‐plane)

G which is a

preimage

of_g

under the

_$\xi$

_‐operator. To describe the

_pairing,

the

_elliptic

_expansion

of

_{f\in \mathrm{S}_{2k}}

around

f\in \mathbb{H}

is

_{given by}

f(z)=(z-\displaystyle \overline{f})^{-2k}\sum_{n\gg-\infty}a_{f_{3}},(n)X_{f}^{n}(z)

.

(4.4)

For the

_polar

harmonic Maass form G, we

again

denote the coefficients ofits mero‐

Denoting

32

:={\rm Im}(f)

and

writing

$\omega$_{t} for the sizeofthestabilizer

$\Gamma$_{3}

of3 in

\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})

,

the

_pairing

is

_{given by}

_(see

_[4,

_Proposition

_6.1])

\displaystyle \{f, G\}:=\sum_{\mathrm{r}\in \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})\backslash \mathbb{H}}\frac{ $\pi$}{32$\omega$_{ $\delta$}}\sum_{n\in \mathrm{Z}}a_{f,f}(n)a_{G}^{+}

,ồ

(-n-1)

.

(4.5)

It is

_{again important}

to

_emphasize

that the

_{pairing gives}

a formula for the inner

product

with

_only

_finitely

_manycoefficientsin

_(4.5)

non‐zero. In

comparison,

Petersson

evaluated his inner

_product

_{(3.1) (resp. (4.2))}

on

[15,

_pages

42−43]

via the Fourier

(res.

elliptic)

coefficients of the forms

_f

and_g

_themselves,

but his evaluationis

_given

as an infinite _sum, so one can

only

obtain an

approximation

for the inner

product

by computing

the Fourier

_(resp.

_elliptic)

coefficients. In other

_words,

Petersson’s

constructionsarebetter inthesensethat

they

are

_given

intermsofthecoefficientsof

the

_{original functions,}

while oneis

required

to introducenew functionsto determine

(3.2)

and

_(4.5),

but thesumsin these

pairings

areinstead finite.

4.3.

_Higher

Greens functions. The

_{regularization}

_(4.3)

wasusedin

[5]

to_compute

theinner

_product

between

f_{Q}(z)=f_{k,-D,[Q]}(z):=D^{\frac{k}{2}}\displaystyle \sum_{Q\in[Q]}\mathcal{Q}(z, 1)^{-k}

for

_{positive‐definite integral binary quadratic}

forms

_Q

of discriminant -D. These are

weight

2k

_meromorphic

modular forms which have

_poles

of order k atthe

_unique

zero

$\tau$_{Q} of

Q

in\mathbb{H}. The evaluation of the inner

product

betweentwosuch functions is done

by

again

ưm_\mathrm{g} Stokes Theorem to rewrite the inner

product

as the

pairing

(4.5)

in

terms of the

_elliptic

coefficients of

_{f_{Q}}

and the

_elliptic

coefficients of the

_meromorphic

part

ofa

polar

harmonic Maass form

g_{Q}

associated with

f_{Q}

via the

$\xi$

‐operator.

It

thenremains to

_explicitly

_computethe

_elliptic

coefficients

_occurring

in

_(4.5).

In

_particular,

_choosing

twosuch

_{binary quadratic}

forms

_Q

and

\mathcal{Q}

,theinner

product

between

_{f_{Q}}

and

_{f_{Q}}

is related to the

_higher

Green’s

_{function G_{k}}

: \mathbb{H}\times \mathbb{H}\rightarrow \mathbb{C}, which is

_uniquely

characterized

_by

the

_following

_properties:

(1)

G_{k}

is asmooth real‐valued function on

\mathbb{H}\times \mathbb{H}\backslash \{(z, $\gamma$ z)| $\gamma$\in $\Gamma$, z\in \mathbb{H}\}.

(2)

For

_{$\gamma$_{1},$\gamma$_{2}\in $\Gamma$}

, wehave

G_{k}($\gamma$_{1}z,$\gamma$_{2f)}=G_{k}(z, $\delta$)

.

(3)

Denoting

$\Delta$_{0,z}

:=-4y^{2}\displaystyle \frac{\partial}{\partial z}\frac{\partial}{z}

) wehave

$\Delta$_{0,z}(G_{k}(z,3))=$\Delta$_{0_{i}},(G_{k}(z, $\delta$))=k(1-k)G_{k}(z, $\delta$)

.

(4)

As z\rightarrow 3

G_{k}(z, $\delta$)=2$\omega$_{f}\log(r_{f}(z))+O(1)

.

(5)

Asz

approaches

a_cusp,

G_{k}(z, $\delta$)\rightarrow 0.

These

_higher

Green’s functions havea

long history, appearing

as

special

cases of the

resolvent kernel studied

_{by Fay}

_[8]

and

_{investigated thoroughly by Hejhal}

in

_[10],

for

example.

Gross and

_Zagier

_[9]

_conjectured

that their evaluations at

_CM‐points

are

essentially logarithms

of

_{algebraic numbers,}

which has been sinceproven inanumber

ofcases. Tostatethe connection withinner

_products,

let

_{$\beta$(a, b)}

:=\displaystyle \int_{0}^{1}t^{a-1}(1-t)^{b-1}dt

be the beta

_function,

and let

_{\mathcal{Q}_{-D}}

denote theset of

_{positive‐definite integral binary}

quadratic

forms of discriminant -D<0.

Evaluating

the

elhptic

coefficients in

(4.5)

Theorem 4.1

_(Theorem

1.5 of

_[5]).

For

_Q\in

_{\mathcal{Q}_{-D_{1}}}

and

_{\mathcal{Q}\in}

_{\mathcal{Q}_{-D_{2}}}

_{(-D_{1},}

_{-D_{2}}

<0

discriminants)

with

_{[$\tau$_{Q}]\neq[$\tau$_{Q}]_{f}}

wehave

\displaystyle \{f_{Q}, f_{Q}\rangle=-\frac{ $\pi$(-4)^{1-k}}{(2k-1) $\beta$(k,k)}\frac{G_{k}($\tau$_{Q},$\tau$_{Q})}{$\omega$_{$\tau$_{Q}}$\omega$_{$\tau$_{Q}}}.

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DEPARTMENT OF_MATHEMATICS, UNIVERSITYOF HONG_{KONG, POKFULAM,} HONG KONG E‐mail address: bkaneQmaths.\mathrm{b}\mathrm{k}\mathrm{u}.hk