# Distribution of the successive minima of the Petersson norm on cusp forms

Souparna Purohit

###### Abstract.

Let $\Gamma\subseteq\text{PSL}_{2}(\mathbb{Z})$ be a finite index subgroup.
Let $\mathscr{X}(\Gamma)$ be a regular proper model of the modular curve
associated with $\Gamma$, and let $\overline{\mathscr{L}}^{\otimes k}$ be the
logarithmically singular metrized line bundle on $\mathscr{X}(\Gamma)$
associated to modular forms of level $\Gamma$ and weight $12k$, endowed with
the Petersson metric. For each $k\geq 1$, the sub-lattice
$\mathscr{S}_{k}\subseteq H^{0}(\mathscr{X}(\Gamma),\mathscr{L}^{\otimes k})$
of integral cusp forms of level $\Gamma$ and weight $12k$ is a euclidean
lattice with respect to the Petersson norm. In this paper, we describe the
distribution of the successive minima of the $\mathscr{S}_{k}$ as
$k\to\infty$, generalizing the work of Chinburg, Guignard, and Soulé in
[CGS18], which addressed the case $\Gamma=\text{PSL}_{2}(\mathbb{Z})$.

###### Contents

  1. 1 Introduction
  2. 2 Background on integral modular forms
  3. 3 Proof of theorem 1.1
  4. 4 Comparison of measures - Proof of theorem 1.2

##  1\. Introduction

Let $\mathscr{X}$ be an arithmetic variety (integral, flat, projective scheme)
over $\text{Spec}(\mathbb{Z})$ with smooth generic fiber. Let
$\overline{\mathscr{L}}$ be a possibly singular hermitian line bundle on
$\mathscr{X}$, and for every integer $k\geq 1$, let
$\overline{\mathscr{L}}^{\otimes k}$ denote $\mathscr{L}^{\otimes k}$ endowed
with the product metric. Let $\|\cdot\|_{k,\infty}$ denote the $L^{2}$ norm on
the global sections
$H^{0}(\mathscr{X}_{\mathbb{C}},\mathscr{L}_{\mathbb{C}}^{\otimes
k}):=H^{0}(\mathscr{X},\mathscr{L}^{\otimes k})\otimes_{\mathbb{Z}}\mathbb{C}$
with respect to a fixed smooth volume form on $\mathscr{X}(\mathbb{C})$
invariant under complex conjugation. The finite free $\mathbb{Z}$-modules
$H^{0}(\mathscr{X},\mathscr{L}^{\otimes k})$ endowed with
$\|\cdot\|_{k,\infty}$ are euclidean lattices.

A basic set of invariants of euclidean lattices are their successive minima.
Recall that for $i=1,\dots,d_{k}$, where $d_{k}:=\text{rk}_{\mathbb{Z}}\
H^{0}(\mathscr{X},\mathscr{L}^{\otimes k})$, the $i$th successive minima,
$\mu_{k,i}$, is the smallest real number $\mu$ such that there exist vectors
$s_{1},\dots,s_{i}\in H^{0}(\mathscr{X},\mathscr{L}^{\otimes k})$ that are
linearly independent in $H^{0}(\mathscr{X},\mathscr{L}^{\otimes
k})\otimes_{\mathbb{Z}}\mathbb{Q}$ and such that $\|s_{j}\|_{k,\infty}\leq\mu$
for all $j$. The study of successive minima has a long history, beginning with
Minkowski’s work on the geometry of numbers, to more recent works in Arakelov
theory. From the perspective of Arakelov theory, it turns out to be more
natural to work with $\lambda_{k,i}:=-\log(\mu_{k,i})$, called the successive
maxima.

We are interested in the asymptotic distribution of the successive maxima of
$H^{0}(\mathscr{X},\mathscr{L}^{\otimes k})$ as $k\to\infty$. When the metric
on $\overline{\mathscr{L}}$ is smooth, or even continuous, the work of Chen in
[Che10] shows the existence of a limiting distribution associated to the
successive maxima of the sequence of lattices
$H^{0}(\mathscr{X},\mathscr{L}^{\otimes k})$. More precisely, if

$\nu_{k}:=\frac{1}{d_{k}}\sum_{i=1}^{d_{k}}\delta_{\frac{1}{k}\lambda_{k,i}},$

where $\delta_{x}$ is the Dirac delta function at a real number $x$, then
Chen’s results (in particular, theorem 4.1.8 of [Che10]) show that the
sequence of probability measures $(\nu_{k})_{k}$ on $\mathbb{R}$ converges
weakly to a compactly supported probability measure on $\mathbb{R}$.

In [CGS18], Chinburg, Guignard, and Soulé prove an analogous result when the
metric on $\overline{\mathscr{L}}$ is logarithmically singular (see [Küh25]
definition 3.1, and [BGKK20] chapter 7). In particular, they consider
$\mathscr{X}=\mathscr{X}(1)$ an integral model of the modular curve
$X(1)_{\mathbb{C}}$ for $\text{PSL}_{2}(\mathbb{Z})$, and
$\overline{\mathscr{L}}$ the line bundle of integral modular forms for the
group $\text{PSL}_{2}(\mathbb{Z})$ of weight $12$ with the Petersson metric
(see §3 of [CGS18]). As shown by Kühn in proposition 4.9 of [Küh25], the
Petersson metric on the line bundle of modular forms is logarithmically
singular along the cusps and elliptic points. Due to these singularities, the
$L^{2}$ norms, i.e. the Petersson norms, are no longer defined on the spaces
$H^{0}(\mathscr{X}(1)_{\mathbb{C}},\mathscr{L}_{\mathbb{C}}^{\otimes k})$ of
all modular forms of weight $12k$. But they are defined on the subspaces
$H^{0}(\mathscr{X}(1)_{\mathbb{C}},\mathscr{L}_{\mathbb{C}}^{\otimes
k}(-\infty))$ of cusp forms (here $\infty\in\mathscr{X}(1)_{\mathbb{C}}$
denotes the unique cusp). In this setting, theorem 3.2.2 of [CGS18] describes
the asymptotic behavior of the successive maxima of the lattices of integral
cusp forms for the group $\text{PSL}_{2}(\mathbb{Z})$ of weight $12k$ with the
Petersson norm as $k\to\infty$, in close analogy to Chen’s result mentioned
above.

In this paper, we generalize theorem 3.2.2 of [CGS18] to cusp forms of any
finite index subgroup $\Gamma\subseteq\text{PSL}_{2}(\mathbb{Z})$. We state
our results precisely in the following sub-section.

###  1.1. Statement of main results

We follow the setup in §4.11 of [Küh25]. Let
$\Gamma(1):=\text{PSL}_{2}(\mathbb{Z})$, and identify the complex modular
curve $X(1)_{\mathbb{C}}$ associated to $\Gamma(1)$ with
$\mathbb{P}_{\mathbb{C}}^{1}$ via the modular $j$-function (see example 2.1).
Let $\Gamma\subseteq\Gamma(1)$ be a finite index subgroup. The modular curve
$X(\Gamma)_{\mathbb{C}}$ and the natural map
$\pi_{\Gamma,\mathbb{C}}:X(\Gamma)_{\mathbb{C}}\to\mathbb{P}_{\mathbb{C}}^{1}$
are defined over a number field $E$ (see §2.3). Let $X(\Gamma)$ be a (smooth
projective geometrically connected) model of $X(\Gamma)_{\mathbb{C}}$ over
$E$, let $\pi_{\Gamma,E}:X(\Gamma)\to\mathbb{P}_{E}^{1}$ be the model of
$\pi_{\Gamma,\mathbb{C}}$ over $E$, and let
$\pi_{\Gamma}:X(\Gamma)\to\mathbb{P}_{\mathbb{Q}}^{1}$ denote the composition
of $\pi_{\Gamma,E}$ with the natural map from $\mathbb{P}_{E}^{1}$ to
$\mathbb{P}_{\mathbb{Q}}^{1}$. Let $\infty\in\mathbb{P}_{\mathbb{Q}}^{1}$
correspond to the unique pole of the $j$-function, and let
$D\in\text{Div}(X(\Gamma))$ denote the sum of the points in
$\pi_{\Gamma}^{-1}(\infty)$. We call $D$ the divisor of cusps of $X(\Gamma)$.

Let $\mathscr{X}(\Gamma)$ be an arithmetic surface associated to $\Gamma$ (see
§2.3 and §4.11 of [Küh25]). Then $\mathscr{X}(\Gamma)$ is a regular projective
arithmetic surface with generic fiber $X(\Gamma)$, and comes with a morphism
$\pi_{\Gamma,\mathbb{Z}}:\mathscr{X}(\Gamma)\to\mathbb{P}_{\mathbb{Z}}^{1}$
extending $\pi_{\Gamma}$. Let $\overline{\mathscr{L}}$ denote the metrized
line bundle on $\mathscr{X}(\Gamma)$ associated to modular forms of level
$\Gamma$ and weight $12$ endowed with the Petersson metric (see §2.3 and §2.4,
and §4.12 of [Küh25]). As mentioned above, this metric has logarithmic
singularities at elliptic points and cusps (proposition 4.9 in [Küh25]).

######  Theorem 1.1.

Let $\Gamma\subseteq\Gamma(1)$ be a finite index subgroup. Let
$\mathscr{X}(\Gamma)$ be an arithmetic surface associated to $\Gamma$,
$X(\Gamma)$ its generic fiber, and let $D$ denote the divisor of cusps of
$X(\Gamma)$. Let $\overline{\mathscr{L}}$ be the line bundle on
$\mathscr{X}(\Gamma)$ associated to modular forms of level $\Gamma$ and weight
$12$ endowed with the Petersson metric. For every $k\geq 1$, let

$\mathscr{S}_{k}:=H^{0}(\mathscr{X}(\Gamma),\mathscr{L}^{\otimes k})\cap
H^{0}(X(\Gamma),\mathscr{L}_{\mathbb{Q}}^{\otimes k}(-D))$

denote the euclidean lattice of integral cusp forms of level $\Gamma$ and
weight $12k$ with respect to the Petersson inner product. Let $\mu_{k,i}$
denote the successive minima of $\mathscr{S}_{k}$, and let
$\lambda_{k,i}:=-\log(\mu_{k,i})$ denote the successive maxima. Let
$d_{k}:=\emph{rk}_{\mathbb{Z}}\mathscr{S}_{k}$, and let

$\nu_{k}:=\frac{1}{d_{k}}\sum_{i=1}^{d_{k}}\delta_{\frac{1}{k}\lambda_{k,i}}$

denote the probability measure on $\mathbb{R}$ associated to the normalized
successive maxima of $\mathscr{S}_{k}$. Then the $\nu_{k}$ converge weakly to
a Borel probability measure $\nu$ on $\mathbb{R}$. $\nu$ has support bounded
above and unbounded below.

A natural question is: given finite index subgroups
$\Gamma^{\prime}\subseteq\Gamma$ of $\Gamma(1)$, how do the successive maxima
of their integral cusp forms, and the associated limit measures compare? This
is addressed by the following theorem.

######  Theorem 1.2.

Let $\Gamma^{\prime}\subseteq\Gamma$ be finite index subgroups of $\Gamma(1)$.
Let $\mathscr{X}(\Gamma^{\prime})$ (resp. $\mathscr{X}(\Gamma)$) be an
arithmetic surface associated to $\Gamma^{\prime}$ (resp. $\Gamma$) with
generic fiber $X(\Gamma^{\prime})$ (resp. $X(\Gamma)$). Suppose there is a
$\mathbb{Q}$-morphism $\pi_{\Gamma^{\prime},\Gamma}:X(\Gamma^{\prime})\to
X(\Gamma)$ that is a model over $\mathbb{Q}$ of the natural map
$X(\Gamma^{\prime})_{\mathbb{C}}\to X(\Gamma)_{\mathbb{C}}$ of complex modular
curves.

Let $\nu_{k}^{\prime}$ (resp. $\nu_{k}$) denote the probability measure on
$\mathbb{R}$ associated to the normalized successive maxima of the euclidean
lattice of integral cusp forms of level $\Gamma^{\prime}$ (resp. $\Gamma$) and
weight $12k$ with respect to $\mathscr{X}(\Gamma^{\prime})$ (resp.
$\mathscr{X}(\Gamma)$) as in theorem 1.1. Suppose that
$\nu_{k}^{\prime}\to\nu^{\prime}$ (resp. $\nu_{k}\to\nu$) weakly for a Borel
probability measure $\nu^{\prime}$ (resp. $\nu$) on $\mathbb{R}$ (by theorem
1.1). Then there is a Borel probability measure $\omega$ on $\mathbb{R}$ such
that

$\nu^{\prime}=\frac{1}{\emph{deg}(\pi_{\Gamma^{\prime},\Gamma})}\cdot\nu+\left(1-\frac{1}{\emph{deg}(\pi_{\Gamma^{\prime},\Gamma})}\right)\cdot\omega.$

###  1.2. Remarks

######  Remark 1.3.

As mentioned before, theorem 1.1 generalizes theorem 3.2.2 (i), (ii) of
[CGS18], which addresses the case $\Gamma=\Gamma(1)$ for the associated
arithmetic surface $\mathscr{X}(1)=\mathbb{P}_{\mathbb{Z}}^{1}$
(identification coming from the $j$-function). We remark that our overall
approach is similar to that of [CGS18], with a couple of key differences.
First, the approach in [CGS18] uses various properties of $q$-expansions of
modular forms for $\Gamma(1)$, including integrality of the coefficients of
$q$-expansions of modular forms over the integral modular curve
$\mathscr{X}(1)$ to obtain (lower) bounds on the Petersson norms of integral
cusp forms (see for instance, proposition 3.3.1, lemma 3.3.1, theorem 3.4.1,
and lemma 3.4.2 in [CGS18]). For a general (in particular, non-congruence)
finite index subgroup $\Gamma\subseteq\Gamma(1)$, we do not have access to
such integrality properties for the $q$-expansion coefficients of integral
modular forms.

Instead, following [Che10], we use results from the intersection theory of
logarithmically singular line bundles on arithmetic surfaces as developed by
Kühn in [Küh25], along with height formulas developed by Bost, Gillet, and
Soulé in [BGS94] to obtain corresponding (lower) bounds for the sup norms of
integral cusp forms (see proposition 3.3, which is the counterpart to lemma
3.4.2 in [CGS18]). We then appeal to a “Gromov’s lemma” type result in our
log-singular setting (see proposition 3.2, page 1 of [FJK13], and theorem 1.7
in [AMM16]), comparing sup norms to $L^{2}$ norms for sections in increasing
powers of the line bundle of modular forms, to get analogous (lower) bounds on
the $L^{2}$ (i.e. Petersson) norms of integral cusp forms for $\Gamma$. The
estimates obtained from the intersection theory approach, however, are not as
sharp as those obtained from the $q$-expansions, and this ultimately prevents
us from showing directly that the support of the limit measure $\nu$ in
theorem 1.1 is unbounded below. Instead, we deduce unboundedness (in corollary
4.4) from theorem 1.2 along with the unboundedness in the case
$\Gamma=\Gamma(1)$ proved in part (ii) of theorem 3.2.2 of [CGS18].

Another key difference with [CGS18] is that, unlike the $\Gamma=\Gamma(1)$
case considered there, it is difficult to write down explicit bases for the
spaces of cusp forms for general $\Gamma$, which are then used for getting
bounds on the successive maxima. In proposition 3.9, we construct less
explicit bases that nevertheless have the same general shape as those in
[CGS18] theorem 3.4.1 and lemma 3.4.4, and that turns out to be enough to
yield the desired bounds on the successive maxima in our case.

######  Remark 1.4.

An important feature of theorem 1.1 and theorem 3.2.2 of [CGS18] is the fact
that the limit measures have support unbounded below, in stark contrast to
Chen’s theorem for continuous metrics mentioned before, where the limit
measure has compact support. This is explained, very roughly, as follows. When
$\mathscr{L}$ has continuous metric, then at least if
$\mathscr{L}_{\mathbb{Q}}$ is ample,

$\bigoplus_{k\geq
0}H^{0}(\mathscr{X}_{\mathbb{Q}},\mathscr{L}_{\mathbb{Q}}^{\otimes k})$

is a finitely generated quasi-filtered algebra over $\mathbb{Q}$ (in the sense
of definition 3.2.1 of [Che10]). The convergence result for the measures then
follows from a more general result of Chen (theorem 3.4.3 [Che10]) concerning
the distribution of successive maxima for such algebras, which shows, in
particular, that the limiting distribution has compact support.

In our setting, and that of [CGS18], the Petersson metric on the line bundle
$\mathscr{L}$ of modular forms is logarithmically singular, and hence to make
sense of the norms at the archimedean places, we restricted to the subspace
$H^{0}(\mathscr{X}_{\mathbb{Q}},\mathscr{L}_{\mathbb{Q}}^{\otimes k}(-D))$ of
sections vanishing along the cusps. The graded $\mathbb{Q}$-algebra

$B:=\bigoplus_{k\geq
0}H^{0}(\mathscr{X}_{\mathbb{Q}},\mathscr{L}_{\mathbb{Q}}^{\otimes k}(-D))$

is not finitely generated. However, we can “approximate” $B$ by a sequence of
finite-type quasi-filtered $\mathbb{Q}$ algebras $B_{L}$ indexed by integers
$L$ (as in §3.2, and §3.6 of [CGS18]). Then applying theorem 3.4.3 of [Che10]
to each $B_{L}$, we get a sequence of compactly supported measures
$\nu_{L,\infty}$, where the support of each $\nu_{L}$ is bounded above by a
constant independent of $L$, but the lower bound for the support goes to
$-\infty$ as $L\to\infty$, at least in the case $\Gamma=\Gamma(1)$ considered
in [CGS18]. We then show that the $\nu_{L}$ converge to $\nu$ as $L\to\infty$,
which accounts for the support of $\nu$ being unbounded below.

######  Remark 1.5.

A natural question is whether theorem 1.1 generalizes to automorphic forms for
other groups. More generally, we may ask the following general problem in the
setting of [BM12]. Let $\mathscr{X}$ be an arithmetic variety with regular
generic fiber, $\overline{\mathscr{L}}$ a hermitian line bundle on
$\mathscr{X}$ with logarithmic singularities along a normal crossings divisor
$D\in\text{Div}(\mathscr{X}_{\mathbb{Q}})$. Fix a smooth volume form on
$\mathscr{X}(\mathbb{C})$ invariant under complex conjugation. For each $k\geq
1$, let $\mathscr{S}_{k}:=H^{0}(\mathscr{X},\mathscr{L}^{\otimes k})\cap
H^{0}(\mathscr{X}_{\mathbb{Q}},\mathscr{L}_{\mathbb{Q}}^{\otimes k}(-D))$
denote the euclidean lattice of sections of $\mathscr{L}^{\otimes k}$ that
vanish along $D$ equipped with the $L^{2}$ norm for the given volume form.
Then do the probability measures $\nu_{k}$ associated to the (appropriately)
normalized successive maxima of $\mathscr{S}_{k}$ converge weakly to a
probability measure on $\mathbb{R}$?

The case $\dim\mathscr{X}=2$ (i.e., an arithmetic surface) is still
interesting. One difficulty, even for $\dim\mathscr{X}=2$, is the lack of a
Gromov’s lemma type result for general logarithmically singular line bundles,
though some results are known with specific hypotheses on
$\overline{\mathscr{L}}$ (see, for instance, [AMM16]). We hope to come back to
this problem in the future.

Outline of the paper. In §2, we start with some background on complex modular
forms and modular curves. Then, following [Küh25], we define integral models
for modular curves and the line bundle of modular forms in §2.3, and describe
the Petersson metric on this bundle in §2.4. We then introduce the notion of
adelic vector bundles, as in definition 3.1 of [Gau08], their successive
maxima, and apply these concepts to the space of rational cusp forms of weight
$12k$ for finite index subgroups of $\text{PSL}_{2}(\mathbb{Z})$. This lets us
define a decreasing $\mathbb{R}$-filtration on each $S_{k}$, which enables us
to use results of Chen on quasi-filtered algebras in order to prove our
results. Then in §3 and §4, we prove theorems 1.1 and 1.2, respectively.

Acknowledgements. I would like to thank my advisor, Ted Chinburg, for many
helpful discussions regarding this project.

##  2\. Background on integral modular forms

In this section, we start with some background on complex modular forms and
modular curves. Then, following [Küh25], we define integral models for modular
curves and the line bundle of modular forms in §2.3, and describe the
Petersson metric on this bundle in §2.4. We then introduce the notion of
adelic vector bundles, as in definition 3.1 of [Gau08], their successive
maxima, and apply these concepts to the space of rational cusp forms of weight
$12k$ for finite index subgroups of $\text{PSL}_{2}(\mathbb{Z})$.

###  2.1. Modular curves and modular forms over $\mathbb{C}$

Let $\Gamma$ be a finite index subgroup of
$\Gamma(1):=\text{PSL}_{2}(\mathbb{Z})$. Then $\Gamma$ acts on the (complex)
extended upper half plane
$\mathfrak{h}^{*}:=\mathfrak{h}\cup\mathbb{P}^{1}(\mathbb{Q})$ by linear
fractional transformations

$\Gamma\ni\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\cdot z=\frac{az+b}{cz+d}.$

The quotient

$X(\Gamma)_{\mathbb{C}}:=\Gamma\backslash(\mathfrak{h}\cup\mathbb{P}^{1}(\mathbb{Q})).$

is a compact Riemann surface called the modular curve associated to $\Gamma$.
The inclusion $\Gamma\subseteq\Gamma(1)$ induces a holomorphic map
$\pi_{\Gamma,\mathbb{C}}:X(\Gamma)_{\mathbb{C}}\to X(\Gamma(1))_{\mathbb{C}}$
of Riemann surfaces of degree $[\Gamma(1):\Gamma]$.

For a point $z\in\mathfrak{h}^{*}$, let

$\Gamma_{z}:=\\{\tau\in\Gamma:\tau\cdot z=z\\}$

denote the stabilizer of $z$ in $\Gamma$. This group is either trivial,
finite, or infinite cyclic, and we call the image of $z$ in
$X(\Gamma)_{\mathbb{C}}$ an ordinary point, elliptic point, or cuspidal point
(or just a cusp) in these cases, respectively. For $\Gamma(1)$, the elliptic
points are the images of $i$ and $e^{2\pi i/3}$ and there is a unique cusp
corresponding to the image of (any) point of $\mathbb{P}^{1}(\mathbb{Q})$. For
a general $\Gamma$, the elliptic points and cusps of $X(\Gamma)_{\mathbb{C}}$
map to elliptic points and cusp of $X(\Gamma(1))_{\mathbb{C}}$, respectively.

Given $\gamma\in\text{PSL}_{2}(\mathbb{R})$, the weight $k$ slash operator
$(\cdot|_{k}\gamma)$ on holomorphic functions on $\mathfrak{h}$ is given by

$f(z)\mapsto(f|_{k}\gamma)(z):=(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right)$

for $\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$. For
$\Gamma\subseteq\Gamma(1)$, a meromorphic modular form of level $\Gamma$ and
weight $k$ is a meromorphic function $f:\mathfrak{h}\to\mathbb{C}$ such that

  1. (a)

$(f|_{k}\gamma)(z)=f(z)$ for all $\gamma\in\Gamma$, and

  2. (b)

$f$ is meromorphic at the cusps of $\Gamma$. This means that at each cusp, if
$t$ denotes a local parameter, then $f$ admits a Fourier expansion of the form
$f(t)=\sum_{n}a_{n}t^{n}$, with $a_{n}=0$ for sufficiently small $n$.

Meromorphic modular forms of weight $0$ are called modular functions \- these
are simply the rational functions of $X(\Gamma)_{\mathbb{C}}$. A modular form
of weight $k$ for $\Gamma$ is a meromorphic modular form that is holomorphic
everywhere, including at the cusps. A cusp form of weight $k$ for $\Gamma$ is
a modular form that is zero at every cusp (i.e., the $a_{0}$ coefficient in
the local Fourier expansion at every cusp is zero). The space of modular
(resp. cusp) forms of level $\Gamma$ and weight $k$ is denoted by
$M_{k}(\Gamma)_{\mathbb{C}}$ (resp. $S_{k}(\Gamma)_{\mathbb{C}}$) - these are
finite dimensional vector spaces over $\mathbb{C}$.

######  Example 2.1.

  1. (i)

The classical $j$-invariant function is a modular function for $\Gamma(1)$
given by

$j(z)=\frac{1}{q}+744+196884q+\dots,$

where $q=e^{2\pi iz}$ is the local parameter at the unique cusp $S_{\infty}$
of $X(1)_{\mathbb{C}}:=X(\Gamma(1))_{\mathbb{C}}$. Note that $j$ has a unique
pole of order $1$ at $S_{\infty}$, so that the induced morphism
$j:X(1)_{\mathbb{C}}\to\mathbb{P}_{\mathbb{C}}^{1}$ is an isomorphism.
Henceforth, we identify $X(1)_{\mathbb{C}}$ with $\mathbb{P}_{\mathbb{C}}^{1}$
via $j$.

  2. (ii)

The classical discriminant function $\Delta$ is a cusp form of weight $12$ for
$\Gamma(1)$ given by

$\Delta(z)=q\prod_{n=1}^{\infty}(1-q^{n})^{24},$

where as before, $q=e^{2\pi iz}$.

The functional equations satisfied by meromorphic modular forms show that they
are sections of a line bundle $\mathfrak{M}_{k}(\Gamma)_{\mathbb{C}}$ on
$X(\Gamma)_{\mathbb{C}}$, and the meromorphic modular forms that are zero at
every cusp form a sub-bundle $\mathfrak{S}_{k}(\Gamma)_{\mathbb{C}}$. The
following is proposition 4.7 in [Küh25]:

######  Proposition 2.2.

Let $\Gamma\subseteq\Gamma(1)$ be a finite index subgroup, and let $k\geq 1$
be a positive integer. Let $S_{1},\dots,S_{t}\in X(\Gamma)_{\mathbb{C}}$
denote the cusps, and let
$\pi_{\Gamma,\mathbb{C}}:X(\Gamma)_{\mathbb{C}}\to\mathbb{P}_{\mathbb{C}}^{1}$
denote the canonical map. Let
$D_{\mathbb{C}}:=S_{1}+\dots+S_{t}\in\emph{Div}(X(\Gamma)_{\mathbb{C}})$. Then
we have an isomorphism of line bundles:

$\mathfrak{M}_{12k}(\Gamma)_{\mathbb{C}}\to\pi_{\Gamma,\mathbb{C}}^{*}\mathcal{O}_{\mathbb{P}_{\mathbb{C}}^{1}}(\infty)^{\otimes
k}:f\mapsto f/\Delta^{k},$

which restricts to an isomorphism

$\mathfrak{S}_{12k}(\Gamma)_{\mathbb{C}}\to\left(\pi_{\Gamma,\mathbb{C}}^{*}\mathcal{O}_{\mathbb{P}_{\mathbb{C}}^{1}}(\infty)^{\otimes
k}\right)(-D_{\mathbb{C}}).$

###  2.2. Petersson metric on $\mathfrak{M}_{12k}(\Gamma)$

The Petersson metric on $\mathfrak{M}_{12k}(\Gamma)$ is defined as follows.
Given a section $f$ of $\mathfrak{M}_{12k}(\Gamma)$ over the open subset
$U\subseteq X(\Gamma)_{\mathbb{C}}$ and a point $z\in\mathfrak{h}$
corresponding to a point of $U$, we set

$|f|_{\text{Pet}}(z)^{2}:=|f(z)|^{2}(4\pi\operatorname{im}(z))^{12k}.$

This differs from the classical Petersson metric by a factor of
$(4\pi)^{12k}$. See [Küh25] p. 227-228 for the reason for this factor. This
metric has logarithmic singularities in the sense of definition 3.1 of [Küh25]
at the elliptic points and cusps of $\Gamma$ (see proposition 4.9 in [Küh25]).

Let $d_{\Gamma}:=[\Gamma(1):\Gamma]$, and let $\mathcal{F}_{\Gamma}$ denote a
fundamental domain for the action of $\Gamma$ on $\mathfrak{h}$. Define the
Petersson inner product on $S_{12k}(\Gamma)_{\mathbb{C}}$ by

$\langle
f,g\rangle_{\text{Pet}}:=\frac{1}{d_{\Gamma}}\int_{\mathcal{F}_{\Gamma}}f(z)\overline{g(z)}(4\pi\operatorname{im}(z))^{12k}\
\frac{dxdy}{y^{2}}\qquad(f,g\in S_{12k}(\Gamma)_{\mathbb{C}}),$

where $z=x+iy\in\mathfrak{h}$. This is a Hermitian inner product on
$S_{12k}(\Gamma)_{\mathbb{C}}$, and the norm of $f\in
S_{12k}(\Gamma)_{\mathbb{C}}$ is given by $\|f\|_{\text{Pet}}:=\langle
f,f\rangle_{\text{Pet}}^{1/2}$, which we note is simply the $L^{2}$ norm of
the Petersson metric (with respect to the hyperbolic volume form on
$X(\Gamma)_{\mathbb{C}}$).

###  2.3. Integral models of modular curves and modular forms

We follow the setup in §4.11 of [Küh25] (with slightly modified notation). Let
$\Gamma\subseteq\Gamma(1)$ be a finite index subgroup, and let
$\pi_{\Gamma,\mathbb{C}}:X(\Gamma)_{\mathbb{C}}\to\mathbb{P}^{1}_{\mathbb{C}}$
denote the natural map given by the $j$-function. The branch points of
$\pi_{\Gamma,\mathbb{C}}$ are contained in
$\\{0,1728,\infty\\}\subseteq\mathbb{P}^{1}_{\mathbb{C}}$, and hence
$X(\Gamma)_{\mathbb{C}}$ and $\pi_{\Gamma,\mathbb{C}}$ are defined over a
number field $E$. For such an $E$, let $X(\Gamma)_{E}$ be a smooth projective
geometrically connected curve over $E$ with base change to $\mathbb{C}$ (via
the implicit embedding of $E$ into $\mathbb{C}$) isomorphic to
$X(\Gamma)_{\mathbb{C}}$, and let
$\pi_{\Gamma,E}:X(\Gamma)_{E}\to\mathbb{P}_{E}^{1}$ be an $E$-morphism such
that its base change to $\mathbb{C}$ (again, via the implicit embedding)
coincides with $\pi_{\Gamma,\mathbb{C}}$. To simplify notation, we will refer
to $X(\Gamma)_{E}$ by $X(\Gamma)$. Let
$\pi_{\Gamma}:X(\Gamma)\to\mathbb{P}_{\mathbb{Q}}^{1}$ denote the composition
of $\pi_{\Gamma,E}$ with the natural map
$\mathbb{P}_{E}^{1}\to\mathbb{P}_{\mathbb{Q}}^{1}$.

Let $X(\Gamma)_{\mathbb{Z}}$ denote the normalization of
$\mathbb{P}^{1}_{\mathbb{Z}}$ in $X(\Gamma)$ under the natural map
$X(\Gamma)\to\mathbb{P}_{\mathbb{Q}}^{1}\to\mathbb{P}_{\mathbb{Z}}^{1}$. Then
$X(\Gamma)_{\mathbb{Z}}$ is a normal arithmetic surface with $X(\Gamma)$ as
generic fiber. By results of Lipman (see [Lip78]), there exists a
desingularization of $X(\Gamma)_{\mathbb{Z}}$. Namely, there exists a regular
projective arithmetic surface with a proper birational morphism to
$X(\Gamma)_{\mathbb{Z}}$. Let $\mathscr{X}(\Gamma)\to X(\Gamma)_{\mathbb{Z}}$
denote such a desingularization, and let the composite map
$\mathscr{X}(\Gamma)\to X(\Gamma)_{\mathbb{Z}}\to\mathbb{P}_{\mathbb{Z}}^{1}$
be denoted by
$\pi_{\Gamma,\mathbb{Z}}:\mathscr{X}(\Gamma)\to\mathbb{P}_{\mathbb{Z}}^{1}$.
As in §4.11 in [Küh25], we call any such $\mathscr{X}(\Gamma)$ an arithmetic
surface associated to $\Gamma$. We remark that this definition differs
slightly from that in §4.11 of [Küh25] in that we do not require the field of
constants $E$ of the generic fiber $X(\Gamma)$ of $\mathscr{X}(\Gamma)$ to be
of minimal degree here.

Let $\infty\in\mathbb{P}_{\mathbb{Q}}^{1}$ correspond to the unique pole of
the $j$-function, and let
$\overline{\infty}\subseteq\mathbb{P}_{\mathbb{Z}}^{1}$ be its Zariski
closure. Let
$\mathscr{L}:=\pi_{\Gamma,\mathbb{Z}}^{*}\mathcal{O}_{\mathbb{P}^{1}_{\mathbb{Z}}}(\overline{\infty})$.

Now,

$\mathscr{X}(\Gamma)_{\mathbb{C}}:=\mathscr{X}(\Gamma)\otimes_{\mathbb{Z}}\mathbb{C}=\bigsqcup_{\sigma:E\to\mathbb{C}}X(\Gamma)\otimes_{E,\sigma}\mathbb{C},$

where the disjoint union is over all embeddings $\sigma$ of $E$ into
$\mathbb{C}$. For each $\sigma$, the base change
$X(\Gamma)\otimes_{E,\sigma}\mathbb{C}$ is also a modular curve, say
associated to the group $\Gamma_{\sigma}\subseteq\Gamma(1)$. Then

$\mathscr{X}(\Gamma)_{\mathbb{C}}\cong\bigsqcup_{\sigma:E\to\mathbb{C}}X(\Gamma_{\sigma})_{\mathbb{C}}.$

For each $k\geq 1$,

$\mathscr{L}^{\otimes
k}_{\mathbb{C}}=\bigoplus_{\sigma:E\to\mathbb{C}}\pi_{\Gamma_{\sigma},\mathbb{C}}^{*}\mathcal{O}_{\mathbb{P}_{\mathbb{C}}^{1}}(\infty)^{\otimes
k}$

is identified with the sum of the line bundles of modular forms of level
$\Gamma_{\sigma}$ and weight $12k$ by proposition 2.2. Hence, we call
$\mathscr{L}$ the line bundle on $\mathscr{X}(\Gamma)$ associated to modular
forms of level $\Gamma$ and weight $12$.

Now let $\\{T_{1},\dots,T_{r}\\}\subseteq X(\Gamma)$ denote the preimage of
$\infty\in\mathbb{P}_{\mathbb{Q}}^{1}$ under $\pi_{\Gamma}$, and let
$D:=T_{1}+\dots+T_{r}\in\text{Div}(X(\Gamma))$ be the divisor of cusps of
$X(\Gamma)$. Then, again by proposition 2.2,

$\mathscr{L}^{\otimes
k}_{\mathbb{C}}(-D_{\mathbb{C}}):=\mathscr{L}_{\mathbb{Q}}^{\otimes
k}(-D)\otimes_{\mathbb{Q}}\mathbb{C}$

is the sum of the bundles of cusp forms for the $\Gamma_{\sigma}$.

######  Remark 2.3.

We will be interested in studying the submodule of
$H^{0}(\mathscr{X}(\Gamma),\mathscr{L}^{\otimes k})$ associated to cusp forms
(see §2.6). As such, it does not matter which desingularization
$\mathscr{X}(\Gamma)\to X(\Gamma)_{\mathbb{Z}}$ we choose, since for any such
desingularization, and for any vector bundle $E$ on $X(\Gamma)_{\mathbb{Z}}$,
if $\mathscr{E}$ denotes its pullback to $\mathscr{X}(\Gamma)$, then the
natural map $H^{0}(X(\Gamma)_{\mathbb{Z}},E)\to
H^{0}(\mathscr{X}(\Gamma),\mathscr{E})$ is an isomorphism.

###  2.4. Petersson metric on $\mathscr{L}^{\otimes k}$

For $k\geq 1$, we endow $\mathscr{L}^{\otimes k}$ with the Petersson metric as
described in §2.2. Explicitly: for a section $f=(f_{\sigma})_{\sigma}$ of
$\mathscr{L}^{\otimes k}_{\mathbb{C}}$ over the open
$\bigsqcup_{\sigma}U_{\sigma}\subseteq\bigsqcup_{\sigma}X(\Gamma_{\sigma})_{\mathbb{C}}$,

$|f|_{\text{Pet}}^{2}(z):=|(f_{\sigma}\Delta^{k})(z)|^{2}(4\pi\operatorname{im}(z))^{12k},$

where $z\in\mathfrak{h}^{*}$ corresponds to a point of $U_{\sigma}\subseteq
X(\Gamma_{\sigma})_{\mathbb{C}}$. We remark that even though there are choices
involved in picking the groups $\Gamma_{\sigma}$ such that
$X(\Gamma)\otimes_{E,\sigma}\mathbb{C}\cong X(\Gamma_{\sigma})_{\mathbb{C}}$,
the resulting metric on $\mathscr{L}$ is independent of these choices. As
already mentioned in §2.2, this metric is logarithmically singular at the
elliptic points and cusps of the $\Gamma_{\sigma}$. We refer to
$\mathscr{L}^{\otimes k}$ endowed with the Petersson metric by
$\overline{\mathscr{L}}^{\otimes k}$.

For each $\sigma$, denote by $d\mu_{\sigma}$ the invariant measure on
$X(\Gamma_{\sigma})_{\mathbb{C}}$ induced by $\frac{dxdy}{y^{2}}$ on the upper
half plane, where we use the coordinate $z=x+iy$, and let $d\mu$ denote the
measure $\bigsqcup_{\sigma}d\mu_{\sigma}$ on
$\bigsqcup_{\sigma}X(\Gamma_{\sigma})_{\mathbb{C}}$.

For $f\in
H^{0}(\mathscr{X}(\Gamma)_{\mathbb{C}},\mathscr{L}_{\mathbb{C}}^{\otimes
k}(-D_{\mathbb{C}}))$ a global section, define

$\|f\|_{k,\infty}^{2}:=\frac{1}{[E:\mathbb{Q}]d_{\Gamma}}\int_{\mathscr{X}(\Gamma)(\mathbb{C})}|f|_{\text{Pet}}^{2}\
d\mu$

to be the normalized $L^{2}$-norm of the Petersson metric. Note that for
$f=(f_{\sigma})$,

$\|f\|_{k,\infty}^{2}=\frac{1}{[E:\mathbb{Q}]}\sum_{\sigma}\|f_{\sigma}\Delta^{k}\|_{\text{Pet}}^{2},$

where $\|\cdot\|_{\text{Pet}}$ refers to the Petersson norm from §2.2.

###  2.5. Adelic vector bundles, heights, and successive maxima

Let $K$ be a number field, and let $\Sigma_{K}$ denote the set of all places
of $K$. For $v\in\Sigma_{K}$, let $\mathbb{C}_{v}$ denote the completion of an
algebraic closure of $K_{v}$. For $v$ finite, let $|\cdot|_{v}$ denote the
norm on $\mathbb{C}_{v}$ that extends the $p$-adic norm on $\mathbb{Q}_{p}$,
where $p$ is the rational prime lying under $v$. For $v$ an archimedean place,
we let $|\cdot|_{v}$ denote the usual norm on $\mathbb{C}_{v}=\mathbb{C}$.

A finite dimensional $K$-vector space $V$ is called an adelic vector bundle
over $K$ if for each $v\in\Sigma_{K}$, $V\otimes_{K}\mathbb{C}_{v}$ is
equipped with a norm $\|\cdot\|_{v}$ subject to the following conditions:

  1. (i)

There exists a $K$-basis $(s_{1},\dots,s_{r})$ of $V$ such that for all but
finitely many finite places $v\in\Sigma_{K}$,

$\|\alpha_{1}s_{1}+\dots+\alpha_{r}s_{r}\|_{v}=\max(|\alpha_{1}|_{v},\dots,|\alpha_{r}|_{v}).$

  2. (ii)

For every $v\in\Sigma_{K}$, $\|\cdot\|_{v}$ is invariant under the action of
the group $\textrm{Gal}(\mathbb{C}_{v}/K_{v})$. Namely, if
$(s_{1},\dots,s_{r})$ is a $K_{v}$-basis of $E\otimes_{K}K_{v}$, and if
$\alpha_{1},\dots,\alpha_{r}\in\mathbb{C}_{v}$, then

$\|\tau(\alpha_{1})s_{1}+\dots+\tau(\alpha_{r})s_{r}\|_{v}=\|\alpha_{1}s_{1}+\dots+\alpha_{r}s_{r}\|_{v}$

for all $\tau\in\textrm{Gal}(\mathbb{C}_{v}/K_{v})$.

  3. (iii)

For $v\in\Sigma_{K}$ finite, $\|\cdot\|_{v}$ satisfies
$\|s+s^{\prime}\|_{v}\leq\max(\|s\|_{v},\|s^{\prime}\|_{v})$.

Let $V$ be an adelic vector bundle over $K$. The naive adelic height function
on $V$ is given by

$\displaystyle\lambda:V$ $\displaystyle\to\mathbb{R}$ $\displaystyle s$
$\displaystyle\mapsto-\sum_{v\in\Sigma_{K}}k_{v}\log\|s\|_{v},$

where $k_{v}:=[K_{v}:\mathbb{Q}_{p}]$ for $p$ the rational place lying under
$v$. Using $\lambda$, we equip $V$ with the following filtration: given
$a\in\mathbb{R}$, set

$V^{a}:=\text{span}_{K}\\{s\in V:\lambda(s)\geq a\\}.$

This is a decreasing filtration on $V$ indexed by the real numbers with the
property that $V^{a}=V$ for $a\ll 0$, and $V^{a}=0$ for $a\gg 0$. If
$\dim_{K}V\geq 1$, the naive adelic successive maxima of $V$ are the real
numbers $\lambda_{1},\dots,\lambda_{\dim_{K}V}$, where

$\lambda_{i}:=\sup\\{a\in\mathbb{R}:\dim_{K}V^{a}\geq i\\}.$

######  Example 2.4.

Suppose $\mathscr{V}$ is a finite, free $\mathbb{Z}$-module of rank $d\geq 1$,
equipped with a norm $\|\cdot\|_{\infty}$ on the $\mathbb{C}$-vector space
$\mathscr{V}\otimes_{\mathbb{Z}}\mathbb{C}$. Then
$V:=\mathscr{V}\otimes_{\mathbb{Z}}\mathbb{Q}$ has a natural structure of an
adelic vector bundle over $\mathbb{Q}$ as follows. For $v=\infty$, we use the
given norm $\|\cdot\|_{\infty}$ on $V\otimes_{\mathbb{Q}}\mathbb{C}$, and for
$v=p$ finite, define
$\|\cdot\|_{p}:V\otimes_{\mathbb{Q}}\mathbb{C}_{p}\to\mathbb{R}$ by

(2.1)
$\|s\|_{p}:=\inf_{\alpha\in\mathbb{C}_{p}}\\{|\alpha|_{p}:s\in\alpha\left(\mathscr{V}\otimes_{\mathbb{Z}}R_{p}\right)\\},$

where $R_{p}$ is the closed unit ball of $\mathbb{C}_{p}$. If
$s_{1},\dots,s_{d}$ denotes any $\mathbb{Z}$-basis of $\mathscr{V}$, then if
we express $s\in V\otimes_{\mathbb{Q}}\mathbb{C}_{p}$ as
$s=\sum_{i}\alpha_{i}s_{i}$ with $\alpha_{i}\in\mathbb{C}_{p}$, we have
$\|s\|_{p}=\max_{i}(|\alpha_{i}|_{p})$.

If $\lambda_{1},\dots,\lambda_{d}$ denote the successive maxima for $V$ with
respect to the filtration induced by the naive adelic height as above, then

$\lambda_{i}=-\log(\mu_{i}),$

where $\mu_{1},\dots,\mu_{d}$ are the successive minima for the lattice
$\mathscr{V}\subseteq\mathscr{V}\otimes_{\mathbb{Z}}\mathbb{R}$, where
$\mathscr{V}\otimes_{\mathbb{Z}}\mathbb{R}$ is equipped with the norm induced
from that on $\mathscr{V}\otimes_{\mathbb{Z}}\mathbb{C}$.

###  2.6. Adelic $\mathbb{Q}$-vector bundle structure on rational cusp forms

Let $\Gamma\subseteq\Gamma(1)$ be a finite index subgroup and let
$X(\Gamma),\mathscr{X}(\Gamma),\mathscr{L}$, and $D$ be as in §2.3. Let

$\displaystyle\mathscr{M}_{k}$
$\displaystyle:=H^{0}(\mathscr{X}(\Gamma),\mathscr{L}^{\otimes k}),$
$\displaystyle M_{k}$
$\displaystyle:=\mathscr{M}_{k}\otimes_{\mathbb{Z}}\mathbb{Q},$ $\displaystyle
S_{k}$ $\displaystyle:=H^{0}(X(\Gamma),\mathscr{L}_{\mathbb{Q}}^{\otimes
k}(-D)),$ $\displaystyle\mathscr{S}_{k}$ $\displaystyle:=\mathscr{M}_{k}\cap
S_{k},$

where the last intersection takes place in $M_{k}$. These correspond to the
spaces of integral modular forms, rational modular forms, rational cusp forms,
and integral cusp forms of level $\Gamma$ and weight $12k$, respectively.

Note that $\mathscr{S}_{k}$ is a finite free $\mathbb{Z}$-module with
$S_{k}=\mathscr{S}_{k}\otimes_{\mathbb{Z}}\mathbb{Q}$. By example 2.4, $S_{k}$
has a natural structure of an adelic vector bundle over $\mathbb{Q}$, with
respect to the norms $\|\cdot\|_{k,p}$, as in equation 2.1 for a finite place
$p$, and $\|\cdot\|_{k,\infty}$, the normalized $L^{2}$ norm from §2.4. We
denote the naive adelic height function on $S_{k}$ by $\lambda_{k}$, the
induced filtration by $(S_{k}^{a})_{a\in\mathbb{R}}$, and the associated
successive maxima by $\lambda_{k,i}$ (for $i=1,\dots,\dim_{\mathbb{Q}}S_{k}$)
(as in §2.5). These maxima are equal to the ones in theorem 1.1 by example
2.4.

##  3\. Proof of theorem 1.1

A key ingredient in the proof of theorem 1.1 is theorem 3.4.3 in [Che10]
concerning quasi-filtered graded algebras. We apply this result to the graded
algebras $B_{L}$ in proposition 3.6. To do so, we need to get a uniform upper
bound on the normalized height $\lambda_{k}(f)/k$ of any non-zero $f\in
S_{k}$, and show that the algebra $B_{L}$ is quasi-filtered with respect to an
appropriate function (see definition 3.2.1 of [Che10]).

The uniform upper bound is proved in §3.1. The key ingredients for this are
formulas for the intersection numbers of logarithmically singular line bundles
as in [Küh25] and heights of cycles with respect to a smoothly metrized line
bundle as in [BGS94] (see proposition 3.3), along with a version of “Gromov’s
lemma” proved in [FJK13] for cusp forms with respect to the Petersson norm and
later improved (and generalized) in [AMM16] (see proposition 3.2). Proposition
3.2 is also key to lemma 3.5, which in turn is used to show that the $B_{L}$
are quasi-filtered. Using all of this, we show that the $\nu_{k}$ converge
vaguely to a sub-probability Borel measure $\nu$ on $\mathbb{R}$.

Finally, in §3.3, we derive explicit lower bounds on the successive maxima
$\lambda_{k,i}$ for $k$ large by constructing a basis for $S_{k}$ of a
particular shape, and doing explicit calculations with it. These lower bounds
then imply that $\nu$ is a probability measure (and hence the vague
convergence is also weak convergence).

We follow the general structure of §3.6 and §3.7 of [CGS18] in §3.2 and §3.3.

Throughout this section, we keep the notation from §2.6.

###  3.1. Upper bound on heights

######  Lemma 3.1.

Let $f\in S_{k}$ be a non-zero element. There exists a rational number $\beta$
such that $\beta f\in\mathscr{S}_{k}$ and $\beta f\not\in
B\cdot\mathscr{S}_{k}$ for all positive integers $B\geq 2$. Furthermore, for
any $f\in\mathscr{S}_{k}$ such that $f\not\in B\cdot\mathscr{S}_{k}$ for all
integers $B>1$, we have $\|f\|_{k,p}=1$ for all finite places $p$, and hence,
$\lambda_{k}(f)=-\log\|f\|_{k,\infty}$.

###### Proof.

Take any $\mathbb{Z}$-basis $f_{1},\dots,f_{d}$ of $\mathscr{S}_{k}$, and let
$f=\sum_{i}\alpha_{i}f_{i}$ with $\alpha_{i}\in\mathbb{Q}$. Then it is clear
that there is a rational number $\beta$ such that
$\beta\alpha_{i}\in\mathbb{Z}$ for all $i$ and
$\gcd(\beta\alpha_{1},\dots,\beta\alpha_{d})=1$. It is also clear that $\beta
f\in\mathscr{S}_{k}$, and $\beta f\not\in B\cdot\mathscr{S}_{k}$ for all
integers $B>1$.

Finally given any $f\in\mathscr{S}_{k}$ such that $f\not\in
B\cdot\mathscr{S}_{k}$ for all integers $B>1$, writing
$f=\sum_{i}\alpha_{i}f_{i}$, we have $\alpha_{i}\in\mathbb{Z}$ and
$\gcd(\alpha_{1},\dots,\alpha_{d})=1$. By example 2.4, we conclude that
$\|f\|_{k,p}=\max_{i}(|\alpha_{i}|_{p})=1$ for all finite places $p$. ∎

Let $\Gamma\subseteq\Gamma(1)$ be a finite index subgroup, and let
$S_{2k}(\Gamma)_{\mathbb{C}}$ denote the space of complex cusp forms for
$\Gamma$ of weight $2k$. Let $\langle\cdot,\cdot\rangle_{\text{Pet}}$ denote
the Petersson inner product on $S_{2k}(\Gamma)_{\mathbb{C}}$ as in §2.2. For
$f\in S_{2k}(\Gamma)$, denote by

$\|f\|_{\sup}:=\sup_{z\in\Gamma\backslash\mathfrak{h}}|f(z)|(4\pi\operatorname{im}(z))^{k}$

the sup-norm of $f$ with respect to the Petersson metric.

######  Proposition 3.2.

Let $\Gamma\subseteq\Gamma(1)$ be a finite index subgroup, and let
$S_{2k}(\Gamma)_{\mathbb{C}}$ denote the space of complex cusp forms for
$\Gamma$ of weight $2k$. There exist positive constants $c_{1}$ and $c_{2}$,
with $c_{2}$ independent of $\Gamma$, such that for any $0\neq f\in
S_{2k}(\Gamma)_{\mathbb{C}}$, we have

$c_{1}\|f\|_{\emph{Pet}}\leq\|f\|_{\sup}\leq c_{2}k^{3/4}\|f\|_{\emph{Pet}}.$

###### Proof.

Let $\mu_{\Gamma}$ be the measure on $X(\Gamma)_{\mathbb{C}}$ induced from the
hyperbolic volume form $\frac{dxdy}{y^{2}}$ on $\mathfrak{h}$ with coordinate
$z=x+iy$. For the first inequality, note that

$\|f\|_{\text{Pet}}^{2}=\frac{1}{d_{\Gamma}}\int_{X(\Gamma)_{\mathbb{C}}}|f(z)|^{2}(4\pi\operatorname{im}(z))^{2k}\
d\mu_{\Gamma}(z)\leq\|f\|_{\sup}^{2}\left(\frac{1}{d_{\Gamma}}\int_{X(\Gamma)_{\mathbb{C}}}d\mu_{\Gamma}\right),$

so we may let
$c_{1}=(\frac{1}{d_{\Gamma}}\int_{X(\Gamma)_{\mathbb{C}}}d\mu_{\Gamma})^{-1/2}$.

For the other direction, let $\\{f_{1},\dots,f_{d}\\}$ be an orthonormal basis
for $S_{2k}(\Gamma)_{\mathbb{C}}$ for the Petersson inner product. Note that
by our normalization of the Petersson inner product (i.e. the inclusion of the
$(4\pi)^{2k}$ factor), $\\{(4\pi)^{k}f_{1},\dots,(4\pi)^{k}f_{d}\\}$ is an
orthonormal basis for the classical Petersson inner product. Then, for
$z\in\mathfrak{h}$, define

$B_{k}^{\Gamma}(z):=\sum_{j=1}^{d}|(4\pi)^{k}f_{j}(z)|^{2}\operatorname{im}(z)^{2k}=\sum_{j=1}^{d}|f_{j}(z)|^{2}(4\pi\operatorname{im}(z))^{2k}$

as in [AMM16]. Note that for any $0\neq f\in S_{2k}(\Gamma)_{\mathbb{C}}$, if
$f=\sum_{j=1}^{d}\alpha_{j}f_{j}$, then
$\|f\|_{\text{Pet}}^{2}=\sum_{j=1}^{d}|\alpha_{j}|^{2}$, and for any
$z\in\mathfrak{h}$,

$\displaystyle|f(z)|^{2}(4\pi\operatorname{im}(z))^{2k}$
$\displaystyle=\left|\sum_{j=1}^{d}\alpha_{j}f_{j}(z)\right|^{2}(4\pi\operatorname{im}(z))^{2k}$
$\displaystyle\leq\left(\sum_{j=1}^{d}|\alpha_{j}|^{2}\right)\left(\sum_{j=1}^{d}|f_{j}(z)|^{2}\right)(4\pi\operatorname{im}(z))^{2k}.$

Hence,

$\frac{|f(z)|^{2}(4\pi\operatorname{im}(z))^{2k}}{\|f\|_{\text{Pet}}^{2}}\leq
B_{k}^{\Gamma}(z),$

and

$\frac{\|f\|_{\sup}^{2}}{\|f\|_{\text{Pet}}^{2}}=\sup_{z\in\Gamma\backslash\mathfrak{h}}\frac{|f(z)|^{2}(4\pi\operatorname{im}(z))^{2k}}{\|f\|_{\text{Pet}}^{2}}\leq\sup_{z\in\Gamma\backslash\mathfrak{h}}B_{k}^{\Gamma}(z).$

By theorem 1.7 in [AMM16],

$\sup_{z\in\Gamma\backslash\mathfrak{h}}B_{k}^{\Gamma}(z)=\left(\frac{k}{\pi}\right)^{3/2}+O(k).$

In particular, there exists a constant $c_{2}$ (independent of $k$ and
$\Gamma$) which gives us the desired bound. ∎

We follow very closely the proof of lemma 4.1.7 in [Che10] for the following
result.

######  Proposition 3.3.

There is a constant $C$ such that for any $0\neq f\in S_{k}$,
$\lambda_{k}(f)\leq Ck$.

###### Proof.

Since $\lambda_{k}(f)=\lambda_{k}(\alpha f)$ for any non-zero
$\alpha\in\mathbb{Q}^{\times}$, we may suppose, after appropriately scaling
$f$, that $f\in\mathscr{S}_{k}$ and $f\not\in B\cdot\mathscr{S}_{k}$ for any
positive integer $B>1$. Then by lemma 3.1,
$\lambda_{k}(f)=-\log\|f\|_{k,\infty}$.

Taking any projective (closed) embedding
$\Phi_{\mathbb{Z}}:\mathscr{X}(\Gamma)\hookrightarrow\mathbb{P}^{N}_{\mathbb{Z}}$,
we let
$\overline{L}:=\Phi_{\mathbb{Z}}^{*}\overline{\mathcal{O}_{\mathbb{P}^{N}_{\mathbb{Z}}}(1)}$,
where $\overline{\mathcal{O}_{\mathbb{P}^{N}_{\mathbb{Z}}}(1)}$ refers to
$\mathcal{O}_{\mathbb{P}^{N}_{\mathbb{Z}}}(1)$ endowed with the Fubini-Study
metric. Then $\overline{L}$ is arithmetically ample with
$c_{1}(\overline{L})>0$. Now take any global section $\ell\in
H^{0}(\mathscr{X}(\Gamma),L)$ such that $\text{div}_{L}(\ell)$ and
$\text{div}_{\mathscr{L}^{\otimes k}}(f)$ don’t share any common horizontal
divisors (i.e. their divisors on the generic fiber $X(\Gamma)$ have disjoint
support). Then the generalized intersection number
$\overline{L}\cdot\overline{\mathscr{L}}^{\otimes k}$ in the sense of equation
(3.10) in [Küh25] is given by

$\overline{L}\cdot\overline{\mathscr{L}}^{\otimes k}=(\ell\cdot
f)_{\text{fin}}+\langle\ell\cdot f\rangle_{\infty},$

where $(\ell\cdot
f)_{\text{fin}}=(\text{div}_{L}(\ell)\cdot\text{div}_{\mathscr{L}^{\otimes
k}}(f))_{\text{fin}}$ is equal to

$(\text{div}_{L}(\ell)\cdot\text{div}_{\mathscr{L}^{\otimes
k}}(f))_{\text{fin}}:=\sum_{i,j=0}^{1}(-1)^{i+j}\log\\#H^{i}\left(\mathscr{X}(\Gamma),\text{Tor}_{j}^{\mathcal{O}_{\mathscr{X}(\Gamma)}}(\mathcal{O}_{\text{div}_{L}(\ell)},\mathcal{O}_{\text{div}_{\mathscr{L}^{\otimes
k}}(f)})\right)$

($\mathcal{O}_{D}$ denotes the structure sheaf for an effective divisor
$D\subseteq\mathscr{X}(\Gamma)$), and $\langle\ell\cdot f\rangle_{\infty}$ is
given in our case by

$\displaystyle\langle\ell\cdot
f\rangle_{\infty}=-\sum_{P\in\mathscr{X}(\Gamma)(\mathbb{C})}n_{P}\log|\ell(P)|-\int_{\mathscr{X}(\Gamma)(\mathbb{C})}\log|f|_{\text{Pet}}\cdot
c_{1}(\overline{L}),$

if $\text{div}_{\mathscr{L}^{\otimes
k}_{\mathbb{C}}}(f)=\sum_{P\in\mathscr{X}(\Gamma)(\mathbb{C})}n_{P}P$ (lemma
3.9 in [Küh25]). Note that this expression makes sense since
$\text{div}_{\mathscr{L}^{\otimes k}_{\mathbb{C}}}(f)$ and
$\text{div}_{L_{\mathbb{C}}}(\ell)$ have disjoint support. Then, using
bilinearity of the intersection pairing, we have

$(\ell\cdot
f)_{\text{fin}}-\sum_{P\in\mathscr{X}(\Gamma)(\mathbb{C})}n_{P}\log|\ell(P)|=k(\overline{L}\cdot\overline{\mathscr{L}})+\int_{\mathscr{X}(\Gamma)(\mathbb{C})}\log|f|_{\text{Pet}}\cdot
c_{1}(\overline{L}),$

where the expression on the left is the height
$h_{\overline{L}}(\text{div}_{\mathscr{L}^{\otimes k}}(f))$ of the cycle
$\text{div}_{\mathscr{L}^{\otimes k}}(f)$ with respect to $\overline{L}$, as
defined in 3.1.1 in [BGS94] (see also §2.3.4 of [BGS94], in particular,
equation 2.3.17). Since $\overline{L}$ is arithmetically ample and
$\text{div}_{\mathscr{L}^{\otimes k}}(f)$ is effective,
$h_{\overline{L}}(\text{div}_{\mathscr{L}^{\otimes k}}(f))\geq 0$ (proposition
3.2.4 in [BGS94]). Hence

$0\leq
k(\overline{L}\cdot\overline{\mathscr{L}})+\int_{\mathscr{X}(\Gamma)(\mathbb{C})}\log|f|_{\text{Pet}}\cdot
c_{1}(\overline{L}),$

and since $c_{1}(\overline{L})>0$, we get

$\log\sup_{z\in\mathscr{X}(\Gamma)(\mathbb{C})}|f|_{\text{Pet}}(z)\geq
C^{\prime}k$

for
$C^{\prime}:=-(\overline{L}\cdot\overline{\mathscr{L}})/\int_{\mathscr{X}(\Gamma)(\mathbb{C})}c_{1}(\overline{L})$.

Denote the image of $f$ under the map
$H^{0}(\mathscr{X}(\Gamma),\mathscr{L}^{\otimes k}(-D))\to
H^{0}(\mathscr{X}(\Gamma)_{\mathbb{C}},\mathscr{L}^{\otimes
k}_{\mathbb{C}}(-D_{\mathbb{C}}))$ by $(f^{\sigma})_{\sigma}$. Suppose
$|f|_{\text{Pet}}(z)$ achieves its supremum on the component
$X(\Gamma_{\sigma^{\prime}})_{\mathbb{C}}$, so that

$\sup_{z\in\mathscr{X}(\Gamma)(\mathbb{C})}|f|_{\text{Pet}}(z)=\sup_{z\in
X(\Gamma_{\sigma^{\prime}})_{\mathbb{C}}}|(f^{\sigma^{\prime}}\Delta^{k})(z)|(4\pi\operatorname{im}(z))^{6k}=\|f^{\sigma^{\prime}}\Delta^{k}\|_{\sup}.$

Then

$\|f\|_{k,\infty}^{2}=\frac{1}{[E:\mathbb{Q}]}\sum_{\sigma}\|f^{\sigma}\Delta^{k}\|_{\text{Pet}}^{2}\geq\frac{1}{[E:\mathbb{Q}]}\|f^{\sigma^{\prime}}\Delta^{k}\|_{\text{Pet}}^{2}\geq\frac{1}{[E:\mathbb{Q}]}c_{2}^{-2}k^{-3/2}\|f^{\sigma^{\prime}}\Delta^{k}\|_{\sup}^{2},$

where the last inequality uses proposition 3.2. We conclude that

$\lambda_{k}(f)=-\log\|f\|_{k,\infty}\leq Ck$

for a constant $C$ independent of $k$. ∎

###  3.2. Cusp forms vanishing to increasing orders at the cusps

For an integer $L\geq 1$, define

$\mathcal{B}_{L,k}:=\mathscr{L}_{\mathbb{Q}}^{\otimes k}\\!\left(-\lceil
k/L\rceil D\right)\subseteq\mathscr{L}_{\mathbb{Q}}^{\otimes k}(-D).$

Then for each $\sigma:E\to\mathbb{C}$,

$(\mathcal{B}_{L,k})_{\mathbb{C}}|_{X(\Gamma_{\sigma})_{\mathbb{C}}}=\mathscr{L}_{\mathbb{C}}^{\otimes
k}\\!\left(-\lceil
k/L\rceil)D_{\mathbb{C}}\right)|_{X(\Gamma_{\sigma})_{\mathbb{C}}}$

is identified with the line bundle of weight $12k$ cusp forms for
$X(\Gamma_{\sigma})_{\mathbb{C}}$ that vanish to order at least $k/L$ at every
cusp.

Let $B_{L,k}:=H^{0}(X(\Gamma),\mathcal{B}_{L,k})$. We use the inclusion
$B_{L,k}\subseteq S_{k}$ to define a filtration on $B_{L,k}$. Namely, for
$a\in\mathbb{R}$, we set the $a$th filtered piece of $B_{L,k}$ to be

$B_{L,k}^{a}:=S_{k}^{a}\cap B_{L,k}.$

This is a decreasing $\mathbb{R}$-filtration on $B_{L,k}$. For
$i=1,\dots,\dim_{\mathbb{Q}}B_{L,k}$, let

$\lambda_{L,k,i}=\sup\\{a\in\mathbb{R}:\dim_{\mathbb{Q}}B_{L,k}^{a}\geq i\\}$

denote the $i$th successive maxima of $B_{L,k}$. Since these are the
successive maxima associated to the subspace filtration, the multi-set
$\\{\lambda_{L,k,i}\\}_{i=1}^{\dim_{\mathbb{Q}}B_{L,k}}$ is a sub multi-set of
$\\{\lambda_{k,i}\\}_{i=1}^{\dim_{\mathbb{Q}}S_{k}}$.

Furthermore, let

$\widetilde{\lambda}_{k}:S_{k}\to\mathbb{R}\cup\\{\infty\\},\quad\widetilde{\lambda}_{L,k}:B_{L,k}\to\mathbb{R}\cup\\{\infty\\}$

be defined by $\widetilde{\lambda}_{k}(f):=\sup\\{a\in\mathbb{R}:f\in
S_{k}^{a}\\}$, and $\widetilde{\lambda}_{L,k}(f):=\sup\\{a\in\mathbb{R}:f\in
B_{L,k}^{a}\\}$. These are called the index functions of $S_{k}$ and $B_{L,k}$
for their respective filtrations (as in §2 of [Che07]). For $f\in B_{L,k}$, we
have $\widetilde{\lambda}_{L,k}(f)=\widetilde{\lambda}_{k}(f)$.

######  Remark 3.4.

We could have opted to use the filtration on $B_{L,k}$ obtained from the
restriction of the height function $\lambda_{k}$ on $S_{k}$, so that for
$a\in\mathbb{R}$ the $a$th filtered piece of $B_{L,k}$ would be

$B_{L,k,a}:=\text{span}_{\mathbb{Q}}\\{s\in B_{L,k}:\lambda_{k}(s)\geq a\\}.$

This approach is taken in [CGS18], §3.6 (see, in particular, the proof of
lemma 3.6.2) and makes proving various estimates (see [CGS18], lemma 3.7.1)
rather tricky. By contrast, working with the subspace filtration simplifies
these matters significantly - compare lemma 3.7 below to its counterpart,
[CGS18], lemma 3.7.1.

As mentioned at the start of this section, the following lemma is used to show
the algebras $B_{L}$ in proposition 3.6 are quasi-filtered.

######  Lemma 3.5.

Let $\psi(k):=\frac{3}{4}\log(k)+\log(c_{2})-\log(c_{1})/2$ where $c_{1}$ and
$c_{2}$ are the constants from proposition 3.2 for $\Gamma\subseteq\Gamma(1)$.
For any collection of elements $f_{i}\in S_{k_{i}}$ ($i=1,\dots,n$, with
$n\geq 2$), we have

$\widetilde{\lambda}_{k_{1}+\dots+k_{n}}(f_{1}\cdots
f_{n})\geq\sum_{i=1}^{n}\left(\widetilde{\lambda}_{k_{i}}(f_{i})-\psi(k_{i})\right).$

###### Proof.

Note that $c_{1}$ was set to be
$(\int_{X(\Gamma)_{\mathbb{C}}}d\mu_{\Gamma})^{-1}$, which only depends on the
index of $\Gamma$ in $\Gamma(1)$. Since
$[\Gamma(1):\Gamma]=[\Gamma(1):\Gamma_{\sigma}]$ for all $\sigma$, we may use
the same $c_{1}$ for all the $\Gamma_{\sigma}$. Hence for any $f\in S_{k}$,
proposition 3.2 gives

$c_{1}^{2}\|f\|_{k,\infty}^{2}\leq\sum_{\sigma}\|f^{\sigma}\Delta^{k}\|_{\sup}^{2}\leq
c_{2}^{2}k^{3/2}\|f\|_{k,\infty}^{2}.$

Then for $f_{i}\in S_{k_{i}}$ ($i=1,\dots,n$), letting $K:=k_{1}+\dots,k_{n}$,
we get

$\displaystyle c_{1}^{2}\|f_{1}\dots,f_{n}\|_{K,\infty}^{2}$
$\displaystyle\leq\sum_{\sigma}\|(f_{1}\dots
f_{n})^{\sigma}\Delta^{K}\|_{\sup}^{2}$
$\displaystyle\leq\sum_{\sigma}\|f_{1}^{\sigma}\Delta^{k_{1}}\|_{\sup}^{2}\cdots\|f_{n}^{\sigma}\Delta^{k_{n}}\|_{\sup}^{2}$
$\displaystyle\leq\left(\sum_{\sigma}\|f_{1}^{\sigma}\Delta^{k_{1}}\|_{\sup}^{2}\right)\cdots\left(\sum_{\sigma}\|f_{n}^{\sigma}\Delta^{k_{n}}\|_{\sup}^{2}\right)$
$\displaystyle\leq c_{2}^{2n}k_{1}^{3/2}\cdots
k_{n}^{3/2}\|f_{1}\|_{k_{1},\infty}^{2}\dots\|f_{n}\|_{k_{n},\infty}^{2},$

and consequently,

$\|f_{1}\dots f_{n}\|_{K,\infty}\leq
e^{\psi(k_{1})+\dots+\psi(k_{n})}\|f_{1}\|_{k_{1},\infty}\cdots\|f_{n}\|_{k_{n},\infty}.$

(We use the fact that $c_{1}<1$ here.)

Pick any $\varepsilon>0$. By definition, $f_{i}\in
S_{k_{i}}^{\widetilde{\lambda}_{k_{i}}(f_{i})-\varepsilon/n}$, and hence
$f_{i}=\sum g_{i,j}$ for $g_{i,j}\in S_{k_{i}}$ with
$\lambda_{k_{i}}(g_{i,j})\geq\widetilde{\lambda}_{k_{i}}(f_{i})-\varepsilon/n$.
It is easy to see that for any finite place $p$,

$\|g_{1,j_{1}}\cdots
g_{n,j_{n}}\|_{K,p}\leq\|g_{1,j_{1}}\|_{k_{1},p}\cdots\|g_{n,j_{n}}\|_{k_{n},p}.$

This combined with the previous paragraph yields

$\displaystyle\lambda_{K}(g_{1,j_{1}}\cdots g_{n,j_{n}})$
$\displaystyle\geq\sum_{i=1}^{n}\left(\lambda_{k_{i}}(g_{i,j_{i}})-\psi(k_{i})\right)$
$\displaystyle\geq\sum_{i=1}^{n}\left(\widetilde{\lambda}_{k_{i}}(f_{i})-\psi(k_{i})\right)-\varepsilon.$

Since $\varepsilon$ is arbitrary, we conclude that

$\widetilde{\lambda}_{K}(f_{1}\cdots
f_{n})\geq\sum_{i=1}^{n}\left(\widetilde{\lambda}_{k_{i}}(f_{i})-\psi(k_{i})\right),$

as required. ∎

Given a Borel measure $\nu$, and an integrable function $f$ on $\mathbb{R}$,
let

$\nu(f):=\int_{\mathbb{R}}f\ d\mu.$

######  Proposition 3.6.

With the notation above, the sequence of probability measures

$\nu_{L,k}:=\frac{1}{\dim_{\mathbb{Q}}B_{L,k}}\sum_{i=1}^{\dim_{\mathbb{Q}}B_{L,k}}\delta_{\frac{1}{k}\lambda_{L,k,i}}$

converges weakly as $k\to\infty$ to a probability measure $\nu_{L,\infty}$
with compact support.

###### Proof.

First, we consider the case $k=qL$ for integers $q\geq 1$. Let

$B_{L}:=\bigoplus_{q\geq 0}B_{L,qL}.$

Since $\deg(\mathcal{B}_{L,L})>0$, $\mathcal{B}_{L,L}$ is ample, and hence
$B_{L}$ is a finitely generated $\mathbb{Q}$-algebra with $B_{L,qL}\neq 0$ for
$q$ large. We endow each $B_{L,qL}$ with the filtration
$(B_{L,qL}^{a})_{a\in\mathbb{R}}$ discussed above. Then by lemma 3.5, $B_{L}$
is $\psi_{L}$-quasi-filtered with $\psi_{L}(q):=\psi(qL)$, where $\psi$ is as
in lemma 3.5. By proposition 3.3,
$\widetilde{\lambda}_{L,qL}(f)=\widetilde{\lambda}_{qL}(f)\leq CqL$ for all
non-zero $f\in B_{L,qL}$. Hence by theorem 3.4.3 of [Che10] (and by rescaling
the measures by $1/L$), the collection of measures

$\nu_{L,qL}:=\frac{1}{\dim_{\mathbb{Q}}B_{L,qL}}\sum_{i=1}^{\dim_{\mathbb{Q}}B_{L,qL}}\delta_{\frac{1}{qL}\lambda_{L,qL,i}}$

converges weakly to a compactly supported probability measure $\nu_{L,\infty}$
on $\mathbb{R}$.

For general $k$, suppose that $k=qL+r$ with $0\leq r<L$. For $f\in
B_{L,k}\subseteq B_{L,(q+1)L}$, lemma 3.5 gives

$\displaystyle\widetilde{\lambda}_{L,(q+1)L}(f)$
$\displaystyle\geq\widetilde{\lambda}_{L,k}(f)+\widetilde{\lambda}_{L,L-r}(1)-\psi(k)-\psi(L-r)$
$\displaystyle\geq\widetilde{\lambda}_{L,k}(f)+c_{3}\log(k),$

for some constant $c_{3}$ independent of $k$. This implies that for all
$i=1,\dots,\dim_{\mathbb{Q}}B_{L,k}$,

$\lambda_{L,(q+1)L,i}\geq\lambda_{L,k,i}+c_{3}\log(k).$

Then

$\dim_{\mathbb{Q}}B_{L,(q+1)L}-\dim_{\mathbb{Q}}B_{L,k}=[E:\mathbb{Q}]d_{\Gamma}(L-r),$

is independent of $q$. Hence, for any bounded increasing continuous function
$f:\mathbb{R}\to\mathbb{R}$, we have

$\nu_{L,k}(f)\leq\nu_{L,(q+1)L}(f)+o(1),$

and hence,

$\limsup_{q\to\infty}\nu_{L,qL+r}(f)\leq\nu_{L,\infty}(f).$

Similarly, using the inclusion $B_{L,qL}\subseteq B_{L,k}$, we deduce that

$\liminf_{q\to\infty}\nu_{L,qL+r}(f)\geq\nu_{L,\infty}(f),$

and hence $\nu_{L,k}(f)\to\nu_{L,\infty}(f)$ for all bounded increasing
continuous functions $f$. Since all measures involved are probability
measures, this shows that $\nu_{L,k}\to\nu_{L,\infty}$ weakly. ∎

######  Lemma 3.7.

For every positive real number $\varepsilon$ and bounded Lipschitz function
$h:\mathbb{R}\to\mathbb{R}$ , there is a constant $L_{0}=L_{0}(\varepsilon,h)$
such that for all $L\geq L_{0}$, and for all $k\geq 1$, we have

$|\nu_{k}(h)-\nu_{L,k}(h)|\leq\epsilon.$

###### Proof.

Let
$|h|_{\text{Lip}}:=\sup_{x\in\mathbb{R}}|h(x)|+\sup_{x,y\in\mathbb{R},x\neq
y}\frac{|h(x)-h(y)|}{|x-y|}=M$. Recall that since $B_{L,k}$ is equipped with
the subspace filtration coming from $S_{k}$, the multi-set
$\\{\lambda_{L,k,i}\\}_{i=1}^{\dim_{\mathbb{Q}}B_{L,k}}$ is a sub multi-set of
$\\{\lambda_{k,i}\\}_{i=1}^{\dim_{\mathbb{Q}}S_{k}}$. Let
$d_{L,k}:=\dim_{\mathbb{Q}}B_{L,k}$ and $d_{k}:=\dim_{\mathbb{Q}}S_{k}$. Then

$\displaystyle|\nu_{k}(h)-\nu_{L,k}(h)|$
$\displaystyle=\left|\frac{1}{d_{k}}\sum_{i=1}^{d_{k}}h\left(\frac{1}{k}\lambda_{k,i}\right)-\frac{1}{d_{L,k}}\sum_{i=1}^{d_{L,k}}h\left(\frac{1}{k}\lambda_{L,k,i}\right)\right|$
$\displaystyle\leq\frac{1}{d_{k}}\left|\sum_{i=1}^{d_{k}}h\left(\frac{1}{k}\lambda_{i,k}\right)-\sum_{i=1}^{d_{L,k}}h\left(\frac{1}{k}\lambda_{i,L,k}\right)\right|+\left(\frac{1}{d_{L,k}}-\frac{1}{d_{k}}\right)\left|\sum_{i=1}^{d_{L,k}}h\left(\frac{1}{k}\lambda_{i,L,k}\right)\right|$
$\displaystyle\leq\frac{1}{d_{k}}(d_{k}-d_{L,k})M+\frac{d_{k}-d_{L,k}}{d_{k}d_{L,k}}d_{L,k}M$
$\displaystyle\leq\frac{c_{4}\left(\lceil k/L\rceil-1\right)}{d_{k}}$
$\displaystyle\leq\frac{c_{4}}{L},$

for some constant $c_{4}$. The conclusion follows at once. ∎

######  Corollary 3.8.

Suppose $N\geq 2$. Let $h:\mathbb{R}\to\mathbb{R}$ be a bounded Lipschitz
function. Then the sequences $(\nu_{k}(h))_{k}$ and $(\nu_{L,\infty}(h))_{L}$
are convergent and have the same limit.

###### Proof.

Let $\varepsilon>0$ be any positive real number, and let
$L_{0}=L_{0}(\varepsilon,h)$ be as in the previous lemma. For any $L\geq
L_{0}$, and any $k$, we have $|\nu_{k}(h)-\nu_{L,k}(h)|\leq\varepsilon$ from
the lemma above, and hence

$\nu_{L,k}(h)-\varepsilon\leq\nu_{k}(h)\leq\nu_{L,k}(h)+\varepsilon.$

From this we get

$\nu_{L,\infty}(h)-\varepsilon\leq\liminf_{k}\nu_{k}(h)\leq\limsup_{k}\nu_{k}(h)\leq\nu_{L,\infty}(h)+\varepsilon,$

and hence

$0\leq\limsup_{k}\nu_{k}(h)-\liminf_{k}\nu_{k}(h)\leq 2\varepsilon.$

Since $\varepsilon$ is arbitrary, we conclude that $\lim_{k}\nu_{k}(h)$
exists. Moreover, since for all $L\geq L_{0}$, we have

$|\nu_{L,\infty}(h)-\lim_{k}\nu_{k}(h)|\leq\varepsilon,$

we conclude that $\lim_{L}\nu_{L,\infty}(h)=\lim_{k}\nu_{k}(h)$, as required.
∎

Now by the Riesz representation theorem for measures, there exists a sub-
probability Borel measure $\nu$ on $\mathbb{R}$ representing the positive
linear functional $\lim_{k}\nu_{k}=\lim_{L}\nu_{L,\infty}$ on
$C_{c}(\mathbb{R})$. Namely, for all $h\in C_{c}(\mathbb{R})$, we have

(3.1) $\nu(h)=\lim_{k}\nu_{k}(h)=\lim_{L}\nu_{L,\infty}(h).$

Hence, $\nu_{k}$ converges vaguely to $\nu$.

###  3.3. $\nu$ is a probability measure

We obtain lower bound estimates on the successive maxima $\lambda_{k,i}$ of
$S_{k}$ in proposition 3.9, which are then used in proposition 3.10 to show
that $\nu$ is a probability measure.

Recall that $T_{1},\dots,T_{r}$ are the pre-images of
$\infty\in\mathbb{P}_{\mathbb{Q}}^{1}$ under the map
$\pi_{\Gamma}:X(\Gamma)\to\mathbb{P}_{\mathbb{Q}}^{1}$, and
$D=T_{1}+\dots+T_{r}$. Let $e_{i}$ denote the ramification index of $T_{i}$
over $\infty$. Then

$\mathscr{L}_{\mathbb{Q}}^{\otimes
k}(-D)=\mathcal{O}_{X(\Gamma)}\left((ke_{1}-1)T_{1}+\dots+(ke_{r}-1)T_{r}\right).$

Let $g$ denote the genus of $X(\Gamma)_{\mathbb{C}}$. Then the genus of
$X(\Gamma)$ as a curve over $E$, and that of
$X(\Gamma_{\sigma})_{\mathbb{C}}$, for each $\sigma$, is also equal to $g$.

######  Proposition 3.9.

Fix an integer $k_{0}$ such that $k_{0}e_{p}-1>2g-2$ for all $p=1,\dots,r$,
and let $d_{0}:=\dim_{\mathbb{Q}}S_{k_{0}}$. For all $k>k_{0}$, if
$\\{\lambda_{k,i}\\}_{i=1}^{\dim_{\mathbb{Q}}S_{k}}$ denote the successive
maxima of $S_{k}$, then:

  * •

For $1\leq i\leq d_{0}$,

$\lambda_{k,i}\geq 6k\log\left(\frac{k-k_{0}}{k}\right)-C_{4}k.$

  * •

For $d_{0}+1\leq i$,

$\lambda_{i,k}\geq
6k\log\left(\frac{k-k_{0}-\left\lceil\frac{i-d_{0}}{d_{\Gamma}[E:\mathbb{Q}]}\right\rceil}{k}\right)-C_{4}k.$

For $i>d_{0}+(k-k_{0}-1)d_{\Gamma}[E:\mathbb{Q}]$, we interpret the right hand
side above as $-\infty$.

###### Proof.

We construct an $E$-basis for $S_{k}$ when $k\geq k_{0}$. We start by
constructing a basis for $S_{k_{0}+1}/S_{k_{0}}$. Using proposition 1.40 in
[Shi71], one easily checks that for $k\geq 1$,
$\deg(\mathscr{L}_{\mathbb{Q}}^{\otimes k}(-D))>2g-2$, and hence by Riemann-
Roch,

$\dim_{E}S_{k}=\dim_{E}H^{0}(X(\Gamma),\mathscr{L}^{\otimes
k}_{\mathbb{Q}}(-D))=\deg(\mathscr{L}^{\otimes k}_{\mathbb{Q}}(-D))-g+1.$

for $k\geq 1$. Hence, for any $m\geq 0$,

(3.2)
$\dim_{E}S_{k_{0}+m+1}-\dim_{E}S_{k_{0}+m}=\deg\mathscr{L}_{\mathbb{Q}}=d_{\Gamma}.$

For $p=1,\dots,r$, and $q=1,\dots,e_{p}$, since

$\dim_{E}H^{0}(X(\Gamma),\mathcal{O}((k_{0}e_{p}-1+q)T_{p}))-\dim_{E}H^{0}(X(\Gamma),\mathcal{O}((k_{0}e_{p}-1+(q-1))T_{p}))=\deg
T_{p},$

there exist rational functions $b_{q,p,i}$ for $i=1,\dots,\deg T_{p}$ in
$H^{0}(X(\Gamma),\mathcal{O}((k_{0}e_{p}-1+q)T_{p}))$ that are $E$-linearly
independent, and that are regular everywhere on $X(\Gamma)$ except at $T_{p}$,
where they have a pole of order $k_{0}e_{p}-1+q$.

Varying $p,q$, and $i$, there are $d_{\Gamma}$ such functions
$\\{b_{q,p,i}\\}_{q,p,i}$, and $b_{q,p,i}\in S_{k_{0}+1}\setminus S_{k_{0}}$.
It is clear from construction that $\\{b_{q,p,i}\\}_{q,p,i}\subseteq
S_{k_{0}+1}$ are $E$-linearly independent, which in light of equation 3.2
implies that their images constitute an $E$-basis of $S_{k_{0}+1}/S_{k_{0}}$.

For any $m\geq 0$, note that

$\text{ord}_{T_{p}}(j^{m}b_{q,p,i})=-me_{p}-(k_{0}e_{p}-1+q)=-[(k_{0}+m)e_{p}-1+q].$

Since $\\{j^{m}b_{q,p,i}\\}_{q,p,i}\subseteq S_{k_{0}+m+1}\setminus
S_{k_{0}+m}$ are $E$-linearly independent, equation 3.2 again implies that
their images make up an $E$-basis of $S_{k_{0}+m+1}/S_{k_{0}+m}$. Now let
$d_{0}^{\prime}=\dim_{E}S_{k_{0}}$ and fix an $E$-basis
$\\{c_{1},\dots,c_{d_{0}^{\prime}}\\}$ of $S_{k_{0}}$. For $k>k_{0}$, the
above discussion yields the $E$-basis

$\\{c_{1},\dots,c_{d_{0}^{\prime}}\\}\cup\bigcup_{t=0}^{k-k_{0}-1}\\{j^{t}b_{q,p,i}\\}_{q,p,i}$

of $S_{k}$. By scaling the $c_{v}$’s and the $b_{q,p,i}$’s by elements of
$\mathbb{Q}^{\times}$, we may assume that they are integral (i.e., they lie in
$\mathscr{S}_{k_{0}}$ and $\mathscr{S}_{k_{0}+1}$, respectively). Then since
$j\in\mathscr{M}_{1}$ is integral, $j^{t}b_{q,p,i}\in\mathscr{S}_{k}$ are
integral as well. Fixing a $\mathbb{Z}$-basis
$\\{\alpha_{1},\dots,\alpha_{[E:\mathbb{Q}]}\\}$ of the ring of integers
$\mathcal{O}_{E}$, we get the $\mathbb{Q}$-basis of $S_{k}$:

$\bigcup_{u=1}^{[E:\mathbb{Q}]}\left(\\{\alpha_{u}c_{1},\dots,\alpha_{u}c_{d_{0}^{\prime}}\\}\cup\bigcup_{t=0}^{k-k_{0}-1}\\{\alpha_{u}j^{t}b_{q,p,i}\\}_{q,p,i}\right).$

Since each $\mathscr{S}_{k}$ is an $\mathcal{O}_{E}$-module, the
$\alpha_{u}c_{v}$ and $\alpha_{u}j^{t}b_{q,p,i}$ are integral as well.

Let $\alpha_{u}^{\sigma}c_{v}^{\sigma}$ and
$\alpha_{u}^{\sigma}j^{t}b_{q,p,i}^{\sigma}$ denote the images of
$\alpha_{u}c_{v}$ and $\alpha_{v}j^{t}b_{q,p,i}$, respectively, in
$S_{k}\otimes_{E,\sigma}\mathbb{C}$. Each
$\alpha_{u}^{\sigma}c_{v}^{\sigma}\Delta^{k_{0}}\in
S_{12k_{0}}(\Gamma_{\sigma})_{\mathbb{C}}$, and
$\alpha_{u}^{\sigma}b_{q,p,i}^{\sigma}\Delta^{k_{0}+1}\in
S_{12(k_{0}+1)}(\Gamma_{\sigma})_{\mathbb{C}}$. There is a positive constant
$C_{1}$ such that the estimates

$\displaystyle\sup_{z_{\sigma}\in\Gamma_{\sigma}\backslash\mathfrak{h}}|(\alpha_{u}^{\sigma}c_{v}^{\sigma}\Delta^{k_{0}})(z)|$
$\displaystyle\leq C_{1}$
$\displaystyle\sup_{z_{\sigma}\in\Gamma_{\sigma}\backslash\mathfrak{h}}|(\alpha_{u}^{\sigma}b_{q,p,i}^{\sigma}\Delta^{k_{0}+1})(z)|$
$\displaystyle\leq C_{1}$ $\displaystyle|j(z)|$ $\displaystyle\leq
C_{1}e^{2\pi y}$ $\displaystyle|\Delta(z)|$ $\displaystyle\leq C_{1}e^{-2\pi
y}$

hold for all $\sigma,u,v$, and triples $(q,p,i)$. Then

$\displaystyle\|\alpha_{u}^{\sigma}c_{v}^{\sigma}\Delta^{k}\|_{\text{Pet}}^{2}$
$\displaystyle=\frac{1}{d_{\Gamma_{\sigma}}}\int_{\mathcal{F}_{\Gamma_{\sigma}}}|(\alpha_{u}^{\sigma}c_{v}^{\sigma}\Delta^{k})(z)|^{2}(4\pi
y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle\leq\frac{C_{1}^{2}}{d_{\Gamma_{\sigma}}}\int_{\mathcal{F}_{\Gamma_{\sigma}}}|\Delta^{k-k_{0}}(z)|^{2}(4\pi
y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle\leq\frac{C_{1}^{2(k-k_{0})+2}}{d_{\Gamma_{\sigma}}}\int_{\mathcal{F}_{\Gamma_{\sigma}}}e^{-4\pi
y(k-k_{0})}(4\pi y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle\leq\frac{C_{1}^{2(k-k_{0})+2}w}{d_{\Gamma_{\sigma}}}\int_{0}^{\infty}e^{-4\pi
y(k-k_{0})}(4\pi y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle\leq\left(\frac{k}{k-k_{0}}\right)^{12k}e^{C_{2}k},$

for some constant $C_{2}$, and where we pick a connected fundamental domain
$\mathcal{F}_{\Gamma_{\sigma}}$ for $\Gamma_{\sigma}$ contained in a vertical
strip of the form $\\{(x,y):\beta\leq x\leq\beta+w,0<y\\}$ for some
$\beta\in\mathbb{R}$, where $w$ is the width of the cusp $\infty$ for
$\Gamma_{\sigma}$. Similarly, for $t=0,\dots,k-k_{0}-2$,

$\displaystyle\|\alpha_{u}^{\sigma}b_{q,p,i}^{\sigma}j^{t}\Delta^{k}\|_{\text{Pet}}^{2}$
$\displaystyle=\frac{1}{d_{\Gamma_{\sigma}}}\int_{\mathcal{F}_{\Gamma_{\sigma}}}|(\alpha_{u}^{\sigma}b_{q,p,i}^{\sigma}j^{t}\Delta^{k})(z)|^{2}(4\pi
y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle=\frac{1}{d_{\Gamma_{\sigma}}}\int_{\mathcal{F}_{\Gamma_{\sigma}}}|(\alpha_{u}^{\sigma}b_{q,p,i}^{\sigma}\Delta^{k_{0}+1})(z)|^{2}|(j^{t}\Delta^{k-k_{0}-1})(z)|^{2}(4\pi
y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle\leq\frac{C_{1}^{2}}{d_{\Gamma_{\sigma}}}\int_{\mathcal{F}_{\Gamma_{\sigma}}}|(j^{t}\Delta^{k-k_{0}-1})(z)|^{2}(4\pi
y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle\leq\frac{C_{1}^{2t+2k-2k_{0}}}{d_{\Gamma_{\sigma}}}\int_{\mathcal{F}_{\Gamma_{\sigma}}}e^{-4\pi
y(k-k_{0}-t-1)}(4\pi y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle\leq\frac{C_{1}^{2t+2k-2k_{0}}w}{d_{\Gamma_{\sigma}}}\int_{0}^{\infty}e^{-4\pi
y(k-k_{0}-t-1)}(4\pi y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle\leq\left(\frac{k}{k-k_{0}-t-1}\right)^{12k}e^{C_{3}k}$

for some constant $C_{3}$. Then

$\displaystyle\|\alpha_{u}c_{v}\|_{k,\infty}^{2}$
$\displaystyle=\frac{1}{[E:\mathbb{Q}]}\sum_{\sigma}\|\alpha_{u}^{\sigma}c_{v}^{\sigma}\Delta^{k}\|_{\text{Pet}}^{2}\leq\left(\frac{k}{k-k_{0}}\right)^{12k}e^{C_{2}k},$
$\displaystyle\|\alpha_{u}b_{q,p,i}j^{t}\|_{k,\infty}^{2}$
$\displaystyle=\frac{1}{[E:\mathbb{Q}]}\sum_{\sigma}\|\alpha_{u}^{\sigma}b_{q,p,i}^{\sigma}j^{t}\Delta^{k}\|_{\text{Pet}}^{2}\leq\left(\frac{k}{k-k_{0}-t-1}\right)^{12k}e^{C_{3}k}.$

Since $\alpha_{u}c_{v}$ and $\alpha_{u}b_{q,p,i}j^{t}$ are all integral and
hence their $p$-norms are all at most $1$ for every finite place $p$, we have

$\displaystyle\lambda_{k}(\alpha_{u}c_{v})$
$\displaystyle\geq-\log\|\alpha_{u}c_{v}\|_{k,\infty}\geq
6k\log\left(\frac{k-k_{0}}{k}\right)-C_{4}k,$
$\displaystyle\lambda_{k}(\alpha_{u}b_{q,p,i}j^{t})$
$\displaystyle\geq-\log\|\alpha_{u}b_{q,p,i}j^{t}\|_{k,\infty}\geq
6k\log\left(\frac{k-k_{0}-t-1}{k}\right)-C_{4}k,$

for a sufficiently large constant $C_{4}$.

Now, let $d_{0}=\dim_{\mathbb{Q}}S_{k_{0}}=[E:\mathbb{Q}]d_{0}^{\prime}$. For
$1\leq i\leq d_{0}$, any $i$ elements of the set

$\bigcup_{u=1}^{[E:\mathbb{Q}]}\\{\alpha_{u}c_{1},\dots,\alpha_{u}c_{d_{0}^{\prime}}\\}$

are $\mathbb{Q}$-linearly independent. Hence, for $1\leq i\leq d_{0}$,

$\lambda_{k,i}\geq 6k\log\left(\frac{k-k_{0}}{k}\right)-C_{4}k.$

For $d_{0}+t_{0}d_{\Gamma}[E:\mathbb{Q}]+1\leq i\leq
d_{0}+(t_{0}+1)d_{\Gamma}[E:\mathbb{Q}]$, where $t_{0}=0,\dots,k-k_{0}-2$, we
can take the subset

$\bigcup_{u=1}^{[E:\mathbb{Q}]}\left(\\{\alpha_{u}c_{1},\dots,\alpha_{u}c_{d_{0}^{\prime}}\\}\cup\bigcup_{t=0}^{t_{0}-1}\\{\alpha_{u}j^{t}b_{q,p,i}\\}_{q,p,i}\right)\cup
S,$

where
$S\subseteq\bigcup_{u=1}^{[E:\mathbb{Q}]}\\{\alpha_{u}b_{q,p,i}j^{t_{0}}\\}$
is any subset of cardinality $i-d_{0}-t_{0}d_{\Gamma}[E:\mathbb{Q}]$. (If
$t_{0}=0$, then the set
$\bigcup_{t=0}^{t_{0}-1}\\{\alpha_{u}j^{t}b_{q,p,i}\\}_{q,p,i}$ in the above
union should be interpreted as the empty set.) This set of $i$
$\mathbb{Q}$-linearly independent elements shows that for $i$ in the above
range,

$\lambda_{k,i}\geq 6k\log\left(\frac{k-k_{0}-t_{0}-1}{k}\right)-C_{4}k.$

Noting that
$t_{0}+1=\left\lceil\frac{i-d_{0}}{d_{\Gamma}[E:\mathbb{Q}]}\right\rceil$, we
conclude that:

  * •

If $1\leq i\leq d_{0}$, then

$\lambda_{k,i}\geq 6k\log\left(\frac{k-k_{0}}{k}\right)-C_{4}k.$

  * •

If $d_{0}+1\leq i\leq d_{0}+(k-k_{0}-1)d_{\Gamma}[E:\mathbb{Q}]$, then

$\lambda_{k,i}\geq
6k\log\left(\frac{k-k_{0}-\left\lceil\frac{i-d_{0}}{d_{\Gamma}[E:\mathbb{Q}]}\right\rceil}{k}\right)-C_{4}k.$

  * •

If $i>d_{0}+(k-k_{0}-1)d_{\Gamma}[E:\mathbb{Q}]$, the above expression still
holds if we interpret the right hand side to be $-\infty$.

This gives us the lower bounds in the proposition. ∎

We now show that the sequence of measures $(\nu_{k})_{k}$ is uniformly tight:
namely, given any $\varepsilon>0$, there exists a compact set
$K\subseteq\mathbb{R}$ for which $\nu_{k}(\mathbb{R}\setminus
K)\leq\varepsilon$ for all $k$.

######  Proposition 3.10.

The sequence of measures $(\nu_{k})_{k}$ is uniformly tight, and hence $\nu$
is a probability measure and the vague convergence $\nu_{k}\to\nu$ is in fact
weak convergence. Furthermore, $\nu$ has support bounded above.

###### Proof.

By proposition 3.3, there is a constant $C$ (which we may take to be positive)
such that $\lambda_{k,i}/k\leq C$ for all $k$ and all $i$, meaning that the
supports of the measures $\nu_{k}$ are all contained in $(-\infty,C]$. Hence
for any positive real number $a$,
$\nu_{k}(\mathbb{R}\setminus[-a,C])=\nu_{k}((-\infty,-a))+\nu_{k}((C,\infty))=\nu_{k}((-\infty,-a))$.
Thus, to show uniform tightness, it suffices to show that for any
$\varepsilon>0$, there is a positive real number $a_{1}$ and a positive
integer $k_{1}$ such that for all reals $a\geq a_{1}$ and all integers $k\geq
k_{1}$, $\nu_{k}((-\infty,-a))<\varepsilon$. (For each $i<k_{1}$, there is
some compact set $K_{i}$ such that $\nu_{i}(\mathbb{R}\setminus
K_{i})<\varepsilon$. Letting $K^{\prime}=K_{1}\cup\cdots\cup
K_{k_{1}-1}\cup[-a,C]$, we note that $\nu_{k}(\mathbb{R}\setminus
K^{\prime})\leq\varepsilon$ for all $k$.)

We keep the conventions from proposition 3.9. Take any $k>k_{0}$. Given a
positive real number $a$,

$\nu_{k}((-\infty,-a))=\frac{\\#\\{i:\lambda_{k,i}/k<-a\\}}{\dim_{\mathbb{Q}}S_{k}}.$

We first restrict to counting only those $i$ with $d_{0}+1\leq i\leq
d_{0}+(k-k_{0}-1)d_{\Gamma}[E:\mathbb{Q}]$, noting that the remaining $i$’s
contribute at most $d_{0}+d_{\Gamma}[E:\mathbb{Q}]$ to the count (which is a
constant independent of $k$). Then for $i$ in the above range, proposition 3.9
gives

$6\log\left(\frac{k-k_{0}-\left\lceil\frac{i-d_{0}}{d_{\Gamma}[E:\mathbb{Q}]}\right\rceil}{k}\right)-C_{4}\leq\lambda_{k,i}/k<-a,$

which implies that

$i>d_{0}+(k-k_{0}-1-C_{5}ke^{-a/6})d_{\Gamma}[E:\mathbb{Q}]$

for $C_{5}=e^{C_{4}/6}$. Hence,

$\nu_{k}((-\infty,-a))=\frac{\\#\\{i:\lambda_{k,i}/k<-a\\}}{\dim_{\mathbb{Q}}S_{k}}\leq\frac{(C_{5}ke^{-a/6}+2)d_{\Gamma}[E:\mathbb{Q}]+d_{0}+d_{\Gamma}[E:\mathbb{Q}]}{\dim_{\mathbb{Q}}S_{k}}.$

Noting that $\dim_{\mathbb{Q}}S_{k}$ grows linearly with $k$, we may pick
$k_{1}$ and $a_{1}$ large enough so that for any $k\geq k_{1}$ and $a\geq
a_{1}$, $\nu_{k}((-\infty,-a))<\varepsilon$. This concludes the proof of
uniform tightness.

To see that $\nu$ is a probability measure, we note that by Prohorov’s
theorem, uniform tightness implies that $(\nu_{k})_{k}$ admits a weakly
convergent subsequence $(\nu_{k_{m}})_{m}$. If $\nu_{k_{m}}\to\omega$ weakly,
then $\omega$ is also the vague limit of the $\nu_{k_{m}}$. Then by uniqueness
of vague limits, we conclude that $\omega=\nu$. Since $\omega$ is a
probability measure, we conclude that $\nu$ is one as well. It is a standard
result that if the limit measure is also a probability measure, then vague
convergence is equivalent to weak convergence, and hence $\nu_{k}$ converges
weakly to $\nu$. Finally, it is clear that the support of $\nu$ is contained
in $(-\infty,C]$. ∎

##  4\. Comparison of measures - Proof of theorem 1.2

###  4.1. Notations

Given finite index subgroups $\Gamma^{\prime}\subseteq\Gamma$ of $\Gamma(1)$,
we say that $\Gamma^{\prime}\subseteq\Gamma$ is defined over $E$, where $E$ is
a number field, if there exist smooth projective geometrically connected
$E$-curves $X(\Gamma^{\prime})$ and $X(\Gamma)$ such that:

  * •

the base change of $X(\Gamma^{\prime})$ and $X(\Gamma)$ to $\mathbb{C}$ give
the modular curves $X(\Gamma^{\prime})_{\mathbb{C}}$ and
$X(\Gamma)_{\mathbb{C}}$, respectively, and

  * •

there exist $E$-morphisms $\pi:X(\Gamma^{\prime})\to X(\Gamma)$ and
$\pi_{\Gamma}:X(\Gamma)\to\mathbb{P}_{E}^{1}$ with base change to $\mathbb{C}$
equal to the natural maps $X(\Gamma^{\prime})_{\mathbb{C}}\to
X(\Gamma)_{\mathbb{C}}$ and
$X(\Gamma)_{\mathbb{C}}\to\mathbb{P}_{\mathbb{C}}^{1}$ induced by the
inclusion $\Gamma^{\prime}\subseteq\Gamma$, and by the $j$-function,
respectively.

In particular, there is also an $E$-morphism
$\pi_{\Gamma}\circ\pi:X(\Gamma^{\prime})\to\mathbb{P}_{E}^{1}$ with base
change to $\mathbb{C}$ equal to the natural map
$X(\Gamma^{\prime})_{\mathbb{C}}\to\mathbb{P}_{\mathbb{C}}^{1}$ given by the
$j$-function.

Given a finite index subgroup $\Gamma\subseteq\Gamma(1)$, we also say that
$\Gamma$ is defined over $E$ to mean that $\Gamma\subseteq\Gamma(1)$ is
defined over $E$. In this case, the above conditions are equivalent to the
existence of models $X(\Gamma)$ and $X(\Gamma)\to\mathbb{P}_{E}^{1}$ over $E$
of $X(\Gamma)_{\mathbb{C}}$ and the $j$-function map
$X(\Gamma)_{\mathbb{C}}\to\mathbb{P}_{\mathbb{C}}^{1}$, respectively. Note
that if $\Gamma^{\prime}\subseteq\Gamma$ is defined over $E$, then so are
$\Gamma^{\prime}$ and $\Gamma$.

For the rest of this section, suppose that $\Gamma^{\prime}\subseteq\Gamma$ is
defined over a number field $E$, and let $X(\Gamma^{\prime}),X(\Gamma)$, and
$\pi:X(\Gamma^{\prime})\to X(\Gamma)$ be as above. As in §2.3, let
$X(\Gamma)_{\mathbb{Z}}$ denote the normalization of
$\mathbb{P}_{\mathbb{Z}}^{1}$ in $X(\Gamma)$ under the natural map
$X(\Gamma)\to\mathbb{P}_{E}^{1}\to\mathbb{P}_{\mathbb{Z}}^{1}$, and define
$X(\Gamma^{\prime})_{\mathbb{Z}}$ analogously. The natural map
$\pi:X(\Gamma^{\prime})\to X(\Gamma)$ induces a morphism
$X(\Gamma^{\prime})_{\mathbb{Z}}\to X(\Gamma)_{\mathbb{Z}}$ (over
$\mathbb{P}_{\mathbb{Z}}^{1}$). For any choice of a desingularization
$\mathscr{X}(\Gamma)\to X(\Gamma)_{\mathbb{Z}}$, there exists a
desingularization $\mathscr{X}(\Gamma^{\prime})\to
X(\Gamma^{\prime})_{\mathbb{Z}}$ along with a morphism
$\pi_{\mathbb{Z}}:\mathscr{X}(\Gamma^{\prime})\to\mathscr{X}(\Gamma)$ that
extends $X(\Gamma^{\prime})_{\mathbb{Z}}\to X(\Gamma)_{\mathbb{Z}}$.

Let $\mathscr{L}^{\prime}:=\pi_{\mathbb{Z}}^{*}\mathscr{L}$, where
$\mathscr{L}=\pi_{\Gamma,\mathbb{Z}}^{*}\mathcal{O}_{\mathbb{P}_{\mathbb{Z}}^{1}}(\overline{\infty})$
(see §2.3). Keeping the same conventions as in §2.6, we let

$\displaystyle\mathscr{M}_{k}$
$\displaystyle:=H^{0}(\mathscr{X}(\Gamma),\mathscr{L}^{\otimes
k}),\quad\qquad\
\mathscr{M}_{k}^{\prime}:=H^{0}(\mathscr{X}(\Gamma^{\prime}),\mathscr{L}^{\prime\otimes
k})$ $\displaystyle M_{k}$
$\displaystyle:=\mathscr{M}_{k}\otimes_{\mathbb{Z}}\mathbb{Q},\qquad\quad\quad\qquad\
\ M_{k}^{\prime}:=\mathscr{M}_{k}^{\prime}\otimes_{\mathbb{Z}}\mathbb{Q}$
$\displaystyle S_{k}$
$\displaystyle:=H^{0}(X(\Gamma),\mathscr{L}_{\mathbb{Q}}^{\otimes
k}(-D)),\quad\quad
S_{k}^{\prime}:=H^{0}(X(\Gamma^{\prime}),\mathscr{L^{\prime}}_{\mathbb{Q}}^{\otimes
k}(-D^{\prime}))$ $\displaystyle\mathscr{S}_{k}$
$\displaystyle:=\mathscr{M}_{k}\cap
S_{k},\qquad\qquad\qquad\quad\mathscr{S}_{k}^{\prime}:=\mathscr{M}_{k}^{\prime}\cap
S_{k}^{\prime},$

where $D$ and $D^{\prime}$ are the formal sums of the pre-images of
$\infty\in\mathbb{P}_{E}^{1}$ in $X(\Gamma)$ and $X(\Gamma^{\prime})$ under
the natural maps $X(\Gamma)\to\mathbb{P}_{E}^{1}$ and
$X(\Gamma^{\prime})\to\mathbb{P}_{E}^{1}$, respectively. We remark that these
definitions are independent of the choices of the regular models
$\mathscr{X}(\Gamma)$ and $\mathscr{X}(\Gamma^{\prime})$.

Given an extension $F/E$ of number fields, let
$X(\Gamma)_{F}:=X(\Gamma)\otimes_{E}F$ denote the base change, and let
$X(\Gamma)_{F,\mathbb{Z}}$ denote the normalization of
$\mathbb{P}_{\mathbb{Z}}^{1}$ under the natural map
$X(\Gamma)_{F}\to\mathbb{P}_{\mathbb{Z}}^{1}$. The map $X(\Gamma)_{F}\to
X(\Gamma)$ extends to a map between the normalizations
$X(\Gamma)_{F,\mathbb{Z}}\to X(\Gamma)_{\mathbb{Z}}$. Next, let
$\mathscr{X}(\Gamma)_{F}$ be a desingularization of $X(\Gamma)_{F,\mathbb{Z}}$
over $F$ that admits a morphism to $\mathscr{X}(\Gamma)$ extending the natural
map $X(\Gamma)_{\mathbb{Z},F}\to X(\Gamma)_{\mathbb{Z}}$. Let
$\mathscr{M}_{F,k}$ denote the global sections of the pullback of
$\mathscr{L}^{\otimes k}$ to $\mathscr{X}(\Gamma)_{F}$, and define
$M_{F,k},S_{F,k}$, and $\mathscr{S}_{F,k}$ in the obvious manner.

For any place $v$ of $\mathbb{Q}$, we let
$\|\cdot\|_{k,v},\|\cdot\|_{k,v}^{\prime}$, and $\|\cdot\|_{F,k,v}$ denote the
local norms on
$S_{k}\otimes_{\mathbb{Q}}\mathbb{C}_{v},S_{k}^{\prime}\otimes_{\mathbb{Q}}\mathbb{C}_{v}$,
and $S_{F,k}\otimes_{\mathbb{Q}}\mathbb{C}_{v}$, respectively. Finally, let
$\lambda_{k},\lambda_{k}^{\prime}$, and $\lambda_{F,k}$ be the naive adelic
height functions on $S_{k},S_{k}^{\prime}$, and $S_{F,k}$, respectively.

We adopt the convention that we drop the subscript $E$ for objects defined
over $E$ (as in §2.3), but include the subscript for objects over subfields or
field extensions of $E$.

###  4.2. Main results

######  Lemma 4.1.

Let $E$ be a number field, $X$ and $Y$ smooth projective integral curves over
$E$, and $\pi:Y\to X$ a non-constant map of curves over $E$. Suppose
$\mathscr{X}$ and $\mathscr{Y}$ are regular projective models of $X$ and $Y$,
respectively, over $\text{Spec}(\mathcal{O}_{E})$, and let
$\pi_{\mathbb{Z}}:\mathscr{Y}\to\mathscr{X}$ be a
$\text{Spec}(\mathcal{O}_{E})$-morphism extending $\pi$. For any line bundle
$\mathscr{L}$ on $\mathscr{X}$ with $L:=\mathscr{L}|_{X}$, we have the
equality

$H^{0}(\mathscr{X},\mathscr{L})=H^{0}(X,L)\cap
H^{0}(\mathscr{Y},\pi_{\mathbb{Z}}^{*}\mathscr{L}),$

where the intersection takes place in $H^{0}(Y,\pi^{*}L)$.

###### Proof.

First let $\mathscr{X}^{\prime}\xrightarrow{\eta}\mathscr{X}$ denote the
normalization of $\mathscr{X}$ in $Y$ with respect to the map $Y\to
X\to\mathscr{X}$. Then $\mathscr{X}^{\prime}$ is a normal arithmetic surface,
which admits a birational map $\mathscr{Y}\dashrightarrow\mathscr{X}^{\prime}$
(inducing the identity on the generic fiber $Y$). By [Liu02] chapter 9.2,
theorem 2.7, there is a projective birational morphism
$\mathscr{Y}^{\prime}\xrightarrow{\alpha}\mathscr{Y}$, with
$\mathscr{Y}^{\prime}$ regular, and a birational morphism
$\mathscr{Y}^{\prime}\xrightarrow{\beta}\mathscr{X}^{\prime}$ lifting
$\mathscr{Y}\dashrightarrow\mathscr{X}^{\prime}$. The natural pullback
morphisms induce the equality
$H^{0}(\mathscr{X}^{\prime},\eta^{*}\mathscr{L})=H^{0}(\mathscr{Y}^{\prime},\beta^{*}\eta^{*}\mathscr{L})=H^{0}(\mathscr{Y},\pi_{\mathbb{Z}}^{*}\mathscr{L})$
(since $\eta\circ\beta=\pi_{\mathbb{Z}}\circ\alpha$) in $H^{0}(Y,\pi^{*}L)$.
Hence, it suffices to show that $H^{0}(\mathscr{X},\mathscr{L})=H^{0}(X,L)\cap
H^{0}(\mathscr{X}^{\prime},\eta^{*}\mathscr{L})$.

We only need to show $\supseteq$. Take $s\in H^{0}(X,L)\cap
H^{0}(\mathscr{X}^{\prime},\eta^{*}\mathscr{L})$. Let
$\text{Spec}(A)\subseteq\mathscr{X}$ be an affine open subset where
$\mathscr{L}$ is trivial. Then $L$ is trivial on
$A\otimes_{\mathbb{Z}}\mathbb{Q}$, and $\eta^{*}\mathscr{L}$ is trivial on
$\eta^{-1}(\text{Spec}(A))=\text{Spec}(A^{\prime})$, where $A^{\prime}$ is the
integral closure of $A$ in the rational function field $\kappa(Y)$. The
section $s$ then corresponds to an element in
$A^{\prime}\cap(A\otimes_{\mathbb{Z}}\mathbb{Q})=A\subseteq\kappa(X)$, since
$A$ is integrally closed in its fraction field $\kappa(X)$. ∎

######  Lemma 4.2.

Let $\Gamma^{\prime}\subseteq\Gamma$ be finite index subgroups of $\Gamma(1)$
defined over a number field $E$, and let $F/E$ be a finite extension. Then
with the notation in §4.1, the local norms respect the inclusions

  1. (a)

$S_{k}\otimes_{\mathbb{Q}}\mathbb{C}_{v}\to
S_{k}^{\prime}\otimes_{\mathbb{Q}}\mathbb{C}_{v}$, and

  2. (b)

$S_{k}\otimes_{\mathbb{Q}}\mathbb{C}_{v}\to
S_{F,k}\otimes_{\mathbb{Q}}\mathbb{C}_{v}$

for all places $v$ of $\mathbb{Q}$.

###### Proof.

First let $v=p$ be a finite place. The inclusion of finite free
$\mathbb{Z}$-modules $\mathscr{M}_{k}\subseteq\mathscr{M}^{\prime}_{k}$
implies that there exists a $\mathbb{Z}$-basis
$\\{b_{1},\dots,b_{d^{\prime}}\\}$ of $\mathscr{M}^{\prime}_{k}$ for which
there exist integers $n_{1},\dots,n_{d}$ such that
$n_{1}b_{1},\dots,n_{d}b_{d}$ is a $\mathbb{Z}$-basis for $\mathscr{M}_{k}$.
Since $\mathscr{M}_{k}=M_{k}\cap\mathscr{M}^{\prime}_{k}$ by lemma 4.1, we
must have $b_{i}\in\mathscr{M}_{k}$, and hence $n_{i}=1$ for all
$i=1,\dots,d$. Given $s\in M_{k}\otimes_{\mathbb{Q}}\mathbb{C}_{p}$, if
$s=\sum_{i=1}^{d}\alpha_{i}b_{i}$ with $\alpha_{i}\in\mathbb{C}_{p}$, then
$\|s\|_{k,p}=\|s\|_{k,p}^{\prime}=\max\\{|\alpha_{i}|\\}_{i=1}^{d}$, by the
definition of the local norms at $p$. A similar argument also shows that
$\|s\|_{k,p}=\|s\|_{F,k,p}$.

Now suppose $v=\infty$. For any $\sigma:E\to\mathbb{C}$, the base change of
$\pi:X(\Gamma^{\prime})\to X(\Gamma)$ by $\sigma$ gives
$\pi_{\sigma}:X(\Gamma^{\prime}_{\sigma})_{\mathbb{C}}\to
X(\Gamma_{\sigma})_{\mathbb{C}}$ for finite index subgroups
$\Gamma^{\prime}_{\sigma}\subseteq\Gamma_{\sigma}\subseteq\Gamma(1)$. Now
given any $f=(f_{\sigma})_{\sigma}\in
S_{k}\otimes_{\mathbb{Q}}\mathbb{C}\subseteq
S_{k}^{\prime}\otimes_{\mathbb{Q}}\mathbb{C}$, the Petersson norm of the cusp
form $f_{\sigma}\Delta^{k}\in S_{12k}(\Gamma_{\sigma})_{\mathbb{C}}\subseteq
S_{12k}(\Gamma^{\prime}_{\sigma})_{\mathbb{C}}$ is independent of the groups
$\Gamma_{\sigma}$ and $\Gamma^{\prime}_{\sigma}$, we get
$\|f\|_{k,\infty}=\|f\|_{k,\infty}^{\prime}$.

Finally, we note that for $f$ as above,

$\displaystyle\|f\|_{F,k,\infty}^{2}$
$\displaystyle=\frac{1}{[F:\mathbb{Q}]}\sum_{\tau:F\to\mathbb{C}}\|f^{\tau}\Delta^{k}\|_{\text{Pet}}^{2}$
$\displaystyle=\frac{1}{[F:\mathbb{Q}]}\sum_{\sigma:E\to\mathbb{C}}\sum_{\begin{subarray}{c}\tau\\\
\tau|_{E}=\sigma\end{subarray}}\|f^{\tau}\Delta^{k}\|_{\text{Pet}}^{2}$
$\displaystyle=\frac{1}{[E:\mathbb{Q}]}\sum_{\sigma:E\to\mathbb{C}}\|f^{\sigma}\Delta^{k}\|_{\text{Pet}}^{2}=\|f\|_{k,\infty}^{2}.$

∎

Now suppose $\Gamma^{\prime}\trianglelefteq\Gamma$ is normal, and that
$\pi:X(\Gamma^{\prime})\to X(\Gamma)$ is Galois over $E$. Since
$X(\Gamma^{\prime})_{\mathbb{Z}}$ is also the normalization of
$X(\Gamma)_{\mathbb{Z}}$ in $X(\Gamma^{\prime})$, the universal property of
normalization implies that every element
$\tau\in\text{Aut}(X(\Gamma^{\prime})/X(\Gamma))$ lifts uniquely to an element
$\widetilde{\tau}\in\text{Aut}(X(\Gamma^{\prime})_{\mathbb{Z}}/X(\Gamma)_{\mathbb{Z}})$.
Let the elements of $\textrm{Aut}(X(\Gamma^{\prime})/X(\Gamma))$ be denoted
$\tau_{j}$, and let $(\cdot)|_{\tau_{j}}$ denote the pullback map on sections
induced by $\tau_{j}$.

Suppose also that $F/E$ is a Galois extension of number fields. Since
$X(\Gamma)$ is geometrically integral over $E$,
$\textrm{Gal}(X(\Gamma)_{F}/X(\Gamma))=\textrm{Gal}(F/E)^{\text{op}}$ (where
the “op” means that we take the opposite group). For
$\alpha_{j}\in\textrm{Gal}(F/E)$, let $\alpha_{j}^{*}$ denote the
corresponding element of $\textrm{Gal}(X(\Gamma)_{F}/X(\Gamma))$. Again by the
universal property of normalization, $\alpha_{j}^{*}$ extends uniquely to an
automorphism $\textrm{Aut}(X(\Gamma)_{\mathbb{Z},F}/X(\Gamma)_{\mathbb{Z}})$.
We denote by $(\cdot)^{\alpha_{j}}$ the pullback map on sections induced by
$\alpha_{j}^{*}$.

######  Lemma 4.3.

Let $\Gamma^{\prime}\trianglelefteq\Gamma$ be defined over $E$ and suppose
that $\pi:X(\Gamma^{\prime})\to X(\Gamma)$ is Galois over $E$. Let $F/E$ be a
Galois extension of number fields. Then with the conventions of §4.1, we have

  1. (a)

$S_{k}^{a}\subseteq S_{k}^{\prime a}\cap S_{k}\subseteq
S_{k}^{a-\log[\Gamma^{\prime}:\Gamma]}$, and

  2. (b)

$S_{k}^{a}\subseteq S_{F,k}^{a}\cap S_{k}\subseteq S_{k}^{a-\log[F:E]}$

for all $a\in\mathbb{R}$.

###### Proof.

The first containment for both (a) and (b) is the content of lemma 4.2. For
the second containment, take any $f\in S_{k}^{\prime a}\cap S_{k}$ (resp.
$s\in S_{F,k}^{a}\cap S_{k}$). Then $f=\sum_{i}g_{i}$ for $g_{i}\in
S_{k}^{\prime}$ with $\lambda_{k}^{\prime}(g_{i})\geq a$ (resp.
$s=\sum_{i}t_{i}$ for $t_{i}\in S_{F,k}$ with $\lambda_{F,k}(t_{i})\geq a$).
Let $h_{i}:=\sum_{j}g_{i}|_{\tau_{j}}$ (resp.
$u_{i}:=\sum_{j}t_{i}^{\alpha_{j}}$). Then we claim that $h_{i}\in S_{k}$ with
$\lambda_{k}(h_{i})\geq a-\log[\Gamma^{\prime}:\Gamma]$ (resp. $u_{i}\in
S_{k}$ with $\lambda_{k}(u_{i})\geq a-\log[F:E]$) for all $i$, from which the
conclusion follows.

First suppose $v=p$ is a finite place. Given any $\mathbb{Z}$-basis
$\\{b_{k}\\}$ of $\mathscr{M}^{\prime}_{k}$, if
$g_{i}=\sum_{k}\alpha_{k}b_{k}$ with $\alpha_{k}\in\mathbb{Q}$, then
$\|g_{i}\|_{k,p}^{\prime}=\max\\{|\alpha_{k}|\\}_{k}$. Now since $\tau_{j}$
extends uniquely to
$\textrm{Aut}(X(\Gamma^{\prime})_{\mathbb{Z}}/X(\Gamma)_{\mathbb{Z}})$, the
pullback map $(\cdot)|_{\tau_{j}}:M^{\prime}_{k}\to M^{\prime}_{k}$ restricts
to an automorphism of the integral sections $\mathscr{M}^{\prime}_{k}$ (see
the remarks in §2.3). Hence $\\{b_{k}|_{\tau_{j}}\\}$ is also a
$\mathbb{Z}$-basis of $\mathscr{M}^{\prime}_{k}$, from which we conclude that
$\|g_{i}|_{\tau_{j}}\|^{\prime}_{k,p}=\|g_{i}\|^{\prime}_{k,p}$ for all
$\tau_{j}$. Hence,

$\|h_{i}\|_{k,p}=\|h_{i}\|^{\prime}_{k,p}=\left\|\sum_{j}g_{i}|_{\tau_{j}}\right\|^{\prime}_{k,p}\leq\max_{j}\|g_{i}|_{\tau_{j}}\|^{\prime}_{k,p}=\|g_{i}\|^{\prime}_{k,p}.$

A similar argument shows that $\|u_{i}\|_{k,p}\leq\|t_{i}\|_{F,k,p}$.

Next, suppose $v=\infty$. First, we address (a). We have the base change
diagram by $\sigma:E\to\mathbb{C}$

${X(\Gamma^{\prime}_{\sigma})_{\mathbb{C}}}$${X(\Gamma^{\prime}_{\sigma})_{\mathbb{C}}}$${X(\Gamma^{\prime})}$${X(\Gamma^{\prime})}$$\scriptstyle{\gamma_{j}}$$\scriptstyle{\tau_{j}}$

where the top horizontal map
$\gamma_{j}\in\textrm{Aut}(X(\Gamma^{\prime}_{\sigma})_{\mathbb{C}}/X(\Gamma_{\sigma})_{\mathbb{C}})$
corresponds to the automorphism of the modular curve
$X(\Gamma^{\prime}_{\sigma})_{\mathbb{C}}$ induced by the coset
$\gamma_{j}\Gamma^{\prime}\in\Gamma/\Gamma^{\prime}$ for some
$\gamma_{j}\in\Gamma$. Hence for all $i$ and $j$, we have
$(g_{i}|_{\tau_{j}})^{\sigma}=g_{i}^{\sigma}|_{\gamma_{j}}$ (where
$|_{\gamma_{j}}$ denotes the usual slash operator for modular forms, which in
this case is simply pre-composition by $\gamma_{j}$). Then we have

$\displaystyle\|h_{i}\|_{k,\infty}^{2}=\|h_{i}\|_{k,\infty}^{\prime 2}$
$\displaystyle=\frac{1}{[E:\mathbb{Q}]}\sum_{\sigma:E\to\mathbb{C}}\|h_{i}^{\sigma}\Delta^{k}\|_{\text{Pet}}^{2}$
$\displaystyle\leq\frac{1}{[E:\mathbb{Q}]}\frac{1}{d_{\Gamma^{\prime}}}\frac{d_{\Gamma^{\prime}}}{d_{\Gamma}}\sum_{\sigma}\int_{\mathcal{F}_{\Gamma^{\prime}}}\sum_{j}\left|((g_{i}|_{\tau_{j}})^{\sigma}\Delta^{k})(z)\right|^{2}(4\pi
y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle=\frac{1}{[E:\mathbb{Q}]}\frac{1}{d_{\Gamma^{\prime}}}\frac{d_{\Gamma^{\prime}}}{d_{\Gamma}}\sum_{\sigma}\int_{\mathcal{F}_{\Gamma^{\prime}}}\sum_{j}\left|((g_{i}^{\sigma}|_{\gamma_{j}})\Delta^{k})(z)\right|^{2}(4\pi
y)^{12k}\ \frac{dxdy}{y^{2}}$
$\displaystyle=\frac{1}{[E:\mathbb{Q}]}\frac{d_{\Gamma^{\prime}}}{d_{\Gamma}}\sum_{j}\sum_{\sigma}\|g_{i}^{\sigma}\Delta^{k}\|_{\text{Pet}}^{2}$
$\displaystyle=\left(\frac{d_{\Gamma^{\prime}}}{d_{\Gamma}}\right)^{2}\|g_{i}\|_{k,\infty}^{\prime
2}.$

where we use Cauchy-Schwartz inequality in the second line, and the invariance
of the Petersson inner product with respect to the slash operator for modular
forms in the fourth line. We conclude that

$\lambda_{k}(h_{i})=-\sum_{v}\log\|h_{i}\|_{k,v}\geq-\sum_{v}\log\|g_{i}\|_{k,v}^{\prime}-\log\left(\frac{d_{\Gamma^{\prime}}}{d_{\Gamma}}\right)=\lambda^{\prime}_{k}(g_{i})-\log[\Gamma^{\prime}:\Gamma].$

Finally, we address (b). For each $\sigma:E\to\mathbb{C}$, fix a lift
$\widetilde{\sigma}:F\to\mathbb{C}$. Then all lifts of $\sigma$ to $F$ are
given by $\widetilde{\sigma}\circ\alpha_{j}$ as $\alpha_{j}$ varies over
$\textrm{Gal}(F/E)$.

$\displaystyle\|u_{i}\|_{k,\infty}^{2}=\frac{1}{[E:\mathbb{Q}]}\sum_{\sigma:E\to\mathbb{C}}\|u_{i}^{\sigma}\Delta^{k}\|_{\text{Pet}}^{2}$
$\displaystyle=\frac{1}{[E:\mathbb{Q}]}\sum_{\sigma:E\to\mathbb{C}}\|u_{i}^{\widetilde{\sigma}}\Delta^{k}\|_{\text{Pet}}^{2}$
$\displaystyle=\frac{1}{[E:\mathbb{Q}]}\sum_{\sigma:E\to\mathbb{C}}\left\|\sum_{j}(t_{i}^{\alpha_{j}})^{\widetilde{\sigma}}\Delta^{k}\right\|_{\text{Pet}}^{2}$
$\displaystyle\leq\frac{[F:E]}{[E:\mathbb{Q}]}\sum_{\sigma:E\to\mathbb{C}}\sum_{j}\left\|t_{i}^{\widetilde{\sigma}\circ\alpha_{j}}\Delta^{k}\right\|_{\text{Pet}}^{2}$
$\displaystyle=[F:E]^{2}\frac{1}{[F:\mathbb{Q}]}\sum_{\tau:F\to\mathbb{C}}\left\|t_{i}^{\tau}\Delta^{k}\right\|_{\text{Pet}}^{2}$
$\displaystyle=[F:E]^{2}\|t_{i}\|_{F,k,\infty}^{2}.$

Hence, we conclude that

$\lambda_{k}(u_{i})\geq\lambda_{F,k}(t_{i})-\log[F:E],$

as required. ∎

###  4.3. Proof of theorem 1.2

###### Proof of theorem 1.2.

Let $E$ and $E_{0}$ be the fields of constants of $X(\Gamma^{\prime})$ and
$X(\Gamma)$, respectively. Taking global sections of the structure sheaves for
the morphism $\pi_{\Gamma^{\prime},\Gamma}:X(\Gamma^{\prime})\to X(\Gamma)$
yields an inclusion $E_{0}\hookrightarrow E$, and hence an $E$-morphism
$X(\Gamma^{\prime})\to X(\Gamma)\otimes_{E_{0}}E$ such that its base change to
$\mathbb{C}$ is equal to the natural map $X(\Gamma^{\prime})_{\mathbb{C}}\to
X(\Gamma)_{\mathbb{C}}$ by assumption. Hence, $\Gamma^{\prime}\subseteq\Gamma$
is defined over $E$.

Let $\Gamma^{\prime\prime}$ be a finite index subgroup of $\Gamma^{\prime}$
with $\Gamma^{\prime\prime}\trianglelefteq\Gamma$. Let $F$ be a finite
extension of $E$ that is Galois over $E_{0}$, and suppose that
$\Gamma^{\prime\prime}$ is defined over $F$, with model
$X(\Gamma^{\prime\prime})_{F}$. By extending $F$ if necessary, we may also
assume that there exist $F$-morphisms $X(\Gamma^{\prime\prime})_{F}\to
X(\Gamma^{\prime})_{F}\to X(\Gamma)_{F}$ with base change to $\mathbb{C}$
equal to the natural maps $X(\Gamma^{\prime\prime})_{\mathbb{C}}\to
X(\Gamma^{\prime})_{\mathbb{C}}\to X(\Gamma)_{\mathbb{C}}$, and that the maps
$X(\Gamma^{\prime\prime})_{F}\to X(\Gamma^{\prime})_{F}$ and
$X(\Gamma^{\prime\prime})_{F}\to X(\Gamma)_{F}$ are Galois covers of curves.
Then repeated application of lemma 4.3 gives

$\displaystyle S_{E_{0},k}^{a}\subseteq S_{k}^{a}\cap S_{E_{0},k}\subseteq
S_{k}^{\prime a}\cap S_{E_{0},k}$ $\displaystyle\subseteq S_{F,k}^{\prime
a}\cap S_{E_{0},k}$ $\displaystyle\subseteq S_{F,k}^{\prime\prime a}\cap
S_{E_{0},k}\subseteq S_{F,k}^{a-\log[\Gamma^{\prime\prime}:\Gamma]}\cap
S_{E_{0},k}\subseteq
S_{E_{0},k}^{a-\log[\Gamma^{\prime\prime}:\Gamma]-\log[F:E_{0}]}.$

In particular, we have

(4.1) $S_{E_{0},k}^{a}\subseteq S_{k}^{\prime a}\cap S_{E_{0},k}\subseteq
S_{E_{0},k}^{a-\log[\Gamma^{\prime\prime}:\Gamma]-\log[F:E_{0}]}.$

Let $\lambda_{E_{0},k,i}^{\prime}$ denote the successive maxima of
$S_{E_{0},k}$ with respect to the subspace filtration $S_{k}^{\prime a}\cap
S_{E_{0},k}$, and let

$\nu_{E_{0},k}^{\prime}:=\frac{1}{\dim_{\mathbb{Q}}S_{E_{0},k}}\sum_{i=1}^{\dim_{\mathbb{Q}}S_{E_{0},k}}\delta_{\frac{1}{k}\lambda_{E_{0},k,i}^{\prime}}.$

Equation 4.1 gives

$\lambda_{E_{0},k,i}\leq\lambda_{E_{0},k,i}^{\prime}\leq\lambda_{E_{0},k,i}+\log[\Gamma^{\prime\prime}:\Gamma]+\log[F:E_{0}]$

for all $i=1,\dots,\dim_{\mathbb{Q}}S_{E_{0},k}$. Consequently,
$\nu_{E_{0},k}^{\prime}\to\nu_{E_{0}}$ weakly.

Consider now the short exact sequence

$0\to S_{E_{0},k}\to S_{k}^{\prime}\to W_{k}:=S_{k}^{\prime}/S_{E_{0},k}\to
0.$

If we equip $W_{k}$ with the quotient filtration as on page 16 of [Che10],
then by proposition 1.2.5 of [Che10], there is a Borel probability measure
$\omega_{k}$ on $\mathbb{R}$ such that

$(\dim_{\mathbb{Q}}S_{k}^{\prime})\cdot\nu_{k}^{\prime}=(\dim_{\mathbb{Q}}S_{E_{0},k})\cdot\nu_{E_{0},k}^{\prime}+(\dim_{\mathbb{Q}}W_{k})\cdot\omega_{k},$

and hence

$\omega_{k}=\frac{\dim_{\mathbb{Q}}S_{k}^{\prime}}{\dim_{\mathbb{Q}}W_{k}}\cdot\nu_{k}^{\prime}-\frac{\dim_{\mathbb{Q}}S_{E_{0},k}}{\dim_{\mathbb{Q}}W_{k}}\cdot\nu_{E_{0},k}^{\prime}.$

Taking the limit as $k\to\infty$, we get that $\omega_{k}$ converges weakly to
the Borel probability measure

$\omega:=\frac{[E:E_{0}][\Gamma:\Gamma^{\prime}]}{[E:E_{0}][\Gamma:\Gamma^{\prime}]-1}\cdot\nu^{\prime}-\frac{1}{[E:E_{0}][\Gamma:\Gamma^{\prime}]-1}\cdot\nu_{E_{0}}.$

Rearranging, we get

$\nu^{\prime}=\frac{1}{[E:E_{0}][\Gamma:\Gamma^{\prime}]}\cdot\nu_{E_{0}}+\left(1-\frac{1}{[E:E_{0}][\Gamma:\Gamma^{\prime}]}\right)\cdot\omega.$

Finally, since
$[E:E_{0}][\Gamma:\Gamma^{\prime}]=\text{deg}(\pi_{\Gamma^{\prime},\Gamma})$,
we get the desired conclusion. ∎

######  Corollary 4.4.

Assume the setup in theorem 1.1. Then the support of the limit measure $\nu$
is bounded above and unbounded below.

###### Proof.

That the support of $\nu$ is bounded above follows from lemma 3.3. The modular
curve $X(1)_{\mathbb{C}}$ for $\Gamma(1)$ has a model $X(1)_{\mathbb{Q}}$ over
$\mathbb{Q}$ that we identify with $\mathbb{P}_{\mathbb{Q}}^{1}$ via the
$j$-function. We have the morphism
$\pi_{\Gamma}:X(\Gamma)\to\mathbb{P}_{\mathbb{Q}}^{1}$ associated to the
inclusion $\Gamma\subseteq\Gamma(1)$, and hence we may apply theorem 1.2 to
get

$\nu=\frac{1}{\text{deg}(\pi_{\Gamma})}\cdot\nu_{\Gamma(1),\mathbb{Q}}+\left(1-\frac{1}{\text{deg}(\pi_{\Gamma})}\right)\cdot\omega,$

where $\nu_{\Gamma(1),\mathbb{Q}}$ is the limit measure associated to the
successive maxima of the spaces of $\mathbb{Q}$-rational cusp forms of level
$\Gamma(1)$ and weight $12k$ as in theorem 3.2.2 of [CGS18], and $\omega$ is a
Borel probability measure on $\mathbb{R}$. By part (ii) of the same theorem,
the support of $\nu_{\Gamma(1),\mathbb{Q}}$ is not bounded below.
Consequently, the support of $\nu$ is unbounded below as well. ∎

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