# New and old Saito-Kurokawa lifts classically via $L^{2}$ norms and bounds on
their supnorms: level aspect

Pramath Anamby  Department of Mathematics
Indian Institute of Science Education and Research
Pune – 411008, India<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Soumya Das  Department of
Mathematics
Indian Institute of Science
Bengaluru – 560012, India<EMAIL_ADDRESS>
###### Abstract.

In the first half of the paper, we lay down a classical approach to the study
of Saito-Kurokawa (SK) lifts of (Hecke congruence) square-free level,
including the allied new-oldform theory. Our treatment of this relies on a
novel idea of computing ranks of certain matrices whose entries are
$L^{2}$-norms of eigenforms. For computing the $L^{2}$ norms we work with the
Hecke algebra of $\mathrm{GSp}(2)$.

In the second half, we formulate precise conjectures on the $L^{\infty}$ size
of the space of SK lifts of square-free level, measured by the supremum of its
Bergman kernel, and prove bounds towards them using the results from the first
half. Here we rely on counting points on lattices, and on the geometric side
of the Bergman kernels of spaces of Jacobi forms underlying the SK lifts.
Along the way, we prove a non-trivial bound for the sup-norm of a Jacobi
newform of square-free level and also discuss about their size on average.

###### Key words and phrases:

Sup-norm, Bergman kernel, Jacobi forms, Saito-Kurokawa lifts of level $N$,
norm relations

###### 2020 Mathematics Subject Classification:

Primary 11F46, 11F50, Secondary 11F37

##  1\. Introduction

One of the primary objectives of this paper is to initiate a study of the
classical sup-norm problem in the context of holomorphic Siegel cusp forms of
degree $2$ (denoted as $S_{k}^{2}(N)$) in the level aspect. The sup-norm
problem is in a nascent stage for higher rank groups – especially for the
symplectic groups, which are the objects of interest here – and only recently
some results have been obtained: viz. Das-Krishna [19], Kramer-Mandal [39],
Saha et al [18] in the scalar weight aspect, or the restriction-to-compact
setting in the Laplace eigenvalue aspect by Blomer-Pohl [11]. It is well known
that automorphic forms on higher rank groups, which do not have global
Whittaker models, like the case of holomorphic Siegel modular forms, pose
significant obstacles to analytical questions. There are two distinct aspects
of the problem: one is to obtain the correct $L^{\infty}$-size of the space
(referred to as ‘size’ henceforth) measured by the supremum of its Bergman
kernel (BK for short); and second, to obtain a good bound on the size of a
single eigenform. We briefly discuss these topics below in the current
context.

As an initial investigation, in this paper we focus our attention on the
subspace of Saito-Kurokawa lifts (SK lifts in short) of (Hecke congruence)
level $N$ which is a subspace of $S^{(2)}_{k}(N)$ obtained as functorial
lifting from the space of holomorphic elliptic cusp forms, denoted by
$S_{k}(N)$. We denote the space of the SK lifts by $\mathrm{SK}_{k}(N)$. See
the works of R. Schmidt [50] for a representation-theoretic treatment where
they arise via CAP representations, and that of Eichler-Zagier [22] and T.
Ibukiyama [28] for a classical construction. Both approaches will play a role
in this paper, but the classical setting will be predominant. Even though it
is helpful that SK lifts are related to $\mathrm{GL}(2)$ objects; but in the
context of sup-norm questions, some of the standard tools that are needed
(viz. the geometric side of the Bergman kernel), are not available at our
disposal in a useful way. In this way, the problem is nicely balanced.

Also as far as we are aware, nothing is known about the correct size of
$\mathrm{SK}_{k}(N)$. Our first task in this direction is then to formulate a
conjecture about this and test it over certain subspaces. We easily get a
global lower bound (viz. $N^{-2}$), and can check that at least the same
appears as an upper bound for the oldspace. Even this requires some work, see
sections 11 and 12.1.1.

With the above remarks in view, the primary objectives of this paper are
twofold:

  1. (i)

To describe the new-oldform theory in the space of SK lifts in the classical
language for square-free levels, which is interesting in its own right, and

  2. (ii)

Use this description to obtain bounds towards the sizes of various objects
related to SK lifts.

Throughout this paper, we call $F$ to be a Hecke eigenform if it is an
eigenfunction of all Hecke operators $T_{S}(n)$ with $(n,N)=1$. The same
convention will apply to all relevant spaces of modular forms.

###  1.1. Old and new SK lifts vs. old and new EZI lifts

Let $k$ be even. We define the space of SK lifts to be the span of those Hecke
eigenforms $F\in S_{k}^{(2)}(N)$ for whose away-from-level spinor $L$-function
factorizes as $\displaystyle
L^{N}(F,s)=\zeta^{N}(s+1/2)\zeta^{N}(s-1/2)L^{N}(f,s)$. We refer the reader to
Section 4 for more details. New and oldforms therein are just the new and
oldforms in $S^{(2)}_{k}(N)$ which are SK lifts. The newforms in
$S_{k}^{(2)}(N)$ are as described in [49]. The classical Hecke equivariant
liftings of Eichler-Zagier-Ibukiyama (EZI in short, see e.g., Figure 1 and
subsection 4.3 for more details) provide a convenient description of most of
the SK lifts but unfortunately do not account for all of the oldforms (cf.
Figure 2; subsection 4.7). We make some efforts to clarify this picture below
in the classical setting. As far as we are aware, the classical theory of SK
lifts for higher levels is not written down in the literature.

The first part of the paper is devoted to the treatment of (i). To summarize:
we will show that the EZI lifts of newforms (from either elliptic or Jacobi
forms) account for all the newforms in the SK space; however not so for all
the oldforms. Here corresponding to each Jacobi form $\phi$ of level $N$ there
are three oldforms in level $Np$ for $p\nmid N$, one of which is not an EZI
lift.111It is an open problem to construct this ‘exceptional’ oldform (e.g.,
to construct $F|W_{p}$, where $F$ SK lift of level $1$), as a lifting from a
suitable space of Jacobi forms – in the example, perhaps those come from
Jacobi forms on the lattice $p\mathbf{Z}+\mathbf{Z}$. These may be visualized
from the diagrams given below, where $\mathcal{S}$, $\mathrm{EZ}$,
$\mathrm{EZI}$ denote the Shintani, Eichler-Zagier, Eichler-Zagier-Ibukiyama
maps respectively.

${f}$${h}$${\phi}$${{F_{f}}}$$\scriptstyle{\mathrm{EZ}}$$\scriptstyle{\mathrm{EZI}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\mathrm{SK}}$
Figure 1.  The SK and EZI maps on the newspace with $f\in S_{2k-2}^{new}(N)$.
${f}$${h}$${\phi}$${{F_{f}}}$${{F_{f}|W_{p}}}$${f}$${{f|U(p)}}$${{h|U(p^{2})}}$${{\phi|U_{J}(p)}}$${{F_{f}|U_{S}(p)}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\mathrm{EZ}}$$\scriptstyle{\mathrm{EZ}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\mathrm{EZI}}$$\scriptstyle{W_{p}}$$\scriptstyle{\mathrm{EZI}}$$\scriptstyle{\mathrm{SK}}$$\scriptstyle{\mathrm{SK~{}but~{}not~{}EZI}}$$\scriptstyle{\mathrm{SK}}$
Figure 2. The SK and EZI maps on the oldspace with $f\in S_{2k-2}(N/p)$.

As discussed in [50], there are four oldforms in level $Np$ in the old-class
of any cusp form $F$ of level $N$ where $(p,N)=1$, which reduces to three for
SK lifts. This feature is nicely captured in Theorem 1.2 (3) given below. In
fact, there are $3$ or $4$ oldforms in the old class of $F$ according as $F$
is an SK lift or not! This gives a new characterization of SK lifts (cf. [47,
23]). We will also produce an explicit orthonormal basis for the old and new
spaces. Even though the above information is probably in principle available
from the representation theoretic context; we believe that the classical
treatment is not, even though desirable222Ibukiyama, personal communication;
and more importantly such a description is crucial to us.

###  1.2. Maaß relations characterize EZI lifts

From Figure 2 it is evident that not all SK lifts are EZI lifts. This leads us
to wonder if one can characterize the EZI lifts inside the space of SK lifts
uniquely. It is quite pleasant that the answer turns out to be the so-called
Maaß relations, first considered by H. Maaß in his proof of the Maaß
conjectures (see [43], [44]). In our setting, it is as follows. Let $G\in
S^{(2)}_{k}(N)$. Then $G$ satisfies the Maaß relations if for all
$T=\left(\begin{smallmatrix}n&r/2\\\ r/2&m\end{smallmatrix}\right)$

(1.1)
$A_{G}(T):=A_{G}(n,r,m)=\sum_{d|(n,r,m),(d,N)=1}d^{k-1}A_{G}(nm/d^{2},r/d,1).$

Then in Theorem 4.10, we show that $G$ is an EZI lift if and only if it
satisfies (1.1). The proof depends on the description of oldforms discussed
above. Let us mention here the related work of B. Heim [26] which is useful
for us.

###  1.3. The matrix $M_{p}$ that controls SK lifts

We now say a few words about our treatment of (i). Usually, the classical
treatment of new-oldform theory, say for elliptic modular forms, goes via
their Fourier expansion á la Atkin-Lehner [8]. However in the present
situation, especially for the oldspace, this entails knowledge of Fourier
expansions at various cusps. For instance, one has to look at the Fourier
expansion of $F|U_{S}(p)W_{p}$ with $F$ being of level $N$ and $p\nmid N$ –
which may not be easy to obtain. See Remark 4.6 for more on this point. To get
around this, we consider instead an approach based solely on $L^{2}$ norms –
this forms the backbone of the part (i) discussed above. Thus this method
works only for cusp forms, but for the Eisenstein series, we believe a direct
Fourier expansion-based approach should work. To describe this idea, we need a
bit of notation.

Throughout this paper $k$ is even and all of our Petersson norms are not
normalized by volume (cf. (2.3)).

As will be discussed in detail in Section 4, there are four types of
operators, denoted as $R_{i}=R_{i}(p)$, $i=1,\ldots,4$ with
$R_{1}=\mathrm{Id.},R_{2}=U_{S}(p),R_{3}=W_{p},R_{4}=U_{S}(p)W_{p}$, via which
oldforms arise in level $Np$ from level $N$ – see subsection 4.2. Put
$F_{j,p}:=F|R_{j}(p)$. With this data, we associate a matrix $M_{p}$, (which
we call the ‘inner-product’ matrix) as follows.

######  Definition 1.1.

(1.2) $M_{p}=M_{p}(F):=\left(\langle F_{i,p},F_{j,p}\rangle_{Np}\right)_{1\leq
i,j\leq 4}.$

In fact, the new-oldform theory would follow from the following
characterization of the space of SK lifts in terms of the $4\times 4$ matrix
$M_{p}$, which therefore plays a central role.

######  Theorem 1.2.

Let $N\geq 1$ be square-free and $F\in S^{(2)}_{k}(N)$ be a eigenform with
eigenvalues $\lambda_{F}(n)$ for $(n,N)=1$. Then the following are equivalent.

  1. (1)

$F$ is an SK lift of a elliptic eigenform of level $N$ and weight $2k-2$.

  2. (2)

The inner product matrix $M_{p}$ has rank $3$ for some prime $p\nmid N$.

  3. (3)

There exists a prime $p\nmid N$, such that
$\displaystyle\lambda_{F}(p^{2})=\lambda_{F}(p)^{2}-(p^{k-1}+p^{k-2})\lambda_{F}(p)+p^{2k-2}$.

Let us mention here that Theorem 1.2 (3) was known in level one from [47],
[23], whereas we mainly work with (2). From Theorem 1.2 we can obtain an
explicit orthonormal basis for the oldspace since we can show that the leading
$3\times 3$ minor of $M_{p}$ is non-zero. Thus $\\{F_{1},F_{2},F_{3}\\}$ is a
basis for the old-class of $F$. This then enables us to study its $L^{\infty}$
size. Further, one needs to suitably choose a $W_{p}$ invariant orthonormal
basis from this set. The $W_{p}$ invariance helps in localizing the study of
the sup-norm problem in suitable regions. Also a byproduct of Theorem 1.2, we
see that for non-lifts, the oldclass of an eigenform $F$ is $4$ dimensional,
and is spanned by $F_{j}$, $j=1,\ldots,4$. This is in conformity with the
investigations in [49].

###  1.4. Computations in the Hecke algebra

In order to obtain Theorem 1.2 for the treatment of new-oldform theory, we
need extensive information about various inner product relations. This is the
content of Section 3 where we express the entries $\displaystyle\langle
F_{i,p},F_{j,p}\rangle_{Np}$ of $M_{p}$ as

(1.3) $\displaystyle\langle
F_{i,p},F_{j,p}\rangle_{Np}=P(\lambda_{F}(p),\lambda_{F}(p^{2}))\cdot\langle
F,F\rangle_{N},$

where $P$ is a polynomial in $\mathbf{Q}[x,y]$. This leads us to the
discussion about obtaining such relations. A by-now standard method (which was
our initial approach) is to use the Kohnen-Skoruppa Dirichlet series
$\displaystyle D(F,G,s)=\sum\nolimits_{m\geq
1}\langle\phi_{F,m},\phi_{G,m}\rangle m^{-s}$, where $\phi_{F,m}$ denotes the
$m$-th Fourier-Jacobi coefficient of $F$. The idea is to compare the residues
of this series at $s=k$ in two ways. Indeed, for a newform $F$, this is the
approach taken in [2]. However, there seem to be some inaccuracies in this
paper – this will be discussed in some detail in Section 7. Most importantly,
we just mention here that the method loc. cit. works only if at least one of
the components is a newform, and not otherwise. Since we have to work with
oldforms also, we have to take a different route.

Namely, we work with the operators $R_{j}$ introduced above and related
objects viewed inside the Hecke algebra over $\mathbf{Q}$ for
$\Gamma^{(2)}_{0}(N)$ and thus most of our calculations here are algebraic.
Even though we sometimes have used computer for computations (codes available)
in this part; we feel it is quite remarkable that the Hecke relations in
Theorem 1.2 (3) precisely show up in the factorization of the intricate
expression of $\det(M_{p})$ etc. In a sense, this also serves as a cross-check
for all of our inner product calculations. Some of the calculations in the
Appendix to [11] are quite useful to us here.

As final remarks, let us mention that our method can be adapted to study
various problems about $S_{k}^{(2)}(N)$; for instance, it seems likely that
one can reprove Böcherer’s result that $U_{S}(p)$ is injective on
$S_{k}^{(2)}(N)$ for all $p|N$ ($N$ square-free, cf. [12]) by this method.
Moreover, this approach is expected to work for other spaces, e.g., Hermitian
modular forms, etc.

###  1.5. Some background on the sup-norm problem in the level aspect

The second part of the paper depends on (i) and addresses the sup-norm
questions for various spaces related to $\mathrm{SK}_{k}(N)$ mentioned in (ii)
above. We begin with a brief discussion of the problem in degree $1$ on
average.

Let $S_{k}(N)$ denote the space of holomorphic cusp forms of weight $k$ and
level $\Gamma_{0}(N)$. Denote by $\mathcal{B}_{k}(N)$ an orthonormal basis for
$S_{k}(N)$. Then the Bergman kernel for the space $S_{k}(N)$ is given by

(1.4)
$\mathbb{B}_{k,N}(\tau):=\sum\nolimits_{f\in\mathcal{B}_{k}(N)}v^{k}|f(\tau)|^{2}.$

The size of the space $S_{k}(N)$ is then measured by

(1.5)
$\displaystyle\sup(S_{k}(N)):=\sup\nolimits_{\tau\in\mathbf{H}}\mathbb{B}_{k,N}(\tau).$

From the work of Michel-Ullmo [45] and later by Kramer-Jorgensson [30] we know
that for $k=2$ and square-free $N$, the bounds
$\sup(S_{2}(N))\ll\tau_{5}(N)\log N$ and $\ll 1$ respectively. Whereas Michel-
Ullmo use the spectral Large sieve, Kramer-Jorgensson use the heat kernel
method to arrive at the above bounds. Steiner has shown for $k>2$ the hybrid
result $\sup(S_{k}(N))\ll_{A}1+\nu_{N}\,k^{-A}$ for any $A>0$. Here
$\nu_{N}\ll N\log N$ denotes the index of $\Gamma_{0}(N)$ in
$\mathrm{SL}_{2}(\mathbf{Z})$. From this, it is not clear whether one gets a
bound $O_{k}(1)$ (as obtained when $k=2$), as the implied constant depends on
$A$ and $k$ is fixed. However, we note that Steiner’s result is quite
general—valid for all real weights and congruence subgroups.

In an appendix to this paper – see the Introduction, subsection 1.11 and
Section 13 – we will prove in particular the bound
$\displaystyle\sup(S_{\kappa}(N))\asymp_{k}1$ e.g., for all $N$ square-free
and for all $\kappa\geq 5/2$, which could be half-integral.

###  1.6. Conjectures and results on sup-norms

In the remaining part of the introduction, we will discuss our results about
the sizes of various spaces that are intertwined with SK lifts. First, we
discuss about Jacobi forms, whose sizes have a direct bearing on that of SK
lifts, apart from their interest.

###  1.7. Size of Jacobi forms of index 1

Let $\phi\in J^{cusp}_{k,1}(N)$, space of Jacobi forms of index $1$ and level
$N$. It is well-known that such Jacobi forms can be uniquely expressed as

(1.6)
$\displaystyle\phi=h_{0}(\tau)\theta_{0}(\tau,z)+h_{1}(\tau)\theta_{1}(\tau,z),$

called the theta decomposition, and both $h_{0},h_{1}$ are determined by a
single $h\in S_{k-1/2}(4N)$ – in fact the assignment $\phi\leftrightarrow h$
is a Hecke equivariant isomorphism. Here $\theta_{j}$ are certain Jacobi theta
series, see subsection 6.1. This provides the link to pass to half-integral
weights, but there is one catch. Namely from (1.6) one can expect that, say
when $v\approx N^{-1}$, (with $\widetilde{\phi}$ being the invariant function
related $\phi$)

(1.7)
$\displaystyle\sum\nolimits_{\phi}\widetilde{\phi}(\tau,z)^{2}\approx\mathbb{B}_{k-1/2,N}(\tau)\cdot
v^{1/2}(|\theta_{0}|^{2}+|\theta_{1}|^{2})\ll 1\cdot v^{-1/2}\ll N^{1/2},$

since the theta series behave like $v^{-1/2}$ for small $v$. This extra
contribution from the theta series is the “catch” mentioned above, without
which one can just get the bound $O_{k}(1)$ confirming the Conjecture 1.4
below. Technicalities aside, we indeed prove that the bound in (1.7) is true.

######  Theorem 1.3.

Let $k\geq 4$ be even and $N$ square-free. Then,

(1.8) $\displaystyle 1\ll\sup(J^{cusp}_{k,1}(N))\ll N^{1/2}.$

Given the lower bound $1$ it seems reasonable to expect that the size of
$J^{cusp}_{k,1}(N)$ is $1$, but we can not corroborate this at the moment.

######  Conjecture 1.4.

Let $k\geq 4$ be even. Then for large $N$, one has
$\displaystyle\sup(J^{cusp}_{k,1}(N))\asymp 1$.

###  1.8. Non-trivial bound for a Jacobi newform

However, quite interestingly, we are able to prove a non-trivial bound for the
sup-norm of an individual Jacobi form, relying upon the corresponding result
for half-integral weights by [32]. The ‘trivial bound’ is $O_{k}(1)$, as
follows from Conjecture 1.4. Alternatively, we can refer to Proposition 10.1
for the bound $O(N^{\epsilon})$, which we also call the trivial bound. Here
also we encounter the “catch” mentioned around (1.7), and have to work to
overcome it. We prove the following.

######  Theorem 1.5.

Let $N$ be an odd, square-free integer and $\phi\in J^{cusp}_{k,1}(N)$ be an
$L^{2}$-normalized eigenform of all Hecke operators away from the level $N$.
Then

(1.9)
$\sup\nolimits_{(\tau,z)\in\mathbf{H}\times\mathbf{C}}\widetilde{\phi}(\tau,z)\ll_{k,\epsilon}N^{-1/36+\epsilon}.$

The main point behind the success (see subsection 6.1.4) here is that even
though one obtains an ‘extra’ $v^{1/2}$ from $\displaystyle
v^{k}|\phi|^{2}\approx v^{k-1/2}|h|^{2}\cdot v^{1/2}|\theta|^{2}$ after
reduction to a convenient region (where $v\gg 1/N$), one still has some room
to ‘absorb’ the factor $v^{1/2}$ into the bounds for $h$ (via its Fourier
expansion) and thus can proceed further. Note that the weight of $h$ is
$k-1/2$. Without this feature, we wouldn’t have been able to succeed, since
the theta series doesn’t decay with $N$ in the region $v\gg 1/N$. The reader
might be curious why the same idea can’t be used in the previous subsection:
the answer is that the room available for a single $\phi$ is lost for the sum
of squares, since this produces a factor $\approx N$ from the number of
summands.

The bulk of our calculations are focused on a single $\phi$ in Section 6. The
average result follows along the same lines. We end this discussion with a
conjecture.

######  Conjecture 1.6.

Let $k\geq 4$ be even and $\phi\in J^{cusp}_{k,1}(N)$ be an $L^{2}$-normalized
newform. Then for large $N$ and any $\epsilon>0$, one has
$\left\lVert\phi\right\rVert_{\infty}\ll_{\epsilon}N^{-1/2+\epsilon}$.

###  1.9. Size of spaces of SK lifts

We now move to the next level of complexity and consider the space of Saito-
Kurokawa lifts of elliptic modular forms of level $N$ – the main objects of
interest in this paper. In the weight aspect in level $1$, such a problem was
studied by V. Blomer [9] under generalized Lindelöf hypothesis, by relying on
the Fourier expansion directly. In [20] an approach using the Fourier-Jacobi
expansion was laid down. The size of $\mathrm{SK}_{k}$ was precisely computed
in [4]: $\displaystyle\sup(\mathrm{SK}_{k})\asymp k^{5/2}$.

Let $B_{k}^{SK}(N)$ denote an orthonormal basis for the space
$\mathrm{SK}_{k}(N)$, and put

(1.10) $\displaystyle\mathbb{B}_{N}(Z):=\sum\nolimits_{F\in
B_{k}^{SK}(N)}\det(Y)^{k}|F(Z)|^{2};\quad\sup(\mathrm{SK}_{k}(N)):=\sup\nolimits_{Z\in\mathbf{H}_{2}}\mathbb{B}_{N}(Z).$

For SK lifts of level $p$, we prove the following result (restriction to prime
levels is mainly for convenience and length).

######  Theorem 1.7.

Let $k>5$ be even and $p$ be a prime. Then for any $\epsilon>0$,

(1.11) $\displaystyle
p^{-2}\ll\sup(\mathrm{SK}_{k}(p))\ll_{k,\epsilon}p^{-1+\epsilon}.$

Going by the above lower bound, and the size of the oldspace at level $p$,
viz. $\displaystyle\sup(\mathrm{SK}_{k}(p)^{old})\ll p^{-2}$ (see Sections 11
and 12.1.1) – which surprisingly seems not at all trivial – leads us to
propose the following conjecture about the size of $\mathrm{SK}_{k}(N)$.

######  Conjecture 1.8.

For all even $k\geq 4$ and $N\geq 1$, one has
$\displaystyle\sup(\mathrm{SK}_{k}(N))\asymp k^{5/2}N^{-2}$ .

For a newform, we propose the following.

######  Conjecture 1.9.

Let $k\geq 4$ be even and $N\geq 1$. For an $L^{2}$-normalized newform
$F\in\mathrm{SK}_{k}(N)$ one has $\left\lVert F\right\rVert_{\infty}\ll
N^{-3/2}$.

These conjectures are consistent with the expected size of a Hecke eigenform
in $S_{k}^{(2)}(N)$, viz., $\left\lVert F\right\rVert_{\infty}\ll N^{-3/2}$,
where the latter is inspired from the expected bound
$\sup(S_{k}^{(2)}(N))\asymp_{k}1$. In a way, testing the global sup-norm over
smaller and smaller subspaces, say e.g., the SK lifts as in this paper, lends
support towards its plausibility.

Perhaps it is worth remarking here that in order to arrive at the statement of
Conjecture 1.8, one invariably has to consider norm relations between, say, a
Jacobi newform $\phi$ and its EZI lift $F$. However various conflicting
normalizations in the literature, and in some cases, ambiguous statements on
this point (cf. [2], [27]) makes the path misleading. To us, arriving at the
statement of Conjecture 1.8 was as subtle as compared to working on it. We
refer the reader to Section 7 for a more detailed discussion.

###  1.10. Discussion about the proofs

The lower bound in Theorem 1.7 follows from bounding below the size of the
newspace. Here we first use the norm relation between the SK newforms and the
corresponding Jacobi newforms. There are some complications here already (cf.
Section 7). In a few words, the norm relations (cf. (1.3)) are much more
subtle for the oldspace, and need to be worked out from scratch, see
Proposition 5.7. In fact, this is one of the other reasons to write Section 3
in the first place.

Next, we use Waldspurger’s formula to relate the Fourier coefficients to the
twisted central $L$-values of the corresponding elliptic newforms. This
reduces the problem of obtaining a lower bound for $\sup(\mathrm{SK}_{k}(p))$
to one such for the average of these central $L$-values: cf. (7.8). This is
already available from [4, Theorem 1.8] and gives us the desired global lower
bound $p^{-2}$. This picture most likely generalizes to square-free levels,
see Remark 7.2. In fact, all of our results on the sizes of SK lifts should in
principle generalize to square-free levels at the cost of some more delicate
technicalities, and most of the ingredients are already present in this paper.
We choose to work with prime levels mainly because the paper is already long
and technical.

####  1.10.1. The three regions: Type 0, 1, 2

To obtain an upper bound for the size of $\mathrm{SK}_{k}(p)$, it is necessary
to choose a suitable fundamental domain for
$\Gamma_{0}^{(2)}(p)\backslash\mathbf{H}_{2}$. This is described in Section 8.
This essentially leads to three types of regions, which we call Type 0,1,2 –
and the analysis proceeds differently for each of these. Moreover, we estimate
the sizes of the newspace and oldspace separately, as we can’t fruitfully use
the geometric side of the Bergman kernel. Moreover, we have to pass via Jacobi
forms – for this, we need norm relations between $F$ and $\phi$. But due to
the non-uniform nature of the norm relations (see Section 7), we again need to
handle the oldforms separately.

####  1.10.2. Type 0 – Fourier expansion

This is the easiest region, where $Y:=\mathrm{Im}(Z)\gg 1$, and even the upper
bound $O(p^{-2})$ (cf. (7)) follows from the Fourier expansion itself. The
method used here is the same as that in [19] or [4] – viz. looking at the
second moment of Fourier coefficients, which are controlled by Fourier
coefficients of certain Poincaré series. See Section 9, which also contains
some general results about estimating the Fourier expansion in the level
aspect. These may be useful elsewhere as well.

####  1.10.3. Type 1 – Fourier-Jacobi expansion

This is at the next level of difficulty. Here we resort to the Fourier-Jacobi
(FJ) expansion of $F$: namely, we look at

(1.12) $F(Z)=\sum\nolimits_{m}\phi_{F,m}(\tau,z)e(m\tau^{\prime}),$

where $Z=\left(\begin{smallmatrix}\tau&z\\\
z&\tau^{\prime}\end{smallmatrix}\right)$. We then transfer the problem to the
FJ coefficients $\phi_{m}=\phi|V_{m}$ (here $V_{m}$ is a certain Hecke
operator, see subsection 2.1) and ultimately to one on $\phi_{F,1}$ via the
Cauchy-Schwarz inequality – i.e., to the corresponding question about the size
of $J_{k,1}^{cusp}(p)$, which was discussed in the Introduction, subsection
1.7. In this region we obtain the bound $O(p^{-3/2})$ unconditionally and
$O(p^{-2})$ assuming Conjecture 1.4. We refer the reader to Section 12 for
more details.

####  1.10.4. Type 2 – counting matrices

This is the hardest of the three regions, where we have $Y\gg 1/p$. It is
worth mentioning here that having an old-basis that is invariant under $W_{p}$
at our disposal is crucial, to be able to reduce to this region. This is done
in subsection 5.2. Recall that we bound the oldspace separately.

It is well known that the Fourier expansion is hopeless in this region. Thus,
we again fall back to the FJ expansion of $F$. We view the FJ coefficients
(cf. (1.12)) akin to Fourier coefficients and start with a Rankin-Selberg type
argument: roughly we write, after suitably truncating the $m$ sum in (1.12),
that

(1.13) $|F|^{2}\ll(\sum\nolimits_{m}|\phi_{F,m}|^{2})(\sum\nolimits_{m}e(-4\pi
mv^{\prime})),$

and then sum over $F$ on both sides and appeal to the geometric side of the
Bergman kernel for the sum over $F$ i.e., $\phi$. This leads to a fairly
involved counting problem for discrete matrices, which finally gives the bound
$O(p^{-1})$ in this region for the newspace. The counting problem can be
thought of as the usual one weighted by certain upper triangular matrices:
e.g., in we have to count

(1.14)
$\mathcal{C}(\tau,m,\delta):=\\#\\{(\gamma,\alpha_{m},\beta_{m}):\gamma\in\Gamma_{0}(N),|A_{\beta_{m}^{-1}\gamma\alpha_{m}}(\tau)|<\delta\\}.$

Here $\alpha_{m},\beta_{m}$ are certain upper triangular matrices,
$\displaystyle
A_{g}(\tau):=(g\tau-\overline{\tau})j(g,\tau)\mathrm{Im}(\tau)^{-1}$,
$\mathrm{Im}(\tau)\gg p^{-1}$, and $m$ runs over an interval of length
$\approx p$, see subsection 10.4.1. It may be worth noting that the counting
arguments used here are written for any square-free $N$.

Along the way, we prove various miscellaneous results. For instance we show in
Proposition 10.1 the ‘trivial bound’ $\left\lVert\phi\right\rVert_{\infty}\ll
N^{\epsilon}m^{1+\epsilon}$ for any $L^{2}$-normalized $\phi\in
J_{k,m}^{cusp}$ in the $N$ aspect. Improving the $N$ aspect is probably
possible by inserting an amplifier. This bound (in fact any polynomial bound)
is sufficient for our purposes of truncating a certain Fourier-Jacobi
expansion, see subsections 10.3, 10.6.

####  1.10.5. Contribution of oldforms.

As stated earlier, we bound the oldspace separately. This is due to two
reasons. First, as mentioned before, the non-uniform nature of the norm
relations. Second, the basis of oldspace is not an EZI lift of a basis of
Jacobi oldforms (see Section 5), and thus is not captured fully by the BK of
the space of Jacobi forms.

As in the case of newspace, the size of the oldspace is estimated separately
in Type 0, 1, and Type 2 regions. In fact, in regions 1 and 2 we have to work
with a bit more complicated version of the counting argument while dealing
with the oldspace $\mathrm{SK}_{k}|U_{S}(p)$, see sections 11 and 12.1.1. When
$n=1$, estimating the contribution of the subspaces arising due to the $U(p)$
operator is quite straightforward because of the relation between the Hecke
operators $T(p)$, $U(p)$ and $B_{p}$ (see subsection 13.3). However, no such
useful relation exists for the $U_{S}(p)$ operator. The other methods such as
using the Fourier expansion of $F|U_{S}(p)$ or using the double coset
decomposition of $U_{S}(p)$ do not seem to yield the desired bounds. Thus, the
use of BK of Jacobi forms of level $1$ and the counting argument becomes
necessary in this case.

###  1.11. Appendix: the case $n=1$

In an Appendix (Section 13) we provide optimal bounds for the size of
$S_{k}(N)$ for all $k>2$ and $N$, including $k$ half-integral ($N$ square-
free). For this, we give a direct argument relying on Poincaré series and mean
squares of Fourier coefficients. When the level is not square-free some
delicate arguments involving various cusps will be used, however. This is more
complicated than the square-free case, and we rely on a lemma of Saha [48]
that allows us to work in convenient regions, and then we can follow our
original trajectory of proof. When $k=2$ and $N$ is square-free, the following
is known from [30], but it seems nothing is known for non-square-free levels.

######  Theorem 1.10.

Let $N$ be any integer and $k>2$ be any positive integer. Then

(1.15) $\sup(S_{k}(N))\asymp_{k}1.$

The same result holds for all principal congruence subgroups and for half-
integral weights $\geq 5/2$ and square-free level.

The implied constant can be shown to be polynomial in $k$. The main issue for
half-integral weights is that the theory of Atkin-Lehner operators is subtle.
Our arguments use no (arithmetic) diagonalization of oldforms (cf. [45]), nor
Deligne’s bound on Fourier coefficients. We essentially relate the problem to
majorants of suitable Eisenstein series, which are well understood.

### Acknowledgments

It is a pleasure for the authors to thank Abhishek Saha and Tomoyoshi
Ibukiyama for their comments on the topic of oldforms for SK lifts. S.D.
thanks IISc. Bangalore, UGC Centre for Advanced Studies, DST India for
financial support. P.A. thanks IISER, Pune, where he is a postdoctoral fellow
and NBHM, DAE for the financial support through NBHM postdoctoral fellowship.

##  2\. Notations and preliminaries

In this paper, we will mostly use standard notation, some of which are
collected below, and the rest will be introduced as and when it is necessary.
We note that $A\ll B$ and $A=O(B)$ are the Vinogradov and Landau notations,
respectively. By $A\asymp B$ we mean that there exists a constant $c\geq 1$
such that $B/c\leq A\leq cB$. Any subscripts under them (e.g., $A\ll_{n}B$)
indicate the dependence of any implicit constants on those parameters.
Throughout, $\epsilon$ will denote a small positive number, which can vary
from one line to another.

Let $\mathbf{Z},\mathbf{Q},\mathbf{R}$, and $\mathbf{C}$ denote the integers,
rationals, reals, and complex numbers respectively. For a commutative ring $R$
with unit, $\mathrm{M}_{n,m}(R)$ and $\mathrm{M}_{n}(R)$ denote the set of
$n\times m$ and $n\times n$ matrices over $R$, respectively.
$\mathrm{GL}_{n}(R)$ denotes the group of invertible elements in
$\mathrm{M}_{n}(R)$. $\mathrm{Sym}_{n}(R)$ denotes the set of all $n\times n$
symmetric matrices over $R$. $\mathrm{Sp}_{n}(R)$ denotes the symplectic group
of degree $n$ over $R$. $\Lambda_{n}$ (resp. $\Lambda_{n}^{+}$) denotes the
set of all $n\times n$ real symmetric, positive semi–definite (resp.
positive–definite), half–integral matrices (that is $T=(t_{ij})$ with
$2t_{ij},\;t_{ii}\in\mathbf{Z}$). We will denote the transpose of $A$ by
$A^{t}$. For matrices $A$ and $B$ of appropriate size, we write
$A[B]:=BAB^{t}$. Moreover, $1_{n}$ and $0_{n}$ will be the $n\times n$
identity and zero matrices, respectively. We drop the subscripts when the
order of these matrices is obvious. For standard facts about Siegel modular
forms and Jacobi forms, we refer the reader to [24], [34], [22].

###  2.1. Jacobi forms of higher level.

Let $k$, $m$ and $N$ be positive integers. A holomorphic function
$\phi:\mathbf{H}\times\mathbf{C}\rightarrow\mathbf{C}$ is a Jacobi form of
weight $k$, index $m$ and level $N$ if:

  1. (1)

$\phi\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right)(c\tau+d)^{-k}e^{m}\left(\frac{-cz^{2}}{c\tau+d}\right)=\phi(\tau,z)$
for all $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma_{0}(N)$.

  2. (2)

$\phi(\tau,z+\lambda\tau+\mu)e^{m}(\lambda^{2}\tau+2\lambda z)=\phi(\tau,z)$
for all $(\lambda,\mu)\in\mathbf{Z}^{2}$.

  3. (3)

$\phi$ is bounded at the cusps of $\Gamma_{0}(N)\backslash\mathbf{H}$.

If the Fourier coefficients corresponding to $4mn-h_{M}r^{2}=0$ vanish for all
$M\in\mathrm{SL}_{2}(\mathbf{Z})$, then $\phi$ is called a Jacobi cusp form.
We denote by $J_{k,m}(N)$ the space of Jacobi forms of weight $k$, index $m$
and level $N$. We denote the space of Jacobi cusp forms by
$J_{k,m}^{cusp}(N)$.

The operators $V_{m}$ are defined on Jacobi forms of index $1$ and level $N$
(even though they can be defined for any index). For $\phi\in
J^{cusp}_{k,1}(N)$ define (cf. [22, § 4 (2)], [28, Section 3])

(2.1)
$\displaystyle(\phi|V_{m})(\tau,z)=m^{k-1}\sum\nolimits_{\gamma}(c\tau+d)^{-k}e(-mcz^{2}/(c\tau+d)^{2})\phi\big{(}\frac{a\tau+b}{c\tau+d},\frac{mz}{c\tau+d}\big{)},$

where $\gamma=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$
runs over a set of representatives $\Gamma_{0}(N)\backslash
M_{2,m}(\mathbf{Z})$. Here $M_{2,m}(\mathbf{Z})$ denotes the set of size $2$
integral matrices $\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)$ with determinant $m$ and $(a,N)=1$.

###  2.2. Siegel modular forms.

Let $\mathbf{H}_{n}$ denote the Siegel’s upper half space of degree $n$. For
any integer $k$ and any function $F$ on $\mathbf{H}_{n}$,
$M=\left(\begin{smallmatrix}A&B\\\
C&D\end{smallmatrix}\right)\in\mathrm{GSp}^{+}_{n}(R)$ acts as

(2.2) $(F|_{k}M)(Z):=r(M)^{k}\det(CZ+D)^{-k}F(MZ),$

where $MZ:=(AZ+B)(CZ+D)^{-1}$ and $MJM^{t}=r(M)J$.

Let $\Gamma_{0}^{(n)}(N)=\left\\{\left(\begin{smallmatrix}A&B\\\
C&D\end{smallmatrix}\right)\in\mathrm{Sp}_{n}(\mathbf{Z}):C\equiv 0\mod
N\right\\}$ denote the Hecke congruence subgroup of level $N$. A holomorphic
function $F:\mathbf{H}_{n}\rightarrow\mathbf{C}$ is called a Siegel modular
form of degree $n$, weight $k$ and level $N$ if $F|_{k}M=F$ for all
$M\in\Gamma_{0}^{(2)}(N)$. When $n=1$, $F$ needs to satisfy an additional
condition of boundedness at cusps. In addition, if $F$ vanishes at all the
cusps we say that $F$ is a Siegel cusp form. The space of of Siegel cusp forms
of degree $n$ is denoted by $S_{k}^{(n)}(N)$.

Throughout this paper, all Petersson inner products are non-normalised, so
that the volume of the fundamental domain is proportional to the index of the
group. To be precise, when $n=2$, the inner products are given by

(2.3) $\displaystyle\langle
F,G\rangle_{N}=\int_{\Gamma_{0}^{(2)}(N)\backslash\mathbf{H}_{2}}F(Z)\overline{G(Z)}\det(Y)^{k}d\mu(Z)\quad(F,G\in
S_{k}^{(2)}(N)).$

We denote by $\mathcal{F}_{j}(N)$ to be the Siegel’s fundamental domain for
the action of $\Gamma_{0}^{(j)}(N)$ on $\mathbf{H}_{j}$ consisting of those
$Z\in\mathbf{H}_{j}$ satisfying the conditions that
$|\det(c_{\gamma}Z+d_{\gamma})|\geq 1$ for all $\gamma\in\Gamma_{0}^{(j)}(N)$,
$Y$ Minkowski reduced and $X\bmod 1$.

####  2.2.1. Double coset operators and the Hecke algebras:

Let $g\in\mathrm{GSp}^{+}_{2}(\mathbf{Q})\cap M_{4}(\mathbf{Z})$ be such that
$gJg^{t}=r(g)J$. Let
$\Gamma_{1},\Gamma_{2}\subseteq\mathrm{Sp}_{2}(\mathbf{Z})$ be any two
congruence subgroups such that $[\Gamma_{2}:\Gamma_{2}\cap
g^{-1}\Gamma_{1}g]<\infty$ and $[\Gamma_{1}:\Gamma_{1}\cap
g\Gamma_{2}g^{-1}]<\infty$. Let
$\Gamma_{1}g\Gamma_{2}=\cup_{i=1}^{d}\Gamma_{1}\alpha_{i}$. We define the
double coset operator
$\Gamma_{1}g\Gamma_{2}:S_{k}^{(2)}(\Gamma_{1})\longrightarrow
S_{k}^{(2)}(\Gamma_{2})$ as

(2.4)
$F|_{k}(\Gamma_{1}g\Gamma_{2}):=r(g)^{k-3}\sum\nolimits_{i}F|_{k}\alpha_{i}.$

The degree of the operator $\Gamma_{1}g\Gamma_{2}$ is defined to be the number
of cosets $\Gamma_{1}\alpha$ contained in $\Gamma_{1}g\Gamma_{2}$. It is
denoted by $\deg(\Gamma_{1}g\Gamma_{2})$ and it is equal to
$d=[\Gamma_{2}:\Gamma_{2}\cap g^{-1}\Gamma_{1}g]$ in this case.

Let $G:=\mathrm{GSp}_{2}^{+}(\mathbf{Z})$ be the semigroup of positive
integral similitudes for the symplectic group. We will mainly work with the
rational Hecke algebra $\mathscr{H}_{\mathbf{Q}}(G,\Gamma)$ for the Hecke pair
$(G,\Gamma)$, which as we may recall is the $\mathbf{Q}$ vector space spanned
by the double cosets $\Gamma g\Gamma$ ($\Gamma$ being a congruence subgroup)
and the multiplication is more generally defined as follows: for two double
cosets $\Gamma_{1}g_{1}\Gamma_{2}=\cup_{i=1}^{d_{1}}\Gamma_{1}\alpha_{i}$ and
$\Gamma_{2}g_{2}\Gamma_{3}=\cup_{i=1}^{d_{2}}\Gamma_{2}\beta_{i}$, the product
is defined as (see [52, Chap. 3])

(2.5) $\Gamma_{1}g_{1}\Gamma_{2}\cdot\Gamma_{2}g_{2}\Gamma_{3}=\sum
m(g)\Gamma_{1}g\Gamma_{3},$

where the sum is over all $g$ such that
$\Gamma_{1}g\Gamma_{3}\subset\Gamma_{1}g_{1}\Gamma_{2}g_{2}\Gamma_{3}$ and

(2.6) $m(g)=\\#\\{(i,j):\Gamma_{1}\alpha_{i}\beta_{j}=\Gamma_{1}g\\}.$

###  2.3. Half-integral weight modular forms.

Let $k\in\mathbf{Z}$ and $\Gamma_{0}(4N):=\Gamma_{0}^{(1)}(4N)$. A holomorphic
function $f:\mathbf{H}\rightarrow\mathbf{C}$ is called a half-integral weight
modular form of weight $\kappa=k+1/2$, if
$f(\gamma\tau)=\left(\frac{c}{d}\right)^{2\kappa}\epsilon_{d}^{-2\kappa}(c\tau+d)^{\kappa}f$
for all $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma_{0}(4N)$ and $f$ is holomorphic at the
cusps of $\Gamma_{0}(4N)$. Here $\left(\frac{\cdot}{d}\right)$ denotes the
Legendre symbol and $\epsilon_{d}=1$ if $d\equiv 1\mod 4$ and $\epsilon_{d}=i$
if $d\equiv-1\mod 4$.

For any $d|N$, the Atkin-Lehner operator at $d$ of level $N$, $W(d,N)$ is
defined as

(2.7) $W(d,N)=\begin{pmatrix}ds&b\\\ Nc&dt\\\
\end{pmatrix}\in\mathrm{M}_{2}(\mathbf{Z})\text{ such that }dst-Nb/d=1.$

##  3\. Inner product relations

Let $N$ be any positive integer. Then for any prime $p\nmid N$, we consider
the following operators acting on the space of Siegel cusp forms of level
$Np$.

  1. (1)

The Atkin-Lehner operator $W_{p}$ defined as

(3.1) $W_{p}=\begin{pmatrix}p\alpha 1_{2}&1_{2}\\\ N\beta 1_{2}&p1_{2}\\\
\end{pmatrix},$

where $\alpha$, $\beta$ are integers satisfying $p\alpha-N\beta/p=1$.

  2. (2)

The $U_{S}(p)$ operator given by the double coset
$\Gamma_{0}^{(2)}(Np)\mathrm{diag}(1,1,p,p)\Gamma_{0}^{(2)}(Np)$. The Fourier
expansion of $F|U_{S}(p)$ is given by

(3.2) $F|U_{S}(p)(Z)=\sum\nolimits_{T}A_{F}(pT)e(\mathrm{tr}(TZ)).$

Let $p$ and $N$ be as above. For a diagonal matrix
$\mathrm{diag}(p^{a},p^{b},p^{c},p^{d})$ with $a+c=b+d$, we put

(3.3)
$T_{S}(p^{a},p^{b},p^{c},p^{d})=\Gamma_{0}^{(2)}(N)\mathrm{diag}(p^{a},p^{b},p^{c},p^{d})\Gamma_{0}^{(2)}(N).$

To obtain the inner product relations, we interpret the operators as double
cosets inside suitable Hecke algebras and work with them, e.g., as in [51]. We
take care to present all the calculations as exact norm relations are
crucially required for our study of SK oldforms.

For the sake of simplicity, we write
$\mu(p):=[\Gamma^{(2)}_{0}(N):\Gamma^{(2)}_{0}(Np)]$ for any integer $N$ and
any prime $p\nmid N$. We start with a lemma which is a degree $2$ analogue of
[51, Thm. 6], if we note the slight difference in normalization of the
operators.

######  Lemma 3.1.

Let $N$ be any positive integer and $p\nmid N$. Let $F,F^{\prime}$ be any two
Siegel cusp forms of level $N$ and $F$ an eigenfunction of the operator
$T_{S}(p)$. Then

(3.4) $\displaystyle\langle
F^{\prime},F|W_{p}\rangle_{Np}=p^{3-k}\lambda_{F}(p)\mu(p)^{-1}\cdot\langle
F^{\prime},F\rangle_{Np}.$

###### Proof.

Let us first note that $\gamma W_{p}=B_{p}$ for some $\gamma\in\Gamma_{0}(N)$.
To see this, recall that $W_{p}$ normalizes $\Gamma_{0}(Np)$ and note that

(3.5) $\begin{pmatrix}pa-Npb\beta&pb\alpha-a\\\ N(pc-d\beta)&d\alpha-Nc\\\
\end{pmatrix}\begin{pmatrix}p\alpha&1\\\ Np\beta&p\\\
\end{pmatrix}=\begin{pmatrix}p&0\\\ 0&1\\\ \end{pmatrix}\begin{pmatrix}a&b\\\
Npc&d\\\ \end{pmatrix},$

$\alpha,\beta$ are as in (3.1) and $a,b,c,d\in\mathbf{Z}$ such that $ad-
Npbc=1$. Thus $F|W_{p}=F|B_{p}$. Since the latter has level $Np$ we can
compute

(3.6) $\displaystyle\mu(p)\langle F^{\prime},F|W_{p}\rangle_{Np}=\langle
F^{\prime},F|B_{p}|\mathrm{Tr}_{Np,N}\rangle_{Np},$

where $\mathrm{Tr}_{N,M}\colon S_{k}^{(2)}(N)\rightarrow S_{k}^{(2)}(M)$ is
the trace map for $M|N$ defined by $\displaystyle
H\mapsto\sum\nolimits_{\gamma\in\Gamma^{(2)}_{0}(N)\backslash\Gamma^{(2)}_{0}(M)}H|\gamma$.

First, we note that $F|\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(Np)=p^{k-3}F|B_{p}$. Writing in
terms of double cosets, the rightmost operator in (3.6) is

$\displaystyle\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(Np)\cdot\Gamma^{(2)}_{0}(Np)1_{4}\Gamma^{(2)}_{0}(N)$
$\displaystyle=1\cdot\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(N)$ (3.7)
$\displaystyle=\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}1&0\\\
0&p\end{smallmatrix}\right)\Gamma^{(2)}_{0}(N)=T_{S}(p).$

In the above we used the formula for multiplication of double cosets in the
Hecke algebra [52], especially [52, (3.1.1)] and [52, Prop. 3.3]. Here, only
one double coset appears after multiplication with multiplicity one. Moreover,
the second equality of double cosets follows from the equality
$\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}p&-t\\\
N&d\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\\
0&p\end{smallmatrix}\right)\left(\begin{smallmatrix}pd&t\\\
-N&1\end{smallmatrix}\right)$; where $t,d$ are integers chosen such that
$pd+tN=1$. ∎

######  Lemma 3.2.

For a prime $p\nmid N$, a set of coset representatives in
$\Gamma^{(2)}_{0}(Np^{2})\backslash\Gamma^{(2)}_{0}(Np)$ is given by
$\\{\left(\begin{smallmatrix}1&0\\\ -NpB&1\end{smallmatrix}\right)\\}$ as
$B=B^{t}$ varies $\bmod$ $p$.

###### Proof.

First, we know that $[\Gamma_{0}^{(2)}(Np):\Gamma_{0}^{(2)}(Np^{2})]=p^{3}$
(see for example [33]). Next, the cardinality of the set
$\\{\left(\begin{smallmatrix}1&0\\\ -NpB&1\end{smallmatrix}\right),B\mod
p,B=B^{t}\\}$ is $p^{3}$ and for any two $B$, $B^{\prime}\mod p$,
$\left(\begin{smallmatrix}1&0\\\
-NpB&1\end{smallmatrix}\right)\cdot\left(\begin{smallmatrix}1&0\\\
NpB^{\prime}&1\end{smallmatrix}\right)\in\Gamma_{0}^{(2)}(Np^{2})$ $\iff$
$B\equiv B^{\prime}\mod p$. This completes the proof. ∎

Let $T_{S}^{\prime}(p)$ be the operator defined as below (see also [28, pp.
158])

(3.8)
$T_{S}^{\prime}(p):=T_{S}(p)^{2}-T_{S}(p^{2})=pT_{S}(1,p,p^{2},p)+p(1+p+p^{2})T_{S}(p,p,p,p).$

######  Lemma 3.3.

Let $N$ be any positive integer and $p\nmid N$ and $F,F^{\prime}$ be any two
Siegel cusp forms of level $N$. Further, let $F$ an eigenfunction of the
operators $T_{S}(p)$ and $T_{S}^{\prime}(p)$ with the eigenvalues
$\lambda_{F}(p)$ and $\lambda_{F}^{\prime}(p)$, respectively. Then

(3.9) $\displaystyle\langle
F|U_{S}(p),F^{\prime}|W_{p}\rangle_{Np}=p^{3-k}\mu(p)^{-1}\left(\lambda_{F}(p)^{2}-(1+\frac{1}{p})\lambda_{F}^{\prime}(p)+p^{2k-5}(p+1)\right)\cdot\langle
F,F^{\prime}\rangle_{Np}.$

###### Proof.

First, we note that $F|U_{S}(p)|W_{p}=F|B_{p^{2}}|\mathrm{Tr}_{Np^{2},Np}$.
This follows from the identity

(3.10) $\displaystyle\left(\begin{smallmatrix}1&B\\\
0&p\end{smallmatrix}\right)W_{p}=\gamma\left(\begin{smallmatrix}p^{2}&0\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\\
-pB&1\end{smallmatrix}\right)\quad\text{ for some
}\gamma\in\Gamma^{(2)}_{0}(N),$

and Lemma 3.2. Then one can write

(3.11) $\displaystyle\langle F|U_{S}(p),F^{\prime}|W_{p}\rangle_{Np}$
$\displaystyle=\langle F|U_{S}(p)|W_{p},F^{\prime}\rangle_{Np}$ (3.12)
$\displaystyle=p^{k-3}\langle
F|B_{p^{2}}|\mathrm{Tr}_{Np^{2},Np},F^{\prime}\rangle_{Np}=p^{k-3}\mu(p)^{-1}\langle
F|B_{p^{2}}|\mathrm{Tr}_{Np^{2},N},F^{\prime}\rangle_{Np},$

where the factor $p^{k-3}$ comes from the definition of $U_{S}(p)$ given by
$(F|U_{S}(p))(Z)=p^{k-3}\sum_{B}F|\left(\begin{smallmatrix}1&B\\\
0&p\end{smallmatrix}\right)$.

First, we note that $F|\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p^{2}&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(Np^{2})\cdot\Gamma^{(2)}_{0}(Np^{2})1_{4}\Gamma^{(2)}_{0}(N)=p^{2k-6}F|B_{p^{2}}|\mathrm{Tr}_{Np^{2},N}$.
Next, note that

(3.13) $\displaystyle\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p^{2}&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(Np^{2})\cdot\Gamma^{(2)}_{0}(Np^{2})1_{4}\Gamma^{(2)}_{0}(N)$
$\displaystyle=1\cdot\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p^{2}&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(N)$ (3.14)
$\displaystyle=\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}1&0\\\
0&p^{2}\end{smallmatrix}\right)\Gamma^{(2)}_{0}(N)$
$\displaystyle=T_{S}(1,1,p^{2},p^{2}),$

by the same logic as in the analogous situation in the proof of Lemma 3.1.
Thus we get that

(3.15) $F|B_{p^{2}}|\mathrm{Tr}_{p^{2},1}=p^{6-2k}T_{S}(1,1,p^{2},p^{2}).$

To complete the proof, let $T_{S}^{\prime}(p)=T_{S}(p)^{2}-T_{S}(p^{2})$. be
as in [28]. Then a small calculation shows that

(3.16)
$T_{S}(1,1,p^{2},p^{2})=T_{S}(p)^{2}-(1+\frac{1}{p})T^{\prime}_{S}(p)+p(p+1)T_{S}(p,p,p,p).\qed$

######  Lemma 3.4.

Let $N$ be any positive integer and $p\nmid N$. Let $F,F^{\prime}$ be any two
Siegel cusp forms of level $N$ and $F$ an eigenfunction of the operator
$T_{S}(p)$ with the eigenvalue $\lambda_{F}(p)$. Then

(3.17) $\langle
F|U_{S}(p),F^{\prime}\rangle_{Np}=p^{3}\lambda_{F}(p)\mu(p)^{-1}\cdot\langle
F,F^{\prime}\rangle_{Np}.$

###### Proof.

First, we note that

(3.18) $[\Gamma^{(2)}_{0}(N):\Gamma^{(2)}_{0}(Np)]\langle
F|U_{S}(p),F^{\prime}\rangle_{Np}=\langle
F|U_{S}(p)|\mathrm{Tr}_{Np,N},F^{\prime}\rangle_{Np}.$

Since $F$ is of level $N$, the operator $U_{S}(p)|\mathrm{Tr}_{Np,N}$ can be
written an

(3.19) $\displaystyle\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}1&0\\\
0&p\end{smallmatrix}\right)\Gamma^{(2)}_{0}(Np)\cdot\Gamma^{(2)}_{0}(Np)1_{4}\Gamma^{(2)}_{0}(N)=p^{3}\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}1&0\\\
0&p\end{smallmatrix}\right)\Gamma^{(2)}_{0}(N)=p^{3}T_{S}(1,1,p,p)=p^{3}T_{S}(p).$

The proof now follows. ∎

######  Lemma 3.5.

Let $p\nmid N$ and $F,F^{\prime}\in S^{(2)}_{k}(N)$ be such that $F^{\prime}$
is an eigenfunction of the operator $T_{S}(p)$ with eigenvalue
$\lambda_{F^{\prime}}(p)$. Then we have

(3.20) $\displaystyle\langle
F|U_{S}(p),F^{\prime}|U_{S}(p)\rangle_{Np}=\mu(p)^{-1}\left(p^{2}(p-1)\lambda_{F^{\prime}}(p)^{2}+p^{2k-4}\mu(p)\right)\langle
F,F^{\prime}\rangle_{Np}.$

######  Remark 3.6.

At first glance, the above relation might look asymmetrical, but it is not.
Since the LHS and hence the RHS are non-zero only if $F=F^{\prime}$, as
eigenvectors with distinct $T_{S}(p)$ eigenvalues are orthogonal.

######  Remark 3.7.

In the case of elliptic modular forms, an analogous result holds (see [3,
Theorem 8]).

###### Proof.

Let $K_{1},K_{2}\subset\Gamma^{(2)}$ be congruence subgroups. We begin by
noting that the adjoint of $T=K_{1}gK_{2}$ is given by $T^{*}=K_{2}g^{*}K_{1}$
where $g^{*}=r(g)g^{-1}$, where $r(g)$ is the similitude of $g$. In our
situation, we have $K_{1}=K_{2}=\Gamma_{0}^{(2)}(Np)$ and therefore

(3.21) $\displaystyle\langle
F|U_{S}(p),F^{\prime}|U_{S}(p)\rangle_{Np}=\langle
F,F^{\prime}|U_{S}(p)U_{S}(p)^{*}\rangle_{Np}.$

Since $F$ has level $N$,

(3.22) $\langle
F,F^{\prime}|U_{S}(p)U_{S}(p)^{*}\rangle_{Np}=\mu(p)^{-1}\langle
F,F^{\prime}|U_{S}(p)U_{S}(p)^{*}|\mathrm{Tr}_{Np,N}\rangle_{Np}.$

Since $F^{\prime}$ has level $N$, the operator
$U_{S}(p)U_{S}(p)^{*}\mathrm{Tr}_{Np,N}$ on the RHS is given by the following
double coset operator:

(3.23) $\displaystyle\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}1&0\\\
0&p\end{smallmatrix}\right)\Gamma_{0}^{(2)}(Np)\cdot\Gamma_{0}^{(2)}(Np)p\left(\begin{smallmatrix}1&0\\\
0&p^{-1}\end{smallmatrix}\right)\Gamma_{0}^{(2)}(Np)\cdot\Gamma_{0}^{(2)}(Np)1_{4}\Gamma^{(2)}_{0}(N)$
(3.24) $\displaystyle=\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}1&0\\\
0&p\end{smallmatrix}\right)\Gamma_{0}(Np)\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(N).$

Since there are $3$ distinct double cosets of similitude $p^{2}$ for
$\Gamma^{(2)}_{0}(N)$, from (2.5) we can write

(3.25) $\Gamma^{(2)}_{0}(N)\begin{pmatrix}1&0\\\ 0&p\\\
\end{pmatrix}\Gamma_{0}^{(2)}(Np)\begin{pmatrix}p&0\\\ 0&1\\\
\end{pmatrix}\Gamma^{(2)}_{0}(N)=c_{1}T_{S}(1,p,p^{2},p)+c_{2}T_{S}(1,1,p^{2},p^{2})+c_{3}T_{S}(p,p,p,p).$

First, let $\mu(p^{i})$ denote that index of $\Gamma_{0}^{(2)}(Np^{i})$ in
$\Gamma^{(2)}_{0}(N)$. Let us recall that for a double coset $x=H_{1}gH_{2}$,
$\deg(x)$ denotes the number of cosets of $H_{1}$ in $x$, extended by
linearity. We need to have information about the degrees of the operators
appearing in (3.25). Namely,

(3.26)
$\displaystyle\deg(U_{S}(p)U_{S}(p)^{*}\cdot\mathrm{Tr}_{Np,N})=p^{6}\mu(p);\quad$
$\displaystyle\deg T_{S}(1,p,p^{2},p)=p\mu(p);$ (3.27) $\displaystyle\deg
T_{S}(1,1,p^{2},p^{2})=p^{3}\mu(p);\quad$ $\displaystyle\deg
T_{S}(p,p,p,p)=1.$

Of these, the first equality in (3.26) follows from the multiplicative
relation $\deg(xy)=\deg(x)\deg(y)$ see [52, Sec. 3.1]), that
$\deg(U_{S}(p))=\deg(U_{S}(p)^{*})=p^{3}$, and that
$\deg(\mathrm{Tr}_{Np,N})=\mu(p)$. It can be easily checked that the degrees
of $T=\Gamma g\Gamma$ and $T^{*}$ are the same by applying the map $x\mapsto
x^{*}$ on $T=\cup_{m}\Gamma\,m$. For the rest, see [40, Chap. 6, Lem. 5.1,
Cor. 7.3].

Following Shimura [52], we first calculate the multiplicities $c_{2}$ and
$c_{3}$ from the combinatorics of the coset decomposition of the operators
$T_{S}(1,1,p^{2},p^{2})$ and $T_{S}(p,p,p,p)$ respectively. Keeping (3.25) in
mind, we first note that

(3.28) $\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}1&0\\\
0&p\end{smallmatrix}\right)\Gamma_{0}^{(2)}(Np)=\bigcup\nolimits_{L=L^{t}}\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}1&L\\\
0&p\end{smallmatrix}\right)$

and

(3.29) $\Gamma_{0}^{(2)}(Np)\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(N)=\bigcup\nolimits_{i,r,\mathcal{A}}\Gamma_{0}^{(2)}(Np)\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\\
NpL_{i}&1\end{smallmatrix}\right)w_{r}^{N,p}n(\mathcal{B}_{r})m(\mathcal{A}),$

where $w_{r}^{N,p}\in\Gamma_{0}^{(2)}(N)$ such that $w_{r}^{N,p}\equiv
w_{r}\mod p$ and $w_{r}^{N,p}\equiv 1_{4}\mod q$ for all $q|N$. In (3.29),
$\left(\begin{smallmatrix}1&0\\\ NpL_{i}&1\end{smallmatrix}\right)$ are the
coset representatives of $\Gamma_{0}(Np^{2})$ in $\Gamma_{0}(Np)$ follows from
Lemma 3.2. These are multiplied by the coset representatives of
$\Gamma_{0}(Np)$ in $\mathrm{Sp}_{2}(\mathbf{Z})$, which is justified below.

Let $\mathbf{F}_{p}$ be the finite field with $p$ elements. Since $p\nmid N$,
reduction of $\Gamma_{0}(N)\bmod p$ equals $\mathrm{Sp}_{2}(\mathbf{F}_{p})$
with kernel $\Gamma(Np)$; from which it is easy to verify (3.29), given that
the same holds when $N=1$, as can be found from [15].

Let $\gamma_{i,r,\mathcal{A}}:=\left(\begin{smallmatrix}1&0\\\
NpL_{i}&1\end{smallmatrix}\right)w_{r}^{N,p}n(\mathcal{B}_{r})m(\mathcal{A})$.
We write $\gamma=\gamma_{i,r,\mathcal{A}}$ when there is no ambiguity. Note
also that $\\{\gamma\\}$ (as in (3.29)) also constitute a set of coset
representatives of $\Gamma_{0}(Np^{2})$ in $\Gamma^{(2)}_{0}(N)$.

To compute $c_{3}$ we have to count pairs $(L,\gamma)$ such that

(3.30) $\displaystyle\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}1&L\\\
0&p\end{smallmatrix}\right)\cdot\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)\gamma=\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p&L\\\
0&p\end{smallmatrix}\right)\cdot\gamma=\Gamma^{(2)}_{0}(N)p1_{4}.$

With the above notation, however, this amounts to checking

(3.31) $\displaystyle\begin{pmatrix}1&L/p\\\ 0&1\\\
\end{pmatrix}\gamma\in\Gamma^{(2)}_{0}(N)\text{ or
}\begin{pmatrix}A+LC/p&B+LD/p\\\ C&D\\\
\end{pmatrix}\in\Gamma^{(2)}_{0}(N),\quad(\gamma=\left(\begin{smallmatrix}A&B\\\
C&D\end{smallmatrix}\right)).$

Thus we require $\displaystyle LC,LD\in pM_{2}(\mathbf{Z})$ which forces
$L\equiv 0\bmod p$ and thus $L=0$, since $(C,D)$ is a co-prime symmetric pair.
But when $L=0$, (3.30) holds for all the $\gamma$’s. Therefore

(3.32) $c_{3}=\mu(p^{2}).$

Similarly, to compute $c_{2}$, we have to count pairs $(L,\gamma)$ such that

(3.33) $\displaystyle\begin{pmatrix}p&L\\\ 0&p\\\
\end{pmatrix}\cdot\begin{pmatrix}A&B/p^{2}\\\ C&D/p^{2}\\\
\end{pmatrix}=\begin{pmatrix}pA+LC&B/p+LD/p^{2}\\\ pC&D/p\\\
\end{pmatrix}\in\Gamma^{(2)}_{0}(N).$

From these we obtain the conditions that $L,C\bmod
p\in\mathrm{GL}_{2}(\mathbf{F}_{p})$, $D\equiv 0\bmod p$.

When $r=0$, a matrix multiplication (using SageMath) shows that the lower
right block $D$ of $\gamma_{i,r,\mathcal{A}}$ satisfies $D\equiv 0\mod p$ only
if $m(\mathcal{A})=0$, which is not possible. Similarly, when $r=1$, $D\equiv
0\mod p$ only if $\mathcal{A}=\left(\begin{smallmatrix}0&0\\\
&*\end{smallmatrix}\right)$, which is again not possible since $\mathcal{A}$
is invertible. Thus, we do not have any solutions to (3.33) when $r=0,1$.

For $r=2$, write $L_{i}=\left(\begin{smallmatrix}l_{1}&l_{2}\\\
l_{3}&l_{4}\end{smallmatrix}\right)$ and
$\mathcal{B}_{r}=\left(\begin{smallmatrix}b_{1}&b_{2}\\\
b_{2}&b_{4}\end{smallmatrix}\right)$. Then $\gamma$ satisfies

(3.34) $\gamma\equiv\left(\begin{array}[]{rrrr}0&0&-1&0\\\ 0&0&0&-1\\\
1&0&-pl_{1}+b_{1}&-pl_{2}+b_{2}\\\
0&1&-pl_{2}+b_{2}&-pl_{4}+b_{4}\end{array}\right)\mod p.$

Since $D\equiv 0\mod p$ from (3.33), we see that $b_{1}=b_{2}=b_{4}=0$. Then
$\gamma=\left(\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\right)$ with
$A=0$, $B=-I$, $C=I$ and $D=-pL_{i}$ in (3.33). Thus we can rewrite the
condition in (3.33) as

(3.35) $\begin{pmatrix}L&-1/p-LL_{i}/p\\\ p&-L_{i}\\\
\end{pmatrix}\in\Gamma^{(2)}_{0}(N).$

This is possible only if $LL_{i}+1\equiv 0\mod p$ and both $L$ and $L_{i}$
symmetric, invertible modulo $p$. Clearly, the choice of $L$ fixes $L_{i}$ mod
$p$ and hence globally as well, since $L,L_{i}$ vary mod $p$. Now, $L$ can be
chosen to be the lift of any $2\times 2$, symmetric, invertible matrix with
entries in $\mathbf{F}_{p}$, whose number is given by $p^{3}-p^{2}$ (see [17]
for $p>2$ and for $p=2$ it is clear). Thus we conclude that

(3.36) $c_{2}=p^{3}-p^{2}.$

In principle, one could compute $c_{1}$ by the (more complicated) above
procedure, but we curtail some work by appealing to the ‘degree equation’.
Namely, we know that

(3.37) $\displaystyle\deg(U_{S}(p)U_{S}^{*}(p))$ $\displaystyle=c_{1}\deg
T_{S}(1,p,p^{2},p)+c_{2}\deg T_{S}(1,1,p^{2},p^{2})+c_{3}\deg T_{S}(p,p,p,p);$

and from the degrees in (3.26), (3.27) we see that

(3.38)
$\begin{split}c_{1}&=p^{-1}\mu(p)^{-1}\left(p^{6}\mu(p)-p^{3}\mu(p)-(p^{3}-p^{2})p^{3}\mu(p)\right)=p^{4}-p^{2}.\end{split}$

Thus we conclude that

(3.39)
$U_{S}(p)U_{S}(p)^{*}=p^{2}(p^{2}-1)T_{S}(1,p,p^{2},p)+p^{2}(p-1)T_{S}(1,1,p^{2},p^{2})+p^{3}\mu(p)T_{S}(p,p,p,p).$

Now we have that

(3.40) $T_{S}(1,p,p^{2},p)=p^{-1}T_{S}^{\prime}(p)-(1+p+p^{2})T_{S}(p,p,p,p).$

Combining this with the expression for $T_{S}(1,1,p^{2},p^{2})$ from (3.16),
we see that

$\displaystyle(p+1)T_{S}(1,p,p^{2},p)+T_{S}(1,1,p^{2},p^{2})$
$\displaystyle=T_{S}(p)^{2}+\left(p(p+1)-(p+1)(1+p+p^{2})\right)T_{S}(p,p,p,p)$
(3.41) $\displaystyle=T_{S}(p)^{2}-(p+1)(p^{2}+1)T_{S}(p,p,p,p).$

Thus

(3.42)
$U_{S}(p)U_{S}(p)^{*}\cdot\mathrm{Tr}_{Np,N}=p^{2}(p-1)T_{S}(p)^{2}+\left(-p^{2}(p^{4}-1)+p^{3}\mu(p)\right)T_{S}(p,p,p,p).$

Thus by noting that $\mu(p)=(p+1)(p^{2}+1)$ and that
$F|T_{S}(p,p,p,p)=p^{2k-6}F$, we get that

(3.43) $\langle
F,F^{\prime}|U_{S}(p)U_{S}(p)^{*}\cdot\mathrm{Tr}_{Np,N}\rangle_{Np}=\mu(p)^{-1}\left(p^{2}(p-1)\lambda_{F^{\prime}}(p)^{2}+p^{2k-4}\mu(p)\right)\langle
F,F^{\prime}\rangle_{Np}.\qed$

######  Lemma 3.8.

Let $N$ be any positive integer and $p\nmid N$ and $F,F^{\prime}$ be any two
Siegel cusp forms of level $N$. Further, let $F$ an eigenfunction of the
operators $T_{S}(p)$ and $T_{S}^{\prime}(p)$ with the eigenvalues
$\lambda_{F}(p)$ and $\lambda_{F}^{\prime}(p)$, respectively. Then

(3.44) $\langle
F|U_{S}(p)|W_{p},F^{\prime}|U_{S}(p)\rangle_{Np}=\frac{\lambda_{F}(p)}{p^{k-3}\mu(p)}\left(\lambda_{F}(p)^{2}-(2+\frac{1}{p})\lambda_{F}^{\prime}(p)+p^{2k-5}(p+1)(p+2)\right)\cdot\langle
F,F^{\prime}\rangle_{Np}.$

###### Proof.

As in Lemma 3.3, we have that
$F|U_{S}(p)|W_{p}=F|B_{p^{2}}|\mathrm{Tr}_{Np^{2},Np}$. Thus we get

(3.45) $\displaystyle\langle F|U_{S}(p)|W(p),F^{\prime}|U_{S}(p)\rangle_{Np}$
$\displaystyle=\langle
F|B_{p^{2}}|\mathrm{Tr}_{Np^{2},Np},F^{\prime}|U_{S}(p)\rangle_{Np}$ (3.46)
$\displaystyle=p^{k-3}\mu(p)^{-1}\langle
F|B_{p^{2}}|\mathrm{Tr}_{Np^{2},Np}|U_{S}(p)^{*}|\mathrm{Tr}_{Np,N},F^{\prime}\rangle_{Np}.$

In terms of double cosets, the operator
$B_{p^{2}}|\mathrm{Tr}_{Np^{2},Np}|U_{S}^{*}(p)|\mathrm{Tr}_{Np,N}$ can be
written as

(3.47) $\displaystyle
p^{6-2k}\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p^{2}&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(Np^{2})\cdot\Gamma^{(2)}_{0}(Np^{2})1_{4}\Gamma^{(2)}_{0}(Np)\cdot\Gamma^{(2)}_{0}(Np)p\left(\begin{smallmatrix}1&0\\\
0&p^{-1}\end{smallmatrix}\right)\Gamma^{(2)}_{0}(N)$ (3.48)
$\displaystyle=p^{6-2k}\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p^{2}&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(Np)\begin{pmatrix}p&0\\\ 0&1\\\
\end{pmatrix}\Gamma^{(2)}_{0}(N).$

The double coset operator on the RHS above is of similitude $p^{3}$ and degree
$p^{3}\mu(p^{2})$. We also have that the operator $T_{S}(1,1,p^{3},p^{3})$ is
of similitude $p^{3}$ and degree $p^{3}\mu(p^{2})$. Thus from the “symplectic
divisor theorem” we have

(3.49) $\Gamma^{(2)}_{0}(N)\left(\begin{smallmatrix}p^{2}&0\\\
0&1\end{smallmatrix}\right)\Gamma^{(2)}_{0}(Np)\begin{pmatrix}p&0\\\ 0&1\\\
\end{pmatrix}\Gamma^{(2)}_{0}(N)=T_{S}(1,1,p^{3},p^{3}).$

Thus we get

(3.50) $\langle
F|U_{S}(p)|W_{p},F|U_{S}(p)\rangle_{Np}=p^{3-k}\mu(p)^{-1}\langle
F|T_{S}(1,1,p^{3},p^{3})F\rangle_{Np}.$

Now we express the operator $T_{S}(1,1,p^{3},p^{3})$ in terms of $T_{S}(p)$
and $T_{S}^{\prime}(p)$. First, we use the computations available from [11] to
write

(3.51) $\displaystyle
T_{S}(1,1,p^{3},p^{3})=x_{0}^{3}\left(\mathbf{x}_{3}^{(0,0)}+\frac{p-1}{p}\mathbf{x}_{3}^{(0,1)}+\frac{(p-1)(2p-1)}{p^{2}}\mathbf{x}_{3}^{(1,1)}\right),$

where $\mathbf{x}_{3}^{(a_{1},a_{2})}$ denotes the symmetric polynomial

(3.52) $\displaystyle\mathbf{x}_{3}^{(a_{1},a_{2})}:=\sum_{(b_{1},b_{2})\in
W_{3}(a_{1},a_{2})}x_{1}^{b_{1}}x_{2}^{b_{2}}.$

Here $W_{3}(a_{1},a_{2})$ is the Weyl orbit of $(a_{1},a_{2})$ under the Weyl
group – in our case, since $(a_{1},a_{2})\in\\{0,1\\}$; they are simply

(3.53) $\displaystyle W_{3}(0,0)$
$\displaystyle=\\{(0,0),(0,3),(3,0),(3,3)\\},$ (3.54) $\displaystyle
W_{3}(0,1)$
$\displaystyle=\\{(0,1),(1,0),(3,1),(1,3),(0,2),(2,0),(3,2),(2,3)\\},$ (3.55)
$\displaystyle W_{3}(1,1)$ $\displaystyle=\\{(1,1),(2,1),(1,2),(2,2)\\}.$

This gives us

(3.56) $\displaystyle\mathbf{x}_{3}^{(0,0)}$
$\displaystyle=1+x_{1}^{3}+x_{2}^{3}+x_{1}^{3}x_{2}^{3},$ (3.57)
$\displaystyle\mathbf{x}_{3}^{(0,1)}$
$\displaystyle=x_{1}+x_{2}+x_{1}x_{2}^{3}+x_{1}^{3}x_{2}+x_{1}^{2}+x_{2}^{2}+x_{1}^{2}x_{2}^{3}+x_{1}^{3}x_{2}^{2},$
(3.58) $\displaystyle\mathbf{x}_{3}^{(1,1)}$
$\displaystyle=x_{1}x_{2}+x_{1}x_{2}^{2}+x_{1}^{2}x_{2}+x_{1}^{2}x_{2}^{2}.$

We also need the images for the operators $T_{S}(p)$ and the similitude
$p^{2}$ Hecke operators in the polynomial algebra under the Satake map, see
e.g., [5, 6], [11, Appendix].

(3.59) $\displaystyle T_{S}(p)$ $\displaystyle=x_{0}(1+x_{1})(1+x_{2}),$
(3.60) $\displaystyle T_{S}(1,1,p^{2},p^{2})$
$\displaystyle=x_{0}^{2}\left((1+x_{1}^{2})(1+x_{2}^{2})+\frac{p-1}{p}(x_{1}+x_{2})(1+x_{1}x_{2})+\frac{2(p-1)}{p}x_{1}x_{2}\right),$
(3.61) $\displaystyle T_{S}(1,p,p^{2},p)$
$\displaystyle=x_{0}^{2}\left(\frac{p^{2}-1}{p^{3}}x_{1}x_{2}+\frac{1}{p}(x_{1}+x_{2})(1+x_{1}x_{2})\right),$
(3.62) $\displaystyle T_{S}(p,p,p,p)$
$\displaystyle=\frac{1}{p^{3}}x^{2}_{0}x_{1}x_{2}.$

Thus to express $T_{S}(1,1,p^{3},p^{3})$ as a polynomial in the similitude
$p,p^{2}$ operators, it is enough to do so for their images. We obtain

(3.63) $\displaystyle T_{S}(1,1,p^{3},p^{3})$
$\displaystyle=x_{0}^{3}(1+x_{1})(1+x_{2})\left((1-\frac{1}{p}x_{1}+x_{1}^{2})(1-\frac{1}{p}x_{2}+x_{2}^{2})+\frac{p-1}{p}x_{1}x_{2}\right)$
(3.64)
$\displaystyle=x_{0}^{3}(1+x_{1})(1+x_{2})\left((1+x_{1}^{2})(1+x_{2}^{2})-\frac{1}{p}(x_{1}+x_{2})(1+x_{1}x_{2})+\frac{1-p+p^{2}}{p^{2}}x_{1}x_{2}\right)$
(3.65)
$\displaystyle=T_{S}(p)\left(T_{S}(1,1,p^{2},p^{2})-pT_{S}(1,p,p^{2},p)+p^{2}T_{S}(p,p,p,p)\right)$
(3.66)
$\displaystyle=T_{S}(p)\left((p+1)T_{S}(1,1,p^{2},p^{2})-pT_{S}(p^{2})+p(p+1)T_{S}(p,p,p,p)\right).$

Since
$T_{S}(1,1,p^{2},p^{2})=T_{S}(p)^{2}-(1+\frac{1}{p})T^{\prime}_{S}(p)+p(p+1)T_{S}(p,p,p,p)$,
the terms inside the bracket can be written as

(3.67)
$\displaystyle(p+1)T_{S}(p)^{2}-p^{-1}(p+1)^{2}T^{\prime}_{S}(p)-pT_{S}(p^{2})+p(p+1)(p+2)T_{S}(p,p,p,p)$
(3.68)
$\displaystyle=T_{S}(p)^{2}+pT_{S}^{\prime}(p)-p^{-1}(p+1)^{2}T^{\prime}_{S}(p)+p(p+1)(p+2)T_{S}(p,p,p,p)$
(3.69)
$\displaystyle=T_{S}(p)^{2}-\left(2+\frac{1}{p}\right)T_{S}^{\prime}(p)+p(p+1)(p+2)T_{S}(p,p,p,p).$

Thus we get

(3.70)
$T_{S}(1,1,p^{3},p^{3})=T_{S}(p)\left(T_{S}(p)^{2}-\left(2+\frac{1}{p}\right)T_{S}^{\prime}(p)+p(p+1)(p+2)T_{S}(p,p,p,p)\right).$

The proof is now complete by substituting the above expression for
$T_{S}(1,1,p^{3},p^{3})$ in (3.50). ∎

##  4\. Saito-Kurokawa lifts of higher level

Apart from our immediate (analytic) applications, this part also has its
independent interest and can serve as a classical treatment of SK lifts of
higher levels.

Higher level SK newforms have been described from the viewpoint of
representation theory, see e.g., [50]. Namely Schmidt shows that when $k$ is
even, $N$ is square-free, and $f\in S_{2k-2}^{new}(N)$ is newform there is a
functorial lifting from
$\mathrm{SL}(2)\times\mathrm{SL}(2)\rightarrow\mathrm{PGSp}(4)$ such that
$1\times\pi_{f}$ defines an automorphic form, which is the SK lift in
question. Other local sign conditions lead to paramodular forms.

The definition via $L$-functions which we adopt, is a consequence of this
approach, recalled briefly below. However, for the purpose of this paper, the
classical version is the backbone, and so we reformulate the situation in the
classical language as well.

###  4.1. Definition of SK lifts

Let $k$ be even and $N$ square-free. Let $F\in S_{k}^{(2)}(N)$ be a Hecke
eigenform of all Hecke operators $T_{S}(m)$ for $(m,N)=1$ and $\lambda_{F}(m)$
be the corresponding Hecke eigenvalues. Define

(4.1)
$L^{N}(F,s):=\zeta^{N}(2s+1)\sum\nolimits_{(m,N)=1}\frac{\lambda_{F}(m)}{m^{s}};$

the prime-to-$N$ spinor zeta function of $F$. Then $F$ is called a classical
Saito-Kurokawa lift, if there exists an integral weight newform $f$ of weight
$2k-2$ and level $M|N$ such that

(4.2) $L^{N}(F,s)=\zeta^{N}(s+1/2)\zeta^{N}(s-1/2)L^{N}(f,s).$

Then the subspace of Saito-Kurokawa lifts of level $N$, $\mathrm{SK}_{k}(N)$
is defined as the span of all such Hecke eigenforms $F\in S_{k}^{(2)}(N)$ that
satisfy the property (4.2) (see [21, 50] for more details).

###  4.2. Old and newforms in degree 2

For $N$ square-free, we now briefly recall the theory of oldforms in the
classical language from [49] – namely $F\in S_{k}^{(2)}(N)$ is ‘old’ at level
$N$ if there is a prime $p|N$ and $F$ arises as the image of one of the
following operators acting on modular forms of level $N/p$:

  1. (1)

The identity map $T_{0}(p)$.

  2. (2)

The Atkin-Lehner operator $W_{p}$ defined as in (3.1).

  3. (3)

The $U_{S}(p)$ operator defined as in (3.2).

  4. (4)

The operator $U_{S}(p)W_{p}$.

The span of such ‘old’ forms is the oldspace – denoted by
$S_{k}^{(2),old}(N)$. The space of newforms $S_{k}^{(2),new}(N)$ is defined as
the orthogonal complement of $S_{k}^{(2),old}(N)$. We put

(4.3)
$\displaystyle\mathrm{SK}_{k}^{new}(N):=S_{k}^{(2),new}(N)\cap\mathrm{SK}_{k}(N);\quad\mathrm{SK}_{k}^{old}(N):=S_{k}^{(2),old}(N)\cap\mathrm{SK}_{k}(N).$

To proceed further, it will be necessary for us to describe the above spaces
more concretely – namely as functorial lifts of spaces of Jacobi forms – which
was the classical approach taken by Maaß, Eichler-Zagier [22] and finally
Ibukiyama [28] in level $N$. This is what we discuss next.

###  4.3. Classical lifts from Jacobi forms (EZI lifts):

Let $\mathcal{L}_{N}:J_{k,1}^{cusp}(N)\longrightarrow S_{k}^{(2)}(N)$
($\phi\mapsto F=\mathcal{L}_{N}\phi$) denote the lifting map of level $N$ from
[28] defined via an explicit Fourier expansion as given below:

(4.4)
$\displaystyle\mathcal{L}_{N}\phi(\tau,z,\tau^{\prime}):=\sum\nolimits_{m\geq
1}(\phi|V_{m})(\tau,z)e(m\tau^{\prime}),$

where $V_{m}$ is the operator defined as in (2.1).

In [28], it is shown that $\mathcal{L}_{N}$ is injective and Hecke-equivariant
in a certain sense. Let $\mathrm{SK}_{k}^{J}(N)$ denote the
$\mathcal{L}_{N}$\- image of $J_{k,1}^{cusp}(N)$ inside $S_{k}^{(2)}(N)$. From
the work of Ibukiyama [28] it follows that
$\mathrm{SK}_{k}^{J}(N)\subset\mathrm{SK}_{k}(N)$. More precisely, we refer
the reader to the REMARK after [28, Theorem 4.1] where he shows (4.2) for the
eigenforms in $\mathrm{SK}_{k}^{J}(N)$. The newspaces in each coincide,
however, this is a strict inclusion for the corresponding oldspaces. For
example, it turns out that $\mathrm{SK}_{k}^{J}(p)$ captures all the newforms
in $\mathrm{SK}_{k}(p)$; and all oldforms therein except for one oldform per
$\phi$ of level one (see subsection 4.7). This picture generalises to square-
free $N$.

Let $\phi\in J_{k,1}^{cusp,new}(N)$ and $p\nmid N$. Consider the EZI lift
$F=\mathcal{L}_{N}(\phi)$. Then we have the following relation between the
eigenvalues ([28, Theorem 4.1]).

(4.5) $\lambda_{F}(p)=\lambda_{\phi}(p)+p^{k-1}+p^{k-2}.$

Now $\lambda_{\phi}(p)=\lambda_{f}(p)$. Therefore, note that by Deligne’s
bound, $\displaystyle\lambda_{F}(p)\asymp p^{k-1}$.

###  4.4. SK newforms as EZI lifts

In this subsection, we show that the EZI lifts of Jacobi newforms of level $N$
are precisely those in the sense of R. Schmidt’s definition – i.e., they are
orthogonal to the oldspace. This is known from [46, Lem. 5.2.2], but our proof
is classical, and in the spirit of this paper.

######  Lemma 4.1.

Let $\mathrm{SK}_{k}^{new}(N)$ be the space of newforms in
$\mathrm{SK}_{k}(N)$ as defined in (4.3) and let $\mathrm{SK}_{k}^{J,new}(N)$
be the space of spanned by the EZI lifts of newforms in $J^{cusp}_{k,1}(N)$.
Then $\mathrm{SK}_{k}^{new}(N)=\mathrm{SK}_{k}^{J,new}(N)$ and all newforms in
$\mathrm{SK}_{k}(N)$ are precisely the EZI lifts of the Jacobi newforms as
above.

###### Proof.

The inclusion $\mathrm{SK}_{k}^{J,new}(N)\subset\mathrm{SK}_{k}^{new}(N)$
would follow if we can show that any EZI lift
$F=\mathcal{L}_{N}(\psi)\in\mathrm{SK}_{k}^{J,new}(N)$ is orthogonal to all
oldforms. We prove the inclusion by induction on the number of prime factors
of $N$. When $N=1$, this is true by the classical theory of Eichler-Zagier.
Now assume that the inclusion holds for level $N/p$ with square-free $N$ and
$(p,N/p)=1$. Then we know that
$\mathrm{SK}_{k}^{old}(N)=\sum_{M|N,M<N}\mathrm{SK}_{k}^{new}(M)|\prod_{q|\frac{N}{M}}R(q)$.
Thus by induction hypothesis, we have to check that

(4.6) $\langle\mathcal{L}_{N}(\psi),\mathcal{L}_{M}(\phi)|R(q)\rangle=0\text{
for }q|\frac{N}{M},$

where $\phi\in J_{k,1}^{new}(M)$ and $\psi\in J_{k,1}^{new}(N)$. Note that the
Hecke operators $T_{S}(q^{\prime})$ with $(q^{\prime},N)=1$ act the same way
on both $\mathrm{SK}_{k}^{new}(N)$ and $\mathrm{SK}_{k}^{new}(M)$ and commute
with the operators $R(q)$. The assertion now follows, since $\psi$, $\phi$ are
newforms of different levels, $\langle\psi,\phi\rangle=0$, via multiplicity-
one for $\mathrm{GL}(2)$.

For the other inclusion, we note that the spaces have the same dimension, as
follows on the one hand from the injectiveness of the EZI map
$\mathcal{L}_{N}$ (cf. [28]), and on the other hand from Schmidt’s
construction and uniqueness [50, Thm. 5.2 (ii)] or from [46, Lem.5.2.2]
respectively. ∎

######  Remark 4.2.

The reader may directly verify in the above proof that
$\displaystyle\langle\mathcal{L}_{N}(\psi),\mathcal{L}_{N}(\phi)|R(q)\rangle=c\cdot\langle\psi,\phi\rangle$
for some constant $c$, if $R(q)=Id,U(q)$. This follows for e.g., from the
calculations in [2]: note that the calculations work fine if at least one of
the components is a newform.

###  4.5. Characterization of SK lifts of level.

In this section, we provide a characterization of SK lifts in the level aspect
(see [23] for level $1$) in terms of the inner product matrix $M_{p}$.
Throughout this subsection, let $N$ be a square-free positive integer and $F$
be a newform of level $N$ with eigenvalues $\lambda_{F}(n)$ for $(n,N)=1$. For
any $p\nmid N$, let $F_{1,p}=F$, $F_{2,p}=F|U_{S}(p)$, $F_{3,p}=F|W_{p}$ and
$F_{4,p}=F|U_{S}(p)W_{p}$.

######  Lemma 4.3.

For any $p\nmid N$ and a newform $F\in S_{k}^{(2)}(N)$, the matrix $M_{p}(F)$
defined as in (1.2) has rank at least 3.

###### Proof.

It is enough to show that the leading $3\times 3$ minor is non-zero. Using the
inner product relations from Section 3, the leading minor is given by 333This
can be verified by a SAGE code available here. (see (3.8) and Lemma 3.3 for
$\lambda_{F}^{\prime}(p)$)

(4.7)
$\begin{split}-&p^{-2\,k}\left(p^{2}+1\right)^{-1}[\Gamma_{0}^{(2)}(N):\Gamma_{0}^{(2)}(Np)]^{-2}\left(p\lambda_{F}(p)^{2}-(p^{2}+1)\lambda_{F}^{\prime}(p)-p^{2\,k-2}-p^{2\,k}\right)\\\
&\times\left(p^{5}\lambda_{F}(p)^{2}-p^{4}(p+1)^{2}\lambda_{F}^{\prime}(p)+\left(p^{2}+2\right){\left(p+1\right)}^{2}p^{2\,k}\right)\langle
F,F\rangle_{Np}.\end{split}$

When $F$ is not an SK lift, then $\lambda_{F}(p)\ll p^{k-3/2}$ and
$\lambda_{F}^{\prime}(p)\ll p^{2k-3}$. Thus both terms involving the
eigenvalues in (4.7) are non-zero.

When $F$ is an SK lift, then we have that
$\lambda_{F}^{\prime}(p)=(p^{k-1}+p^{k-2})\lambda_{F}(p)-p^{2k-2}$. Thus (4.7)
reduces to

(4.8)
$\begin{split}-&p^{-2k+6}\left(p^{2}+1\right)^{-1}[\Gamma_{0}^{(2)}(N):\Gamma_{0}^{(2)}(Np)]^{-2}\lambda_{F}(p)\left(\lambda_{F}(p)-2(p^{k-1}+p^{k-2})\right)\\\
&\left(\lambda_{F}(p)-p^{k-3}(p^{2}+1)(p+1)\right)^{2}\langle
F,F\rangle_{Np}.\end{split}$

Now since $F$ is an SK lift (say of $f\in S_{2k-2}(N)$), one has
$\lambda_{F}(p)=p^{k-1}+p^{k-2}+\lambda_{f}(p)$ from (4.2) and by Deligne,
$|\lambda_{f}(p)|\leq 2p^{k-3/2}$. Thus we get the required non-vanishing.

Thus, in both cases the leading minor is non-zero. Thus $M_{p}$ has rank at
least 3. ∎

With the preparations so far, we can now obtain the desired characterization
of SK lifts – stated as Theorem 1.2 in the introduction.

######  Theorem 4.4.

Let $N$ be a square-free positive integer. Then the following are equivalent.

  1. (1)

$F$ is an SK lift (in the sense of subsection 4.1) of an elliptic eigenform of
level $N$ and weight $2k-2$ or equivalently a Jacobi eigenform of square-free
level $N$.

  2. (2)

For all $p\nmid N$,

$\lambda_{F}(p^{2})=\lambda_{F}(p)^{2}-(p^{k-1}+p^{k-2})\lambda_{F}(p)+p^{2k-2}.$

  3. (3)

There exists a prime $p\nmid N$, such that

$\lambda_{F}(p^{2})=\lambda_{F}(p)^{2}-(p^{k-1}+p^{k-2})\lambda_{F}(p)+p^{2k-2}.$

  4. (4)

The inner product matrix $M_{p}:=\left(\langle
F_{i,p},F_{j,p}\rangle_{Np}\right)_{1\leq i,j\leq 4}$ has rank $3$ for some
prime $p\nmid N$.

  5. (5)

For all $p\nmid N$, $M_{p}$ has rank $3$.

###### Proof.

The statements (1)-(3) are the level analogues of the corresponding statements
for level $1$ in [23, Theorem 4.1]. Their equivalence follows from the same
arguments as in [23]. Namely, the main point for level $1$ was that $F$
satisfies the Ramanujan conjecture if and only if it is not an SK lift. This
assertion is also true for level $N$ newforms from the work of Weissauer [55].
Thus our $F$, which shares the same eigenvalues as that of a newform of some
level $M|N$ has the same property away from the level. We do not reproduce the
argument again.

We next show that (5)$\implies$(4)$\implies$(1)$\implies$(2)$\implies$ (5).
The first implication (5)$\implies$(4) is trivial. To show (4)$\implies$ (1),
as in Lemma 4.3, if $F$ is not an SK lift, then we know that
$|\lambda_{F}(p)|\leq 4p^{k-3/2}$ and $|\lambda_{F}(p^{2})|\leq 10p^{2k-3}$
which follows from Weissauer’s proof of the Ramanujan conjecture for $F$, see
[55]. Next, note that for any $p\nmid N$ 444One can use the same SAGE code
linked above to verify this.

(4.9) $\begin{split}\frac{\det(M_{p})}{\langle
F,F\rangle_{Np}}=&p^{-4k}{\left(p^{2}+1\right)}^{-4}{\left(p+1\right)}^{-4}\left(p^{5}\lambda_{F}(p)^{2}-p^{4}(p+1)^{2}\lambda_{F}^{\prime}(p)+\left(p^{2}+2\right){\left(p+1\right)}^{2}p^{2\,k}\right)^{2}\\\
&\times\left(\lambda_{F}^{\prime}(p)+(p^{k-1}+p^{k-2})\lambda_{F}(p)+p^{2\,k-2}\right){\left(\lambda_{F}^{\prime}(p)-(p^{k-1}+p^{k-2})\lambda_{F}(p)+p^{2\,k-2}\right)}.\end{split}$

Thus the terms involving eigenvalues $\lambda_{F}(p)$ and
$\lambda_{F}^{\prime}(p)$ are non-zero in (4.9). Thus $\det(M_{p})\neq 0$.

We have already shown (1)$\implies$(2). To show (2)$\implies$(5), recall from
Lemma 4.3 that the rank of $M_{p}$ is at least $3$ for any $p\nmid N$. Thus to
show (2) $\implies$ (5) it is enough to show that rank is equal to $3$ when
the condition (2) is satisfied. We will now show that the rank of $M_{p}\leq
3$. Indeed, (2) says that
$\lambda_{F}^{\prime}(p)=\lambda_{F}(p)^{2}-\lambda_{F}(p^{2})$. Thus from
(4.9), $\det(M_{p})=0$ as soon as (2) is satisfied. Thus $M_{p}$ has rank $3$
for all $p\nmid N$. This completes the proof. ∎

###  4.6. SK oldforms in degree 2

For a prime $p$ with $p|N$, define, for each $\phi\in J_{k,1}^{new}(N/p)$,

(4.10) $\displaystyle V(\phi;N/p):=\mathrm{Span}<\mathcal{L}_{N/p}\phi|R,\quad
R=Id,W_{p},U_{S}(p),U_{S}(p)W_{p}>.$

We comment here that it is possible to deduce that $\dim V(\phi;N/p)=3$ if we
combine Theorem 4.4 with Lemma 4.1 and replace $N$ with $N/p$.

But here we give an alternate proof of the same fact by expressing
$(L_{N/p}\phi)|U_{S}(p)W_{p}$ as an explicit linear combination of other $3$
lifts of $\phi$. We believe that an explicit linear relation is useful in
various contexts, including the sup-norm problem. Moreover, the following
calculations also serve as a sanity check for the calculations about the
discriminant of $M_{p}$ in the proof of Theorem 4.4.

######  Lemma 4.5.

For each $\phi\in J_{k,1}^{new}(N/p)$, there is an explicit linear relation
(given in (4.15)) among the $4$ oldforms listed in (4.10); in particular, one
has $\dim V(\phi;N/p)=3$.

###### Proof.

Let $\phi\in J_{k,1}^{cusp,new}(N/p)$, and put $F=\mathcal{L}_{N/p}\phi$. For
some constant $c$, let $v=v_{c}:=F+cF|U_{S}(p)$ and $w=w_{c}:=-v|W_{p}$. Now,
it is enough to show that $\langle v+w,v+w\rangle=0$ for some unique $c$. To
achieve this, we make extensive use of the inner product relations obtained in
the previous section.

We first note that $\langle v+w,v+w\rangle=0$ iff $c$ satisfies the quadratic
equation $Ax^{2}+Bx+C$, where

(4.11) $\begin{split}A&=2\left(\langle
F|U_{S}(p),F|U_{S}(p)\rangle_{Np}-\langle
F|U_{S}(p),F|U_{S}(p)W_{p}\rangle_{Np}\right);\\\ B&=4\left(\langle
F,F|U_{S}(p)\rangle_{Np}-\langle F|U_{S}(p),F|W_{p}\rangle_{Np}\right);\\\
C&=2\left(\langle F,F\rangle_{Np}-\langle
F,F|W_{p}\rangle_{Np}\right).\end{split}$

From the inner product calculation it is clear that $A,B\neq 0$, thus showing
that the discriminant $D:=B^{2}-4AC=0$ would give us the unique $c$.

Using the inner product relations, we can see that $D$ factorises into 555One
can use the same SAGE code linked above to verify this.

(4.12)
$\begin{split}16&p^{-2k}[\Gamma_{0}^{(2)}(N):\Gamma_{0}^{(2)}(Np)]^{-2}\langle
F,F\rangle_{Np}\left(-\lambda_{F}^{\prime}(p)+(p^{k-1}+p^{k-2})\lambda_{F}(p)-p^{2k-2}\right)\\\
&\times\left(p^{5}\lambda_{F}(p)^{2}-p^{4}(p+1)^{2}\lambda_{F}^{\prime}(p)+\left(p^{2}+2\right){\left(p+1\right)}^{2}p^{2\,k}\right).\end{split}$

Now on the space of SK lifts of level $N/p$, we have that

(4.13) $\displaystyle
T^{\prime}_{S}(p)=(p^{k-1}+p^{k-2})T_{S}(p)-p^{2k-2}Id.\quad\text{ since
}(p,N/p)=1.$

Thus we have that

(4.14)
$\left(-\lambda_{F}^{\prime}(p)+(p^{k-1}+p^{k-2})\lambda_{F}(p)-p^{2k-2}\right)=0.$

This in turn implies that $D=0$ and gives us the unique $c$. To conclude, we
have thus $v=-w=v|W_{p}$, which means

(4.15) $\displaystyle F_{1}+cF_{2}=F_{3}+cF_{4},\quad
c=-B/2A=\frac{1}{p^{k-2}-\lambda_{F}(p)}.$

The denominator in the expression for $c$ is clearly non-zero from (4.5).
Therefore we have also shown that $\dim V(\phi;N/p)\leq 3$. Lemma 4.5 now
follows from Lemma 4.3. ∎

######  Remark 4.6.

The reader may notice that another natural way to prove Lemma 4.5 would be
simply to compare the the respective Fourier expansions of the oldforms.
However this would entail understanding the Fourier expansions of cusp forms
at various cusps. For instance when $N=p$, one would need a Fourier expansion
of $F|U(p)W_{p}$, where $F$ is a SK lift of level $1$. In principle this may
be handled e.g., by considering the Rankin-Selberg convolution
$R(F|\gamma,\Theta,s)$ for a suitable theta/Eisenstein series $\Theta$ (e.g.,)
and then ‘transferring’ the $\gamma$ via the integral representation onto a
Siegel Eisenstein series, which we understand at the various cusps reasonably
well. In any case, this will involve non-trivial arguments if it works.
Surprisingly, even for $F\in\mathrm{SK}_{k}(1)$, it seems to be quite delicate
to prove that $F,F|B_{p},F|U_{S}(p)$ are linearly independent by only using
their Fourier expansions.

######  Lemma 4.7.

With the above notation, one has

(4.16)
$\displaystyle\mathrm{SK}_{k}^{old}(N)=\sum\nolimits_{p|N}\left(\mathcal{L}_{\frac{N}{p}}(J_{k,1}^{cusp}(\frac{N}{p}))+\mathcal{L}_{\frac{N}{p}}(J_{k,1}^{cusp}(\frac{N}{p}))|W_{p}++\mathcal{L}_{\frac{N}{p}}(J_{k,1}^{cusp}(\frac{N}{p}))|U_{S}(p)\right).$

###### Proof.

Let us call the RHS of (4.16) as $V(\frac{N}{p})$. We notice from the
description of oldforms from subsection 4.2 and the REMARK after [28, Theorem
4.1], that $V(\frac{N}{p})\subset\mathrm{SK}_{k}^{old}(N)$.

For the converse, let $G\in\mathrm{SK}_{k}^{old}(N)$, which we can assume to
be a Hecke eigenform. From definition (4.3), and distinguishing via
eigenvalues, we can write

(4.17) $\displaystyle
G=\sum\nolimits_{M|N,M<N}F_{M}|\prod\nolimits_{p|\frac{N}{M}}R(p),$

where $F_{M}$ runs over newforms of level $M$ and for each $p|NM^{-1}$, $R(p)$
are the $4$ old-operators. Since $F_{M}$ is a newform, it follows from Lemma
4.1 that $F_{M}$ is an EZI lift of a Jacobi newform of level $M$. Thus by
multiplicity-one in the newspace and distinguishing via eigenvalues, we see
that only one $M$ can survive in the above sum. Writing out the terms in
(4.17) and using Lemma 4.5 then gives the lemma. ∎

######  Corollary 4.8.

Let $N$ be square-free. From the above description, we can inductively write

(4.18)
$\displaystyle\mathrm{SK}_{k}(N)=\sum\nolimits_{M|N}\sum\nolimits_{d,e\colon
de|\frac{N}{M}}\mathrm{SK}_{k}^{new}(M)|W_{d}U_{e}$

Therefore by (generalized) Möbius inversion one obtains the relation

$\dim\mathrm{SK}_{k}^{new}(N)=\sum\nolimits_{M|N}\beta(N/M)\dim\mathrm{SK}_{k}(M),$

where $\beta(n)$ are defined by the generating series
$\displaystyle\sum\nolimits_{n\geq 1}\beta(n)n^{-s}=\zeta(s)^{-3}$, and thus
also a formula for $\dim\mathrm{SK}_{k}^{old}(N)$.

###  4.7. Distinguishing EZI lifts via Maaß relations

We know that the space of EZI lifts sit inside the space of SK lifts and
satisfy the so-called Maaß relations (cf. [28, Prop. 3.8]). In this subsection
we show that this condition actually characterizes them among all SK lifts of
level $N$.

######  Lemma 4.9.

Let $p|N$ and $F\in\mathrm{SK}_{k}^{J}(N/p)$ be non-zero. Then
$F|W_{p}\not\in\mathrm{SK}_{k}^{J}(N)$.

###### Proof.

From [26], $G\in\mathrm{SK}_{k}^{J}(N)$ iff the Maaß relations (1.1) are
satisfied. Thus it is enough to show that the Fourier coefficients of
$F|W_{p}$ do not satisfy (1.1). From (3.5) applied to $N/p$, let us recall
that $\gamma W_{p}=B_{p}$ for some $\gamma\in\Gamma_{0}(N/p)$ and thus
$F|W_{p}=F|B_{p}=p^{k}F(pZ)$. Therefore its Fourier expansion can be written
as

(4.19) $F|W_{p}=\sum\nolimits_{T}A^{*}_{F}(T)e(\mathrm{tr}(TZ)),$

where $A^{*}_{F}(T)=p^{k}A_{F}(T/p)$ when $p|\mathbf{c}(T)$ and is zero
otherwise. If $F|W_{p}$ satisfied the Maaß relations, then we would have

(4.20) $\displaystyle
A^{*}_{F}(n,r,m)=\sum_{d|(n,r,m),(d,N)=1}d^{k-1}A^{*}_{F}(\frac{nm}{d^{2}},\frac{r}{d},1).$

By the aforementioned property of $A^{*}_{F}(T)$, each term on the RHS above
is zero. Since this is true for any $A^{*}_{F}(T)$, we see that $F|W_{p}=0$,
which is impossible. This gives us the desired conclusion. ∎

######  Theorem 4.10.

An element $F\in\mathrm{SK}_{k}(N)$ is an EZI lift (i.e., lies in
$\mathrm{SK}_{k}^{J}(N)$) if and only if it satisfies the Maaß relations
(1.1).

###### Proof.

Let $F\in\mathrm{SK}_{k}(N)$ be an EZI lift. Then from [26], $F$ satisfies the
Maaß relations.

For the converse, from Lemma 4.1
$\mathrm{SK}_{k}^{new}(N)=\mathrm{SK}_{k}^{J,new}(N)$. Thus it is enough to
consider the oldspace $\mathrm{SK}_{k}^{old}(N)$. From Lemma 4.5 and 4.7, we
have

(4.21)
$\mathrm{SK}_{k}^{old}(N)=\sum\nolimits_{p|N}\left(\mathcal{L}_{\frac{N}{p}}(J_{k,1}^{cusp}(\frac{N}{p}))+\mathcal{L}_{\frac{N}{p}}(J_{k,1}^{cusp}(\frac{N}{p}))|W_{p}+\mathcal{L}_{\frac{N}{p}}(J_{k,1}^{cusp}(\frac{N}{p}))|U_{S}(p)\right).$

From [28], we know the relation
$\mathcal{L}_{N}(\phi)=\mathcal{L}_{N/p}(\phi)-p^{-1}\mathcal{L}_{N/p}(\phi)|W_{p}$.
Thus combining with the fact that $\mathcal{L}_{N}$ and $U_{S}(p)$ operators
commute, we can write

(4.22)
$\mathrm{SK}_{k}^{old}(N)=\sum\nolimits_{p|N}\left(\mathcal{L}_{N}(J_{k,1}^{cusp}(\frac{N}{p}))+\mathcal{L}_{N}(U_{J}(p)(J_{k,1}^{cusp}(\frac{N}{p})))\right)+\sum\nolimits_{p|N}\mathcal{L}_{\frac{N}{p}}(J_{k,1}^{cusp}(\frac{N}{p}))|W_{p}.$

By definition, the first two components satisfy the Maaß relation (1.1) and
all the EZI lifts are contained inside them. Clearly the third component
$\sum_{p|N}\mathcal{L}_{\frac{N}{p}}(J_{k,1}^{cusp}(\frac{N}{p}))|W_{p}$ is
non-zero. Thus it is enough to show that the third component doesn’t satisfy
the relations and this is given by Lemma 4.9. ∎

##  5\. An orthonormal basis for SK lifts of level $p$.

For any $\phi\in J_{k,1}^{cusp}$, from Lemma 4.5, we know that the old-space
corresponding to $\phi$ is $3$ dimensional. Let us denote this space by
$\mathrm{SK}_{k}^{old}(p;\phi)$. In view of the supnorm problem, our first
goal is to obtain an explicit orthonormal basis for SK lifts of level $p$
which is $W_{p}$ invariant. In this direction, for $\phi\in J_{k,1}^{cusp}$
and operators $R_{i}(p)$ introduced in subsection 1.3 we put

(5.1) $F_{i,\phi}=\mathcal{L}_{1}(\phi)|R_{i}(p)\text{ and
}G_{i,\phi}=\mathcal{L}_{p}(\phi)|R_{i}(p).$

Whenever $\phi$ is fixed and there is no ambiguity, we drop $\phi$ from the
notations and write as $F_{i}$ and $G_{i}$. The following lemma allows us to
freely switch between the forms $F_{j,\phi}$ and $G_{j,\phi}$ in subsequent
calculations.

######  Lemma 5.1.

With the above notation, the $\mathbf{C}$-spans of
$\\{F_{1,\phi},F_{2,\phi},F_{3,\phi},F_{4,\phi}\\}$ and
$\\{G_{1,\phi},G_{2,\phi},G_{3,\phi},G_{4,\phi}\\}$ are equal.

###### Proof.

We first note that $F(pZ)=p^{-k}F|B_{p}$. We claim that
$G_{1}=F_{1}-p^{-1}F_{3},G_{2}=F_{2}-p^{-1}F_{1},G_{3}=F_{3}-p^{-1}F_{1},G_{4}=F_{4}-p^{k-1}F_{3}$.

We start from the observation $F|B_{p}=F|W_{p}$ since $F$ has level $1$ and
from the relation given in [28]:

(5.2) $\displaystyle
G_{1}=F_{1}-p^{k-1}F_{1}(pZ)=F_{1}-p^{-1}F_{1}|W_{p}=F_{1}-p^{-1}F_{3}.$

Next, one computes

(5.3) $\displaystyle
G_{2}=(F_{1}-p^{-1}F_{3})|U_{S}(p)=F_{2}-p^{-1}F_{1}|W_{p}|U_{S}(p)=F_{2}-p^{-1}F_{1}|B_{p}|U_{S}(p)=F_{2}-p^{k-1}F_{1}.$

For $G_{3},G_{4}$ one similarly gets

(5.4) $G_{3}=(F_{1}-p^{-1}F_{3})|W_{p}=F_{3}-p^{-1}F_{1};\quad
G_{4}=(F_{2}-p^{k-1}F_{1})|W_{p}=F_{4}-p^{k-1}F_{3}.\qed$

######  Lemma 5.2.

The set $\\{G_{1,\phi},G_{2,\phi},G_{3,\phi}\\}$ is a basis for the
$\phi$-oldspace $\mathrm{SK}_{k}^{old}(p;\phi)$ in $\mathrm{SK}_{k}(p)$.

###### Proof.

In principle, this follows from Lemma 4.3. But the following proof is more
illuminating – and also helpful while constructing a $W_{p}$-invariant old-
basis.

From Lemma 4.7, we know that $\mathrm{SK}_{k}^{old}(p;\phi)$ is spanned by
$\\{F_{1},F_{2},F_{3},F_{4}\\}$, which equals the span of
$\\{G_{1},G_{2},G_{3},G_{4}\\}$ by Lemma 5.1. Moreover from Lemma 4.5, we know
that $\dim\mathrm{SK}_{k}^{old}(p)(\phi)=3$ and from [28], that $G_{1},G_{2}$
are linearly independent.

If we put $V=\mathbf{C}G_{1}+\mathbf{C}G_{2}$ and
$W=\mathbf{C}G_{1}|W_{p}+\mathbf{C}G_{2}|W_{p}$, then from the above we
conclude that $\dim V=\dim W=2$, and that $\dim(V+W)=3$ – which implies that
$\dim(V\cap W)=1$. Now let $V\cap W=\mathbf{C}v$. We can then write

(5.5) $\displaystyle
v=c_{1}G_{1}+c_{2}G_{2}=d_{1}G_{1}|W_{p}+d_{2}G_{2}|W_{p}$

for some scalars $c_{j},d_{j}$. By applying $W_{p}$ on both sides we obtain,
by the uniqueness of $v$ up to scalars that $d_{j}=\alpha c_{j}$ for some
$\alpha\neq 0$ and $j=1,2$. Then (5.5) reads

(5.6) $\displaystyle v=G_{12}=\alpha
G_{12}|W_{p},\quad\quad(G_{12}:=c_{1}G_{1}+c_{2}G_{2}).$

Clearly $\alpha^{2}=1$. From this one can also write

(5.7) $\displaystyle c_{1}(G_{1}\pm G_{3})=-c_{2}(G_{2}\pm G_{4}).$

Clearly, $c_{2}\neq 0$ as otherwise we must have $F_{1}=\pm F_{3}$, which is
impossible. Then note that $G_{3}\not\in\mathbf{C}G_{1}+\mathbf{C}G_{2}$ as
otherwise from (5.7), we would have $G_{4}\in\mathbf{C}G_{1}+\mathbf{C}G_{2}$
which would imply $\dim(V+W)=2$. Therefore, $\\{G_{1},G_{2},G_{3}\\}$ are
linearly independent. Of course, we can also replace $G_{3}$ with $G_{4}$.
This proves the lemma. ∎

######  Proposition 5.3.

Let $\mathcal{B}^{J}_{k,1}=\\{\phi\\}$ be an orthogonal basis of Hecke
eigenforms for $J_{k,1}^{cusp}$. Then the set
$B^{SK,old}:=\\{G_{1,\phi},G_{2,\phi},G_{3,\phi}:\phi\in\mathcal{B}^{J}_{k,1}\\}$
is a basis for the space $\mathrm{SK}_{k}^{old}(p)$. The spaces
$\mathrm{SK}_{k}^{old}(p;\phi)$ with $\phi$ as above, are mutually orthogonal.

###### Proof.

From Lemma 5.2, we know that $\\{G_{1,\phi},G_{2,\phi},G_{3,\phi}\\}$ is a
basis for $\mathrm{SK}_{k}^{old}(p;\phi)$ and we have that
$\mathrm{SK}_{k}^{old}(p)=\sum_{\phi\in\mathcal{B}^{J}_{k,1}}\mathrm{SK}_{k}^{old}(p;\phi)$.
That this is a direct sum, follows from the statement about orthogonality. But
this in turn, follows immediately from multiplicity-one in degree $1$, the
fact that the operators $U_{S}(p),W_{p}$ commute with Hecke operators
$T_{S}(q)$ with $q\neq p$ and the equivariance relations from [28, Theorem
4.1]. ∎

######  Remark 5.4.

From Proposition 5.3, we can also obtain (the known fact from subsection 4)
that $G_{3,\phi}$ is not an EZI lift of a Jacobi cusp form of level $p$.

###  5.1. An orthonormal basis for oldspace:

To obtain an orthonormal basis for the oldspace of level $p$, we make use of
the basis obtained in Proposition 5.3. In this direction, we first calculate
the inner products $\langle G_{i},G_{j}\rangle$. For this, we use Lemmas 3.1,
3.3, 3.4 and 3.5 and the relations (5.2), (5.3) and (5.4), which say that
$G_{1}=F_{1}-p^{-1}F_{3}$, $G_{2}=F_{2}-p^{k-1}F_{1}$
$G_{3}=F_{3}-p^{-1}F_{1}$. We summarize these inner product relations in the
following Proposition.

######  Proposition 5.5.

Let $\phi\in J_{k,1}^{cusp}$ and let $F_{1}=\mathcal{L}_{1}(\phi)$,
$G_{1}=\mathcal{L}_{p}(\phi)$. Put $G_{2}:=G_{1}|U_{S}(p)$ and
$G_{3}:=G_{1}|W_{p}$. Further, let $F_{1}$ be an eigenfunction of the Hecke
operators $T_{S}(p)$ and $T_{S}^{\prime}(p)$ with the eigenvalue
$\lambda_{F_{1}}(p)$ and $\lambda_{F_{1}}^{\prime}(p)$, respectively. Then we
have

(5.8) $\displaystyle\langle G_{1},G_{1}\rangle_{p}$
$\displaystyle=\left(1+\frac{1}{p^{2}}-\frac{2\lambda_{F_{1}}(p)}{\mu(p)p^{k-2}}\right)\langle
F_{1},F_{1}\rangle_{p};$ (5.9) $\displaystyle\langle G_{1},G_{2}\rangle_{p}$
$\displaystyle=\left(\frac{\lambda_{F_{1}}^{\prime}(p)}{p^{k-1}(p^{2}+1)}-\frac{\lambda_{F_{1}}(p)^{2}}{\mu(p)p^{k-2}}+\frac{p\lambda_{F_{1}}(p)}{p+1}-\frac{p^{k-3}(p^{4}+p^{2}+1)}{p^{2}+1}\right)\langle
F_{1},F_{1}\rangle_{p};$ (5.10) $\displaystyle\langle G_{1},G_{3}\rangle_{p}$
$\displaystyle=\left(\frac{p^{1-k}\lambda_{F_{1}}(p)}{p+1}-\frac{2}{p}\right)\langle
F_{1},F_{1}\rangle_{p};$ (5.11) $\displaystyle\langle G_{2},G_{2}\rangle_{p}$
$\displaystyle=\left(\frac{p^{2}(p-1)}{\mu(p)}\lambda_{F_{1}}(p)^{2}-\frac{2p^{k+2}}{\mu(p)}\lambda_{F_{1}}(p)+p^{2k-4}+p^{2k-2}\right)\langle
F_{1},F_{1}\rangle_{p};$ (5.12) $\displaystyle\langle G_{2},G_{3}\rangle_{p}$
$\displaystyle=\left(\frac{\lambda_{F}(p)^{2}}{\mu(p)p^{k-3}}-\frac{2p^{2}\lambda_{F_{1}}(p)}{\mu(p)}-\frac{\lambda_{F}^{\prime}(p)}{p^{k-2}(p^{2}+1)}+\frac{p^{k-2}(p^{2}+2)}{p^{2}+1}\right)\langle
F_{1},F_{1}\rangle_{p}.$

######  Remark 5.6.

The inner products $\langle G_{i},G_{i}\rangle_{p}$ in Proposition 5.5 are
indeed positive. To see this, recall from (4.5) that
$\lambda_{F_{1}}(p)=p^{k-1}+p^{k-2}+\lambda_{\phi}(p)$. Then one can verify
that $\langle G_{1},G_{1}\rangle_{p}\asymp\langle F_{1},F_{1}\rangle_{p}$
which is clearly positive. For $G_{2}$, we have $\langle
G_{2},G_{2}\rangle_{p}\asymp\mu(p)^{-1}(p^{2}(p-1)\lambda_{\phi}(p)^{2}-2p^{k}\lambda_{\phi}(p)+p^{2k-1})\langle
F_{1},F_{1}\rangle_{p}$. Since $|\lambda_{\phi}(p)|\leq 2p^{k-3/2}$, we see
that $\langle G_{2},G_{2}\rangle_{p}>0$.

###  5.2. $W_{p}$ invariant old-basis

It is desirable to obtain an orthogonal basis consisting of eigenvectors of
$W_{p}$. This is in particular quite useful in the context of the sup-norm
problem, as it helps us to narrow the realm of search for large values in
convenient regions of $\mathbf{H}_{2}$.

As in Proposition 5.3, let $\mathcal{B}_{k,1}^{J}$ be an orthonormal basis of
Hecke eigenforms for $J_{k,1}^{cusp}$. For $\phi\in\mathcal{B}_{k,1}^{J}$, let
$F_{1,\phi}:=\mathcal{L}_{1}(\phi)$ and $G_{1,\phi}:=\mathcal{L}_{p}(\phi)$.
By construction, clearly $W_{p}$ acts on $\mathrm{SK}_{k}^{old}(p)$. Moreover,

(5.13) $G_{\pm,\phi}:=G_{1,\phi}\pm G_{3,\phi}$

are orthogonal and are eigenvectors for $W_{p}$ with eigenvalues $\pm 1$
respectively. To obtain the third eigenvector, let

(5.14) $\displaystyle
H_{\phi}:=G_{2,\phi}-a_{+,\phi}G_{+,\phi}-a_{-,\phi}G_{-,\phi}$

be orthogonal to both $G_{+,\phi},G_{-,\phi}$. Therefore,

(5.15) $\displaystyle a_{\pm,\phi}=\frac{\langle
G_{2,\phi},G_{\pm,\phi}\rangle_{p}}{\langle
G_{\pm,\phi},G_{\pm,\phi}\rangle_{p}}.$

Next note that $H_{\phi}|W_{p}$ is also orthogonal to both
$G_{+,\phi},G_{-,\phi}$. Since the orthogonal complement of
$<G_{+,\phi},G_{-,\phi}>$ is one dimensional, we must have $H_{\phi}|W_{p}=\pm
H_{\phi}$, giving $H_{\phi}$ to be the last of our desired basis elements. We
do not work out the exact sign, as this is not required for us.

From Proposition 5.5, we see that

(5.16) $\displaystyle\langle G_{\pm,\phi},G_{\pm,\phi}\rangle_{p}\asymp\langle
G_{1,\phi},G_{1,\phi}\rangle_{p}\pm\langle
G_{1,\phi},G_{3,\phi}\rangle_{p}\asymp\langle
F_{1,\phi},F_{1,\phi}\rangle_{p};$ (5.17) $\displaystyle|\langle
G_{2,\phi},G_{\pm,\phi}\rangle_{p}|\ll p^{k-3/2}\langle
F_{1,\phi},F_{1,\phi}\rangle_{p};\quad\langle
H_{\phi},H_{\phi}\rangle_{p}\asymp p^{2k-4}\langle
F_{1,\phi},F_{1,\phi}\rangle_{p}.$

We also have the relation [35, pp. 551]: $\langle
F_{1},F_{1}\rangle_{\mathrm{Sp}_{2}(\mathbf{Z})}=c_{k}\langle\phi,\phi\rangle_{\mathrm{SL}_{2}(\mathbf{Z})}$.
Thus, the level $p$ norms can be related as $\langle
F_{1},F_{1}\rangle_{p}=(p^{2}+1)c_{k}\langle\phi,\phi\rangle_{p}$.

This takes care of the oldspace. For the newspace, all SK newforms are EZI
lifts of unique Jacobi newforms of level $p$. That each of these are invariant
under $W_{p}$ follows from multiplicity-one in a standard manner. We summarize
the above discussion in the following proposition.

######  Proposition 5.7.

Let $\mathcal{B}_{k,1}^{J}$ denote an orthonormal basis of Hecke eigenforms
for $J_{k,1}^{cusp}$. Then the set

(5.18)
$\mathcal{B}^{SK,old}(p)=\left\\{G_{+,\phi},G_{-,\phi},H_{\phi}:\phi\in\mathcal{B}_{k,1}^{J}\right\\}$

is an orthogonal basis of eigenvectors of $W_{p}$ for the oldspace
$\mathrm{SK}_{k}^{old}(p)$ with $G_{+,\phi},H_{\phi}$ as in (5.13), (5.14)
respectively. Moreover, we have that

(5.19) $\langle G_{\pm,\phi},G_{\pm,\phi}\rangle_{p}\asymp\langle
F_{1,\phi},F_{1,\phi}\rangle_{p}\asymp
p^{3}\langle\phi,\phi\rangle_{1};\quad\langle
H_{\phi},H_{\phi}\rangle_{p}\asymp p^{2k-4}\langle
F_{1,\phi},F_{1,\phi}\rangle_{p}\asymp p^{2k-1}\langle\phi,\phi\rangle_{1}.$

For the new space,
$\mathcal{B}^{SK,new}(p)=\\{\mathcal{L}_{p}(\phi)\colon\phi\in
J_{k,1}^{cusp}(p)^{new}\\}$ is the $W_{p}$ invariant basis.

##  6\. Sup-norm of Jacobi forms

The purpose of this section is twofold:

(i) To study the size of $J^{cusp}_{k,1}(N)$ – and this is done by naturally
reducing the problem to one about half-integral weight forms; and

(ii) Prove a non-trivial bound for a Hecke eigenform $\phi\in
J^{cusp}_{k,1}(N)$ by the same strategy as (i) above, but there are quite
subtle issues here – things do not follow in a straightforward way, and one
has to go at the root of the half-integral case treated by Kiral [32] and
account for the extra subtlety mentioned above.

###  6.1. Theta decomposition of Jacobi forms

Let $N$ be an odd, square-free integer and $\phi\in J^{cusp}_{k,1}(N)$ be an
eigenform of all Hecke operators away from the level $N$. The periodicity
properties of $\phi$ allows one to write (cf. [22])

(6.1)
$\displaystyle\phi=h_{0}(\tau)\theta_{0}(\tau,z)+h_{1}(\tau)\theta_{1}(\tau,z).$

where the classical Jacobi theta functions $\theta_{i}$ of weight $1/2$ and
index $1$ are given by

(6.2) $\theta_{i}(\tau,z)=\sum\nolimits_{n\equiv i\bmod
2}e(n^{2}\tau/4)e(nz).$

Then the Eichler-Zagier isomorphism is the map $EZ:\phi\mapsto
h(\tau)=h_{0}(4\tau)+h_{0}(4\tau)$. Crucial to our approach is a convenient
description of theta components $h_{j}$ of $\phi$ in terms of the half-
integral weight cusp form $h$. This is clear when $N=1$ from [22]. But it is
not as trivial when $N>1$; we refer the reader to [38, Satz 8]. For
convenience, we recall the result here.

######  Proposition 6.1.

$h_{0}=\frac{1}{2}\big{(}h(\frac{\tau-1}{4})+h(\frac{\tau+1}{4})\big{)};\quad
h_{1}=\frac{1}{2i}\big{(}h(\frac{\tau-1}{4})-h(\frac{\tau+1}{4})\big{)}$.

Further, from [38, Satz 9] we see that $h\in S_{k-1/2}^{+}(4N)$ – the Kohnen’s
plus space of level $4N$ (see [36]). From Proposition 6.1, we can therefore
write

(6.3) $\displaystyle\phi$
$\displaystyle=\frac{1}{2}\left(h(\frac{\tau-1}{4})(\theta_{0}(\tau,z)-i\theta_{1}(\tau,z))+h(\frac{\tau+1}{4})(\theta_{0}(\tau,z)+i\theta_{1}(\tau,z))\right).$

####  6.1.1. A convenient fundamental domain

To start with, note that the invariant function attached to $\phi\in
J_{k,m}^{cusp}(N)$, written as $\widetilde{\phi}$ is defined as

(6.4) $\displaystyle\widetilde{\phi}:=v^{k/2}e^{-2\pi my^{2}/v}|\phi|.$

Let us now fix a fundamental domain $\mathcal{F}_{N}\subset\mathbf{H}$ for the
action of $\Gamma_{0}(N)$ on $\mathbf{H}$ defined by

(6.5) $\displaystyle\mathcal{F}_{N}=\cup_{q\mid N}\cup_{0\leq j\leq
d-1}A_{q,j}\mathcal{F}_{1};$

where $\mathcal{F}_{1}$ is the classical fundamental domain for
$\mathrm{SL}_{2}(\mathbf{Z})$ on $\mathbf{H}$ (contained in the Siegel set
$\\{z\in\mathbf{H}\mid\Im(z)\geq\sqrt{3}/2$), and $A_{q,j}$ are a set of left-
coset representatives of $\Gamma_{0}(N)$ in $\mathrm{SL}_{2}(\mathbf{Z})$
given by (cf. [1]):

$\displaystyle\mathrm{SL}_{2}(\mathbf{Z})=\cup_{q\mid N}\cup_{0\leq j\leq
q-1}\Gamma_{0}(N)A_{q,j};\quad A_{q,j}=\begin{pmatrix}\beta&1\\\
q\gamma&\frac{N}{q}\end{pmatrix}\begin{pmatrix}0&-1\\\
1&j\end{pmatrix}\quad(\beta\frac{N}{q}-q\gamma=1).$

Next, note that for any function $\Psi$ on $\mathbf{H}\times\mathbf{C}$,
$M\in\mathrm{SL}_{2}(\mathbf{R})$ and $k\geq 1$, one has the relation:

(6.6) $\displaystyle\left(v^{k/2}e^{-2\pi
y^{2}/v}|\Psi|\right)\circ(M(\tau,z))=v^{k/2}e^{-2\pi
y^{2}/v}|(\Psi|_{k,1}M)(\tau,z)|.$

This can be checked from [54] for $\Psi=1$ and
$M\in\mathrm{SL}_{2}(\mathbf{R})$, but our case follows immediately from this.

######  Lemma 6.2.

(i) A complete set of coset representatives of $\Gamma_{0}(N)^{J}$ in
$\Gamma_{1}^{J}$ can be taken to be $(\gamma,[0,0])$ where $\gamma$ runs over
$\Gamma_{0}(N)\backslash\Gamma_{1}$.

(ii) As a fundamental domain $\mathcal{F}_{N}^{J}$ can be taken as
$\mathcal{F}^{J}_{N}=\bigcup_{\gamma}(\gamma,[0,0])\cdot\mathcal{F}_{1}^{J}$
with $\gamma$ as in (i) above.

###### Proof.

Clearly it is enough to prove (i). If
$(M^{\prime},X^{\prime})\in\Gamma_{1}^{J}$, then we can always left multiply
it with $(1_{2},X)$, where $X$ is chosen such that $XM^{\prime}+X=0$. Thus we
can assume $X^{\prime}=0$. Then left multiplication by a suitable element
$(g,0)$ with $g\in\Gamma_{0}(N)$ gives (i). ∎

We now come back to our goal of bounding $\tilde{\phi}$. First, by invariance
under $\Gamma_{0}(N)$ along with (6.6) and Lemma 6.2 we have

(6.7)
$\displaystyle\sup\nolimits_{(\tau,z)\in\mathbf{H}\times\mathbf{C}}\widetilde{\phi}(\tau,z)=\sup\nolimits_{(\tau,z)\in\mathcal{F}^{J}_{N}}\widetilde{\phi}(\tau,z)=\max\nolimits_{q,j}\sup\nolimits_{(\tau,z)\in\mathcal{F}^{J}_{1}}T_{q,j},$

where the quantities $T_{q,j}$ are defined by

(6.8) $T_{q,j}:=v^{k/2}e^{-2\pi y^{2}/v}|(\phi|_{k,1}A_{q,j})(\tau,z)|,$

and $A_{q,j}$ runs over a set of coset representatives of $\Gamma_{0}(N)$ in
$\mathrm{SL}_{2}(\mathbf{Z})$. Then $T_{q,j}$ is bounded by

(6.9)
$\displaystyle\sup\nolimits_{(\tau,z)\in\mathcal{F}^{J}_{1}}\,v^{k/2}e^{-2\pi
y^{2}/v}\sum\nolimits_{\pm}\big{|}h(\frac{\tau\pm 1}{4})(\theta_{0}(\tau,z)\pm
i\theta_{1}(\tau,z))|_{k,1}A_{q,j}\big{|}.$

Put $\kappa:=k-1/2$. Then the quantities in $\sum_{\pm}|\cdots|$ above are
bounded by the sums of the type

(6.10) $\displaystyle\big{|}h(\frac{\tau\pm
1}{4})|_{\kappa}A_{q,j}\big{|}\cdot\big{|}(\theta_{0}(\tau,z)\pm
i\theta_{1}(\tau,z))|_{\frac{1}{2},1}A_{q,j}\big{|}.$

Therefore we have reduced the problem to $h$, but first we have to tackle the
theta series.

####  6.1.2. Handling the theta series:

Let us put

(6.11)
$\displaystyle\vartheta(\tau,z)=|\theta_{0}(\tau,z)|^{2}+|\theta_{1}(\tau,z)|^{2}.$

Although the invariant function $v^{1/2}e^{-4\pi y^{2}/v}|\theta_{j}|^{2}$ is
not bounded in $\mathbf{H}\times\mathbf{C}$, we shall now show that the
function $e^{-4\pi y^{2}/v}\vartheta(\tau,z)$ is bounded in $\mathcal{F}_{1}$,
and that will be enough for us.

######  Lemma 6.3.

$e^{-2\pi y^{2}/v}\vartheta(\tau,z)$ is bounded by an absolute constant on
$\mathcal{F}_{1}^{J}$.

###### Proof.

From the calculations in [4, section 4.2], we see that $\displaystyle e^{-2\pi
y^{2}/v}\vartheta(\tau,z)\ll(1+v^{-1/2})$. The lemma now follows, since
$\tau\in\mathcal{F}_{1}$. ∎

In view of (6.9) and (6.10), we have to bound
$\displaystyle\max\nolimits_{q,j}\sup\nolimits_{\tau\in\mathcal{F}_{1}}\left|(\theta_{0}(\tau,z)\pm
i\theta_{1}(\tau,z))|_{k}A_{q,j}\right|$. We now use the transformation
properties of $\theta_{\mu}$ under $\mathrm{SL}_{2}(\mathbf{Z})$ (more
precisely under its double cover $\widetilde{\mathrm{SL}_{2}(\mathbf{Z})}$;
see [38], [53]) as follows:

(6.12)
$\displaystyle\theta_{\mu}(\tau,z)|_{1/2}(A,\omega(\tau))=c_{1}\theta_{0}(\tau,z)+c_{2}\theta_{1}(\tau,z),$

where $A=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbf{Z}),\omega(\tau)^{2}=c\tau+d$
and $c_{1},c_{2}$ are complex numbers with absolute value less or equal to
$1$. From this, and Lemma 6.3 it is now clear that on $\mathcal{F}_{1}^{J}$,

(6.13)
$\displaystyle\max\nolimits_{q,j}\sup\nolimits_{\tau\in\mathcal{F}_{1}}\left|(\theta_{0}(\tau,z)\pm
i\theta_{1}(\tau,z))|_{k}A_{q,j}\right|\ll\left(e^{-4\pi
y^{2}/v}\vartheta(\tau,z)\right)^{1/2}\ll 1.$

In the following subsection, we reduce the problem of bounding the supnorm of
a Jacobi form to that of a half-integral weight form using Proposition 6.1 and
the Atkin-Lehner operators for the group $\Gamma_{0}(4N)$.

####  6.1.3. Handling the half-integral weight form:

We only handle the ‘$+$’ sum in (6.10), the other case is entirely the same.
We have

(6.14) $\displaystyle\quad\big{|}h(\frac{\tau\pm
1}{4})|_{\kappa}M\big{|}\ll\big{|}h|_{\kappa}\left(\begin{smallmatrix}1&1\\\
0&4\end{smallmatrix}\right)\left(\begin{smallmatrix}\beta\frac{N}{q}&1\\\
N\gamma&\frac{N}{q}\end{smallmatrix}\right)\left(\begin{smallmatrix}\frac{q}{N}&0\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}0&-1\\\
1&j\end{smallmatrix}\right)\big{|}=\big{|}h|_{\kappa}\left(\begin{smallmatrix}1&0\\\
0&4\end{smallmatrix}\right)\left(\begin{smallmatrix}\beta\frac{N}{q}&1\\\
N\gamma&\frac{N}{q}\end{smallmatrix}\right)\cdot
g\cdot\left(\begin{smallmatrix}\frac{q}{N}&0\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}0&-1\\\
1&j\end{smallmatrix}\right)\big{|},$

for some $g\in\Gamma_{0}(N)$. At this point, we note that it is possible to
choose $\beta$ such that $\beta\equiv 0\bmod 4$ and $\beta\frac{N}{q}\equiv
1\bmod q$, by the Chinese remainder theorem. Thus, putting
$\beta=4\beta^{\prime}$, we see that $\left(\begin{smallmatrix}1&0\\\
0&4\end{smallmatrix}\right)\left(\begin{smallmatrix}\beta\frac{N}{q}&1\\\
N\gamma&\frac{N}{q}\end{smallmatrix}\right)=\left(\begin{smallmatrix}\beta^{\prime}\frac{4N}{q}&1\\\
N\gamma&\frac{4N}{q}\end{smallmatrix}\right)=W(\frac{4N}{q},4N)$, the Atkin-
Lehner involution at $4N/q$ of level $4N$. This is well-defined since $N$ is
odd and square-free. Moreover, $h$ being a Hecke eigenform, so is

$h_{d}:=h|W(d,4N),\text{ for any }\,d|4N.$

Therefore, (6.14) can be written as

(6.15)
$\displaystyle\big{|}h_{4N/q}|_{\kappa}g\left(\begin{smallmatrix}\frac{q}{N}&0\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}0&-1\\\
1&j\end{smallmatrix}\right)\big{|}.$

The next step is to write $g$ in terms of coset representatives
$\Gamma_{0}(4N)\backslash\Gamma_{0}(N)$. Let us put

(6.16) $\displaystyle A:=\left(\begin{smallmatrix}1&0\\\
N&1\end{smallmatrix}\right);\,B:=\left(\begin{smallmatrix}\beta&1\\\
N\gamma&4\end{smallmatrix}\right),$

where $4\beta-N\gamma=1$ and $\beta$ even.

######  Lemma 6.4.

The set $\\{1,A,A^{2},A^{3},B,A^{2}B\\}$ can be taken as a set of coset
representatives $\Gamma_{0}(4N)\backslash\Gamma_{0}(N)$.

###### Proof.

A set of coset representatives for $\Gamma_{0}(4N)\backslash\Gamma_{0}(2N)$
can be taken to be $\\{1,A^{2}\\}$ and those for
$\Gamma_{0}(2N)\backslash\Gamma_{0}(N)$ as $\\{1,B,A\\}$, respectively. The
lemma follows. ∎

Then (6.15) can be written as (with $s=0,1,2,3$)

(6.17) $\displaystyle\left|h_{4N/q}|_{\kappa}A^{s}C_{j}\right|\text{ or
}\left|h_{4N/q}|_{\kappa}BC_{j}\right|\text{ or
}\left|h_{4N/q}|_{\kappa}A^{2}BC_{j}\right|,$

where $C_{j}=\left(\begin{smallmatrix}\frac{q}{N}&0\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}0&-1\\\
1&j\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&-\frac{q}{N}\\\
1&j\end{smallmatrix}\right)$. Now we observe that

(6.18) $\displaystyle A^{s}C_{j}$
$\displaystyle=W(4N,4N)\begin{pmatrix}\frac{1}{4N}&\frac{j-sq}{4N}\\\
0&\frac{q}{N}\end{pmatrix}\,(0\leq s\leq 3),$ (6.19) $\displaystyle BC_{j}$
$\displaystyle=W(4,4N)W(4N,4N)\left(\begin{smallmatrix}\frac{1}{4N}&\frac{j}{4N}\\\
0&\frac{q}{4N}\end{smallmatrix}\right),$ (6.20) $\displaystyle A^{2}BC_{j}$
$\displaystyle=g^{\prime}W(4,4N)W(4N,4N)\left(\begin{smallmatrix}\frac{1}{4N}&\frac{-2q+j}{4N}\\\
0&\frac{q}{4N}\end{smallmatrix}\right)\text{ for some
}g^{\prime}\in\Gamma_{0}(4N).$

As a consequence, it is enough to consider the first two cases above.
Moreover, $W(4,4N)W(4N,4N)=W(N,4N)$ – Atkin-Lehner operator at $N$ on
$\Gamma_{0}(4N)$.

For $d|4N$, if we put

$h^{\prime}_{d}:=h_{4N/q}|W(d,4N),$

then after multiplication by $v^{\kappa/2}$, (6.15) can be written as

(6.21) $\displaystyle
v^{\frac{\kappa}{2}}\left|h^{\prime}_{4N}|_{\kappa}\begin{pmatrix}\frac{1}{4N}&\frac{j-q}{4N}\\\
0&\frac{q}{N}\end{pmatrix}\right|=(\frac{vq}{4N^{2}})^{\frac{\kappa}{2}}(q/N)^{-\kappa}\left|h^{\prime}_{4N}\big{(}\frac{\tau+(j-q)}{4q}\big{)}\right|=(\frac{v}{4q})^{\frac{\kappa}{2}}\left|h^{\prime}_{4N}\big{(}\frac{\tau+(j-q)}{4q}\big{)}\right|.$

Similarly

(6.22)
$v^{\kappa/2}\left|h^{\prime}_{N}|_{\kappa}\begin{pmatrix}\frac{1}{4N}&\frac{j}{4N}\\\
0&\frac{q}{4N}\end{pmatrix}\right|=(v\frac{q}{(4N)^{2}})^{\kappa/2}(q/4N)^{-\kappa}|h^{\prime}_{N}\big{(}\frac{\tau+j}{q}\big{)}|=(\frac{v}{q})^{\kappa/2}|h^{\prime}_{N}\big{(}\frac{\tau+j}{q}\big{)}|.$

From (6.21) and (6.22) we calculate that

(6.23) $\displaystyle(\frac{v}{4q})^{\kappa/2}\cdot
v^{1/4}\cdot\left|h^{\prime}_{4N}\big{(}\frac{\tau+(j-q)}{4q}\big{)}\right|=(\frac{v}{4q})^{\kappa/2}\cdot(\frac{v}{q})^{1/4}\cdot\left|h^{\prime}_{4N}\big{(}\frac{\tau+(j-q)}{4q}\big{)}\right|\cdot
q^{1/4}$ $\displaystyle\ll\sup\nolimits_{v\geq
1/N}\,v^{\kappa/2+1/4}\left|h^{\prime}_{4N}\right|\cdot q^{1/4}.$

and similarly

(6.24)
$(\frac{v}{q})^{\kappa/2}v^{1/4}|h^{\prime}_{N}\big{(}\frac{\tau+j}{q}\big{)}|\ll\sup\nolimits_{v\geq
1/N}\,v^{\kappa/2+1/4}|h^{\prime}_{N}|\cdot q^{1/4}.$

Further from the properties of the Atkin-Lehner operators $W(d,4N)$, we have
that $\left\lVert h^{\prime}_{4N}\right\rVert=\left\lVert h_{q}\right\rVert$,
$\left\lVert h^{\prime}_{N}\right\rVert=\left\lVert h_{4q}\right\rVert$ and
$\left\lVert h\right\rVert=\left\lVert h_{d}\right\rVert$. Thus, from what has
been done in the above two sections (see especially (6.7)–(6.9), (6.21),
(6.22) and (6.13)), it follows that (recall that $\kappa=k-1/2$)

(6.25) $\displaystyle
T_{q,j}\ll\sup\nolimits_{\tau\in\mathbf{H}}\,v^{\kappa/2+1/4}\left|h_{q}\right|\cdot
q^{1/4}.$

We note here that $h_{q}\in
S_{k-1/2}^{+}\left(4N,\left(\frac{4q}{\cdot}\right)\right)$.

At this point, let us mention that, if the factor $v^{1/4}$ was absent in the
above, we could have directly invoked [32]. Moreover, the bounds obtained
after amplification in [32] are only on the set $\mathcal{S}\subset\mathbf{H}$
defined below in (6.26), so one can’t quote them directly in (6.25).

####  6.1.4. The final step:

The final step for us is to observe that the maximum of the function
$v^{\kappa/2+1/4}\left|h_{q}\right|$ is also achieved on a more convenient
subset $\mathcal{S}\subset\mathbf{H}$, defined as

(6.26)
$\mathcal{S}=\\{\tau\in\mathbf{H}:\mathrm{Im}(\gamma\tau)\leq\mathrm{Im}(\tau)\text{
for all }\gamma\in A_{0}(2N)\\},$

where $A_{0}(2N)$ is the group generated by all the Atkin-Lehner operators and
$\Gamma_{0}(2N)$. The set $\mathcal{S}$ was first defined and studied in [25],
and has been used in [32] as well.

Next, for a Dirichlet character $\chi\bmod 4N$, let $\mathcal{B}(N,\chi)$
denote a basis of Hecke eigenforms for $S_{\kappa}(4N,\chi)$ and consider
$\mathcal{B}(N)=\cup_{\chi}\mathcal{B}(N,\chi)$. Let
$\mathcal{B}^{\prime}(N)=\cup_{A\in A_{0}(2N)}\mathcal{B}(N)|_{\kappa}A$. We
note here that the union is over finitely many $A$ since $A_{0}(2N)$ is
generated by finitely many Atkin-Lehner operators and $\Gamma_{0}(2N)$.

Let $w\in\mathbf{H}$ and let $\delta\in A_{0}(2N)$ be such that
$w=\delta(\tau)$ with $\tau\in\mathcal{S}$ defined above.

(6.27)
$\displaystyle\mathrm{Im}(w)^{\kappa/2+1/4}\left|h_{q}(w)\right|=\mathrm{Im}(\delta\tau)^{1/4}\mathrm{Im}(\tau)^{\kappa/2}\left|h_{q}|\delta(\tau)\right|.$

But we note that $h_{1}=h_{q}|\delta\in\mathcal{B}^{\prime}(N)$. However, for
any $\delta\in A_{0}(2N)$ and any $\tau\in\mathcal{S}$ one has
$\mathrm{Im}(\delta\tau)\leq\mathrm{Im}(\tau)$, whence

(6.28)
$\displaystyle\mathrm{Im}(w)^{\kappa/2+1/4}\left|h_{q}(w)\right|\leq\mathrm{Im}(\tau)^{\kappa/2+1/4}\left|h_{1}(\tau)\right|$

and this proves our assertion. Therefore, if we look at (6.25) and (6.28), our
problem reduces to finding

(6.29)
$\displaystyle\sup\nolimits_{\tau\in\mathcal{S}}v^{\kappa/2+1/4}\left|H\right|\cdot
N^{1/4},$

where $H$ is any element in $\mathcal{B}^{\prime}(N)$. Our next proposition
does this task.

######  Proposition 6.5.

With the above notation, we have
$\sup_{\tau\in\mathcal{S}}v^{\kappa/2+1/4}\left|H\right|\cdot
N^{1/4}\ll_{\epsilon}N^{-\frac{1}{36}+\epsilon}$.

###### Proof.

We quote two inequalities from [32] which we would use: the first is from the
Fourier expansion and states in our situation that for any
$\tau\in\mathbf{H}$,

(6.30) $\displaystyle
v^{\kappa/2+1/4}\left|H\right|\ll\frac{N^{\epsilon}}{N^{1/2}v^{1/4}},$

where we assume $\left\lVert H\right\rVert=1$. The second inequality comes
from the amplified analysis of the geometric side of the Bergman kernel and
states, in our situation, that for all $\tau\in\mathcal{S}$

(6.31) $\displaystyle
v^{\kappa/2+1/4}\left|H\right|\ll\frac{v^{1/4}N^{\epsilon}}{\Lambda^{1-\epsilon}}\left(\Lambda^{1/2}(1+N^{1/2}v)+2(1+\Lambda^{4}N^{1/2}v)\right),$

where $\Lambda>0$ is arbitrary. We choose $\Lambda=N^{\beta}$, $\beta>0$ to be
determined later. Furthermore, for $\alpha>0$, we split $v$ into two regions:
(i) $\geq N^{-\alpha}$ and use it in (6.30) , and (ii) $\leq N^{-\alpha}$ and
use it in (6.31).

Equating the two terms $\Lambda^{1/2}$ and $\Lambda^{4}N^{1/2}v$ in (6.31)
gives us the equation

(6.32) $\displaystyle\frac{\beta}{2}=4\beta+\frac{1}{2}-\alpha\quad\text{ or,
}\quad\alpha-\frac{7\beta}{2}=\frac{1}{2}.$

Next, equating the contribution from (6.30) and that from (6.31) we get the
equation

(6.33)
$\displaystyle-\frac{1}{2}+\frac{\alpha}{4}=-\frac{\alpha}{4}-\beta+\frac{\beta}{2}\quad\text{
or, }\quad\alpha+\beta=1.$

This gives $\alpha=\frac{8}{9},\beta=\frac{1}{9}$. Finally

(6.34) $v^{\kappa/2+1/4}\left|H\right|\cdot N^{1/4}\ll
N^{\frac{\alpha}{4}-\frac{1}{2}+\frac{1}{4}}=N^{-\frac{1-\alpha}{4}}=N^{-\frac{1}{36}+\epsilon}.\qed$

###  6.2. Size of the space of Jacobi forms of index $1$ and level $N$

######  Proposition 6.6.

$\sup(J^{cusp}_{k,1}(N))\gg 1$.

###### Proof.

Given that the proof will follow the argument via Poincaré series, we will be
brief. For the lower bound we note that for any $v_{0}>0,y_{0}$,

(6.35) $\displaystyle\sup(J^{cusp}_{k,1}(N))\geq v_{0}^{k}e^{-4\pi
v_{0}^{-1}y^{2}_{0}}e^{-4\pi v_{0}}\sum\nolimits_{\phi\in
B^{J}(N)}|c_{\phi}(1,0)|^{2}.$

We also have that $\sum\nolimits_{\phi\in
B^{J}(N)}|c_{\phi}(1,0)|^{2}=4^{k-3/2}\pi^{k-3/2}\Gamma(k-3/2)^{-1}\cdot
p_{1,0}(1,0)$, where $p_{n,r}(n,r)$ is the $(n,r)$-th Fourier coefficient of
the Jacobi Poincaré series $P_{n,r}(\tau,z)$. We take $v_{0}=k/4\pi$ and
$y_{0}=0$. This implies that $\sup(J^{cusp}_{k,1}(N))\gg p_{1,0}(1,0)\gg 1$ if
we invoke the bound for the Fourier coefficient of the Jacobi Poincaré series
(see e.g., Lemma 9.1) and this finishes the proof. ∎

####  6.2.1. Upper bound:

We first note that in the previous section, the arguments up to the bound
(6.25) hold verbatim on average, i.e., one has

(6.36) $\displaystyle\sup J^{cusp}_{k,1}(N)\ll\max_{\chi\bmod
N}\max_{q|N}\sup_{v\geq
1/N}v^{\kappa+1/2}\sum\nolimits_{h_{q}\in\mathcal{B}(N,\chi)}\left|h_{q}\right(\tau)|^{2}\cdot
q^{1/2}.$

Here we are tacitly using the facts that $h\mapsto h_{q}\mapsto
h^{\prime}_{q}$ are $L^{2}$ isometries (cf. two lines above (6.25)) between
the spaces $S_{k-1/2}^{+}\left(4N\right)$ and
$S_{k-1/2}^{+}\left(4N,\chi\right)$ and
$S_{k-1/2}^{+}\left(4N,\chi^{\prime}\right)$ respectively for some characters
$\chi,\chi^{\prime}\bmod N$. See [36] for proofs of these facts.

When $v\geq 1$ we rely on the Poincare series for the relevant space and can
show that (see Section 13)

(6.37)
$\displaystyle\sum\nolimits_{h_{q}\in\mathcal{B}(N,\chi)}v^{\kappa+1/2}\left|h_{q}\right(\tau)|^{2}\ll
v^{1/2-1}\ll 1.$

When $1/N\leq v\leq 1$, we simply bound the extra $v^{1/2}$ by $1$ and use the
fact that $\sup(S^{+}_{\kappa}(N,\chi))\ll 1$ from Theorem 1.10. To preserve
the continuity of the arguments, we relegate the proof, which is very much
analogous to integral weights, to Section 13 at the end of the paper. From
(6.36) therefore we get Theorem 1.3.

It is worth mentioning here that if one agrees to sacrifice $N^{\epsilon}$,
then $\sup(S^{+}_{\kappa}(N,\chi))\ll N^{\epsilon}$ follows easily via a
standard application of the relevant large sieve inequality for Fourier
coefficients from [41].

##  7\. Conjecture about the size of the Saito-Kurokawa lifts

For the space $S^{(2)}_{k}(N)$, one can propose a conjecture about its size
rather easily by looking at a global lower bound by transferring the question
to one about average sizes of squares of Fourier coefficients
$\sum_{F}|a_{F}(T)|^{2}$ and using Petersson’s trace formula. One gets this
lower bound to be $1$ and expects this to be the upper bound as well.

Things are more subtle for the thin subspace of SK lifts. In a related
context, one can already see this in the case of index $p$ Jacobi newspace
(cf. [4, Proof of Prop. 5.1]) where asymptotics of averages of central values
of certain $L$-functions were needed as a substitute to the subtle issue of a
Petersson formula for newforms. In this paper also, we (have to) study the new
and oldspaces separately – and thus the above remarks apply.

###  7.1. The lower bound

The lower bound will follow from the well-known inequality (follows from
Parseval’s inequality for instance): for any $Y_{0}>0$ and
$T_{0}\in\Lambda_{2}$ we have

(7.1)
$\displaystyle\sup(SK_{N})\geq\det(Y_{0})^{k}\exp(-4\pi\mathrm{tr}(T_{0}Y_{0}))\sum\nolimits_{F\in
B_{k}(N)}|a_{F}(T_{0})|^{2}.$

Since we are summing over an arbitrary orthonormal basis in the BK, a natural
way to get the lower bound would have been via the Poincaré series. Thus we
might relate the quantities $\sum_{F\in B_{k}(N)}|a_{F}(T_{0})|^{2}$ with the
Fourier coefficients of Jacobi-Poincaré series of index $1$ and level $N$; in
a way similar to [19]. Note that we must pass to Jacobi forms, as there is no
useful theory of Poincaré series on $\mathrm{SK}_{k}(N)$. The passage onto the
space of Jacobi forms of index $1$ would require a relation between the
Petersson norm of $\langle F,F\rangle$ with $\langle\phi_{F},\phi_{F}\rangle$
where $F$ and $\phi_{F}$ correspond to each other. For $F$ varying over an
arbitrary orthogonal basis (or even in a basis consisting of eigenforms),
there is no apparent relation between the Petersson norms as mentioned above.
This is because all known methods to derive such a relation rely upon a
relation in the Hecke algebra: $V_{m}^{*}\cdot
V_{m}=\sum_{d|m,(d,N)=1}i_{d}d^{k-2}T^{J}(m/d)$ for all $m\geq 1$. So one
would require $\phi$ to be an eigenfunction of all $T^{J}(m)$ for our method
to be successful. In addition, we feel that [2, Prop.4.6] is not correct as
stated; it holds only on the newspace.666Indeed taking $M=p$, Prop. 4.6 in [2]
would imply that $U_{J}(p)$ is self-adjoint in level $p$, which may not be
true.

Therefore to obtain the lower bound, we just consider the contribution of the
newforms. For this, first note that if $F,G$ are distinct newforms of level
$N$ in $\mathrm{SK}_{k}(N)$, then they are orthogonal. This can be seen in
various ways. First, $\mathrm{SK}_{k}^{new}(N)$ and $J_{k,1}^{cusp,new}(N)$
are Hecke-isomorphic and (strong) multiplicity-one result holds for both of
them—this implies that $F,G$ are orthogonal. More directly, one can also see
it from the relation between $\langle F,G\rangle$ with
$\langle\phi_{F},\phi_{G}\rangle$ described below, which will also be crucial
in what follows.

Consider the Dirichlet series $D_{F,G,N}(s)$ introduced by Kohnen-Skoruppa
[35]. Put

(7.2) $\displaystyle D_{F,G,N}(s):=\sum\nolimits_{m\geq
1}\langle\phi_{F,m},\phi_{G,m}\rangle m^{-s}.$

Recall that the Petersson norms are not normalized by volume. Quoting from
[16, Theorem 3.72] we see that $D_{F,G,N}(s)$ has a residue at $s=k$, given
by777(7.3) differs by a factor of $\pi^{-k}$ from [16, Theorem 3.72] because
of different normalizations.888We prefer to use [16] as we feel that there are
some inaccuracies in [2] about this residue; e.g., the factor $M^{-\kappa}$ in
formula (11) loc. cit., which does not fit into the calculations – if true
then, say, by taking $\kappa\geq 4$, the sup-norm problem becomes trivial.
Furthermore, [2] refers to a paper by T. Horie, which we could not access
after a lot of efforts.

(7.3)
$\displaystyle\frac{(4\pi)^{k}\,\pi^{2}}{\zeta^{N}(4)\Gamma(k)}\cdot\langle
F,G\rangle\cdot N^{-2}\cdot\sum\nolimits_{l|N}\mu(l)l^{-2}.$

In particular, $D_{F,G,N}(s)$ is holomorphic at $s=0$ if $\langle
F,G\rangle=0$. On the other hand, computations by Aggarwal-Brown [2] show that
the same residue also equals (clearly the calculations in loc. cit. holds for
two distinct newforms $\phi,\psi$)

(7.4)
$\displaystyle\frac{\zeta(2)}{\zeta^{N}(4)}\cdot\left(\prod\nolimits_{p|N}(1-p^{-1})^{2}(1+p^{-1})\right)\cdot\langle\phi_{F},\phi_{G}\rangle\cdot
L(k,f),$

where $f,\phi_{F},F$ correspond to each other uniquely. Our assertion about
the orthogonality of $F,G$ and the relation between the inner products is now
clear.

######  Remark 7.1.

Note that (7.4) applied to two distinct newforms $\phi,\psi$ does not lead to
the bizarre equality $L(k,f)=L(k,g)$, as the RHS of (7.4) is actually zero in
that case.

More precisely, if we compare (7.3) and (7.4), we can write,

(7.5)
$\displaystyle\langle\phi_{F},\phi_{F}\rangle=\frac{(4\pi)^{k}\pi^{2}\zeta(2)^{-1}}{\Gamma(k)L(k,f)}\cdot
N^{-2}\left(\prod\nolimits_{p|N}(1-p^{-1})^{-2}(1+p^{-1})^{-1}\right)\langle
F,F\rangle.$

So we will now approach the lower bound via a suitable application of
Waldspurger’s formula. We start with the lower bound from (7.1) further
restricting it to newforms:

(7.6)
$\displaystyle\sup(\mathrm{SK}_{k}(N))\geq\det(Y_{0})^{k}\exp(-4\pi\mathrm{tr}(T_{0}Y_{0}))\sum\nolimits_{F\in
H^{*}_{k}(N)}|a_{F}(T_{0})|^{2}\big{/}\langle F,F\rangle.$

We can choose any $T_{0}$ such that $\mathrm{disc}(2T_{0})=D_{0}$ is
fundamental. Then $a_{F}(T_{0})=c_{\phi}(|D_{0}|)$. A good choice is (if we
refer to the Waldspurger’s formula and the exponent of $D_{0}$ there)
$T_{0}=1_{2}$ so that $D_{0}=-4$. We write (7.6) in terms of a basis $\phi\in
J_{k,1}^{cusp,new}(N)$.

For the rest of this section, let us assume that $N=p$, an odd prime.

We now appeal to Waldspurger’s formula (cf. [37]) to notice that

(7.7)
$\displaystyle\frac{4^{k-1}}{6}\frac{|c_{\phi}(-4)|^{2}}{\langle\phi_{F},\phi_{F}\rangle}=\frac{|c_{\phi_{F}}(-4)|^{2}}{\langle
h_{F},h_{F}\rangle}=2\frac{\Gamma(k-1)4^{k-3/2}}{\pi^{k-1}}\frac{L(1/2,f\otimes\chi_{-4})}{\langle
f,f\rangle},$

where $h_{F}\in S^{+}_{k-1/2}(4p)$ is the unique newform in Kohnen’s plus
space corresponding to $\phi_{F}$, having same Fourier coefficients.

Next we appeal to the following average result from [4, Theorem 1.5 (1)] viz.

(7.8) $\displaystyle\sum\nolimits_{f\in
H^{*}_{k}(p)}\frac{L(1/2,f\otimes\chi_{-4})}{\langle f,f\rangle_{p}}\gg 1,$

for all $p$ large enough, and the implied constant being absolute.

######  Remark 7.2.

(7.8) should also hold for all square-free levels. We however don’t know of a
quotable reference. The method in [4] can be generalised along the same lines
to obtain it. This would then immediately give the lower bound $N^{-2}$ for
$\sup(\mathrm{SK}_{k}(N))$.

We can then use the expression for $\langle F,F\rangle$ from (7.5) in (7.6),
along with (7.7) and (7.8). Hence:

(7.9)
$\displaystyle\sup(\mathrm{SK}_{k}^{new}(p))\gg\det(Y_{0})^{k}\exp(-4\pi\mathrm{tr}(Y_{0}))\frac{4^{k}p^{-2}}{\Gamma(k)}\frac{(4\pi)^{k}\pi^{k}}{\Gamma(k-3/2)}\cdot\frac{1}{p^{2}};$

and then choosing $Y_{0}=k1_{2}\cdot i/4\pi$ gives us

(7.10) $\displaystyle\sup(\mathrm{SK}_{k}^{new}(p))\gg k^{5/2}p^{-2}.$

The genesis of Conjecture 1.8 is now clear from (7.10). Later we will see in
Sections 11 and 12.1.1 that the upper bound $O(p^{-2})$ holds for the
oldspace.

##  8\. Description of the fundamental domain.

In order to understand the size of SK lifts in the fundamental domain
$\mathcal{F}_{2}(p)$, we first recall the set of explicit coset
representatives $\Gamma_{0}^{(2)}(p)\backslash\mathrm{Sp}_{2}(\mathbf{Z})$. We
follow the notation used in [14] and for $0\leq j\leq 2$ put

(8.1) $w_{j}=\begin{pmatrix}0_{j}&0&-1_{j}&0\\\ 0&1_{n-j}&0&0_{n-j}\\\
1_{j}&0&0_{j}&0\\\ 0&0_{n-j}&0&1_{n-j}\end{pmatrix}.$

Further, for $A\in\mathrm{GL}_{2}(\mathbf{Z})$ and $B\in M_{2}(\mathbf{Z})$,
we define

(8.2) $m(A):=\left(\begin{smallmatrix}A&0\\\
0&(A^{-1})^{t}\end{smallmatrix}\right)\text{ and
}n(B):=\left(\begin{smallmatrix}1&B\\\ 0&1\end{smallmatrix}\right).$

Let $\mathcal{R}(j)$ denote the set of matrices of the form
$w_{j}n(B_{j})m(A)$, where $B_{j}$ and $A$ are as in [14] (lifts of matrices
in $\mathrm{Sym}_{j}(\mathbf{F}_{p})$ and
$P_{2,j}\backslash\mathrm{GL}_{2}(\mathbf{F}_{p})$ to
$\mathrm{Sp}_{2}(\mathbf{Z})$, respectively). Then a set of right coset
representatives for $\Gamma^{(2)}_{0}(p)\backslash\mathrm{Sp}_{2}(\mathbf{Z})$
is given by

(8.3)
$\Gamma^{(2)}_{0}(p)\backslash\mathrm{Sp}_{2}(\mathbf{Z})=\bigcup\nolimits_{j=0}^{2}\mathcal{R}(j).$

We call the coset representatives as Type 0, Type 1 or Type 2 depending on
whether they belong to $\mathcal{R}(0)$, $\mathcal{R}(1)$ or $\mathcal{R}(2)$
respectively.

Let $\mathcal{F}_{2}$ denote a fundamental domain for the action of
$\mathrm{Sp}_{2}(\mathbf{Z})$ on $\mathbf{H}_{2}$. Then the fundamental domain
$\mathcal{F}_{2}(p)$ for the action of $\Gamma_{0}^{(2)}(p)$ can now be
written as

(8.4)
$\mathcal{F}_{2}(p)=\bigcup\nolimits_{j=0}^{2}\bigcup\nolimits_{\gamma\in\mathcal{R}(j)}\gamma\mathcal{F}_{2}.$

We call $\cup_{\gamma\in\mathcal{R}(j)}\gamma\mathcal{F}_{2}$ as the Type $j$
region.

We would bound the BK for SK as follows:

(8.5)
$\displaystyle\sup(\mathrm{SK}_{k}(p))\ll\sup_{Z\in\mathbf{H}_{2}}\sum_{F\in\mathcal{B}^{SK,new}(p)}\frac{\det(Y)^{k}|F(Z)|^{2}}{\left\lVert
F\right\rVert^{2}_{p}}+\sup_{Z\in\mathbf{H}_{2}}\sum_{F\in\mathcal{B}^{SK,old}(p)}\frac{\det(Y)^{k}|F(Z)|^{2}}{\left\lVert
F\right\rVert^{2}_{p}}.$

Denote the corresponding sizes of new and old spaces in the above equation as
$\sup(\mathrm{SK}_{k}^{new}(p))$ and $\sup(\mathrm{SK}_{k}^{old}(p))$,
respectively. Thus it is enough to bound the new and old spaces separately.

##  9\. Newform contribution in Type 0 region.

In this region, we have $Z\in\mathcal{F}_{2}$ (the standard Siegel’s
fundamental domain of level $1$ inside $\mathbf{H}_{2}$) and thus we have that
$Y\gg 1$. We now recall the formula for the Fourier coefficients of
$F\in\mathrm{SK}_{k}^{new}(p)$ in terms of those of the lifted $\phi\in
J_{k,1}^{cusp,new}(p)$. Throughout the remainder of this section we shall put

(9.1) $F(Z)=\sum\nolimits_{m\geq 1}\phi_{m,F}(\tau,z)e(m\tau^{\prime}).$

and for any $\mathcal{D}\equiv 0,-1\bmod{4}$, $\mathcal{D}>0$, we put
$\displaystyle c_{\phi,F}(\mathcal{D}):=c_{\phi_{1,F}}(\mathcal{D})$.

Then we have, for $T=\left(\begin{smallmatrix}n&r/2\\\
r/2&m\end{smallmatrix}\right)$ and $\mathbf{c}(T)=(n,r,m)$ – the content of
$T$, the following relation holds (where $D=\det(2T)$):

(9.2) $\displaystyle
a_{F}(T)=\sum\nolimits_{a|\mathbf{c}(T),\,(a,N)=1}a^{k-1}c_{\phi,F}\left(D/a^{2}\right).$

We begin by estimating the quantity
$\mathcal{P}_{*}(T):=\sum_{F\in\mathcal{B}^{SK,new}(p)}|a_{F}(T)|^{2}/\left\lVert
F\right\rVert^{2}_{p}$ . If we also put
$p_{*}(D):=\sum_{\phi\in\mathcal{B}^{J,new}_{k,1}(p)}|c_{\phi,F}(D)|^{2}$,
where $\mathcal{B}^{J,new}(p)$ denotes any orthonormal basis of
$J_{k,1}^{cusp,new}(p)$, then we see that

(9.3) $\displaystyle\mathcal{P}_{*}(T)\ll
p^{-2}\,(\sum\nolimits_{a}a^{k-1}p_{*}(D/a^{2})^{1/2})^{2}.$

This follows by using the relations (9.2) and (7.5):

(9.4)
$\begin{split}\mathcal{P}_{*}(T)&=\sum\nolimits_{F}\Big{|}\sum\nolimits_{a|\mathbf{c}(T)}a^{k-1}c_{\phi,F}\left(\frac{D}{a^{2}}\right)\Big{|}^{2}\ll
p^{-2}\Big{(}\sum\nolimits_{a}a^{k-1}\big{(}\sum\nolimits_{\phi}|c_{\phi,F}\left(\frac{D}{a^{2}}\right)|^{2}\big{)}^{1/2}\Big{)}^{2}.\end{split}$

Let $P_{D,r}\in J^{cusp}_{k,1}(N)$ be the $D$-th Poincaré series, it is well-
known that it depends only on $D$. The lemma below gives bounds on its Fourier
coefficients, which follows from [16, Thm. 3.23] and standard estimates.

######  Lemma 9.1.

$c_{D,r}=1+O((D,N)^{1/2}D^{1/2}N^{-1+\epsilon})$.

Now using the bound for the Fourier coefficients of Poincaré series from
Proposition Lemma 9.1 we see that (see [4] for similar arguments in level $1$)

(9.5)
$\displaystyle\mathcal{P}_{*}(T)\ll_{\epsilon}p^{-2}\frac{4^{k}\pi^{2k}}{\Gamma(k)\Gamma(k-3/2)}\cdot\left(D^{k-3/2}c(T)^{1+\epsilon}+p^{-1/2+\epsilon}k^{-1/3}D^{k-1}c(T)^{\epsilon}\right).$

Next, note that we can write by [19, Lem. 4.4] (follows from the Cauchy-
Schwartz inequality) that

(9.6)
$\displaystyle\sum\nolimits_{F\in\mathcal{B}^{SK,new}(p)}\frac{\det(Y)^{k}|F(Z)|^{2}}{\left\lVert
F\right\rVert^{2}_{p}}\leq q_{*}(Y)^{2},$

where for $Y>0$, we have put

(9.7) $\displaystyle
q_{*}(Y):=\sum\nolimits_{T}\mathcal{P}_{*}(T)^{1/2}\det(Y)^{k/2}\exp(-2\pi\mathrm{tr}TY).$

Plugging in the bound for $\mathcal{P}_{*}(T)$ in (9.7) we get that $q_{*}(Y)$
is

(9.8) $\displaystyle\ll
p^{-1}Q(1/2,3/4-\epsilon;Y)+p^{-5/4+\epsilon}Q(\epsilon,1/2;Y),$

where the implied constant depends polynomially on $k$ and

(9.9) $\displaystyle
Q(\alpha,\beta;Y):=\frac{(4\pi)^{k}}{\Gamma(k)}\sum\nolimits_{T}\mathbf{c}(T)^{\alpha}\det(T)^{k/2-\beta+\epsilon}\det(Y)^{k/2}\exp(-2\pi\mathrm{tr}TY).$

###  9.1. Analysis of the Fourier expansion

This section collects various bounds coming from the Fourier expansions of
modular forms in level $N$. We start with a general set-up and consider any
$n\geq 1$. For any $S\subseteq\Lambda_{n}^{+}$ we put

(9.10) $Q_{S}(\beta;Y):=\sum\nolimits_{T\in
S}\det(T)^{k/2-\beta}\det(Y)^{k/2}\exp(-2\pi\mathrm{tr}TY),$

Then we can write $Q_{\Lambda_{n}^{+}}(\beta;Y)$ as

(9.11)
$Q_{\Lambda_{n}^{+}}(\beta;Y)=Q_{\mathcal{C}(Y,N^{\epsilon})}(\beta;Y)+Q_{\Lambda_{n}^{+}\setminus\mathcal{C}(Y,N^{\epsilon})}(\beta;Y),$

where the set $\mathcal{C}(Y,N^{\epsilon})$ is defined as below.

(9.12)
$\displaystyle\mathcal{C}(Y,N^{\epsilon}):=\\{T\in\Lambda^{+}_{n}|\text{ all
the eigenvalues of }TY\text{ are of the magnitude
}\frac{k}{4\pi}+O(k^{1/2+\epsilon}N^{\epsilon})\\}.$

Now we show that the tail in (9.11) has sub-exponential decay. To be precise,
we show that (assuming that $\beta\geq 0$):

######  Lemma 9.2.

For $Y>0$ Minkowski reduced, $N$ large enough, one has

(9.13) $\displaystyle
Q_{\Lambda_{n}^{+}\setminus\mathcal{C}(Y,N^{\epsilon})}(\beta;Y)\ll\exp(-cN^{\epsilon})\det(Y)^{\beta-(n+1)/2}.$

Moreover, if one of the diagonal entries $v,v^{\prime}\gg N^{\epsilon}$, one
has a sub-exponential decay: $\displaystyle
Q_{\Lambda_{n}^{+}}(\beta;Y)\ll\exp(-c_{1}N^{\epsilon})\det(Y)^{\beta-(n+1)/2}$.

###### Proof.

Let $\lambda_{1}(TY)\leq\lambda_{2}(TY)\leq\cdots\leq\lambda_{n}(TY)$ denote
the eigenvalues of $TY$. Then