# The Geometry of Drinfeld Modular Forms

Jesse Franklin  Department of Mathematics and Statistics, University of
Vermont, Burlington VT 05405<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We give a geometric perspective on the algebra of Drinfeld modular forms for
congruence subgroups $\Gamma\leq\mathrm{GL}_{2}(\mathbb{F}_{q}[T])$ by means
of $\Gamma_{2}\leq\Gamma$ the subgroup of matrices which have square
determinants. First, we find an isomorphism between the section ring of a line
bundle on the stacky modular curve for $\Gamma_{2}$ and the algebra of
Drinfeld modular forms for $\Gamma_{2},$ which allows one to compute the
latter ring by geometric invariants using the techniques of Voight, Zureick-
Brown and O’Dorney. We show how to decompose the algebra of modular forms for
$\Gamma_{2}$ into a direct sum of two algebras of modular forms for $\Gamma$
and generalize this result to a larger class of congruence subgroups.

###### Contents

  1. 1 Introduction
    1. 1.1 History and Motivation
    2. 1.2 Main Results
  2. 2 Background
    1. 2.1 Notation and the Drinfeld Setting
    2. 2.2 Drinfeld modules
      1. 2.2.1 Analytic Approach
      2. 2.2.2 Algebraic Approach
    3. 2.3 Stacks and Section Rings
  3. 3 Drinfeld Modular Forms
  4. 4 Drinfeld Modular Curves
  5. 5 Rigid Stacky GAGA
  6. 6 Proof of the Main Theorems
    1. 6.1 Properties of $\Gamma_{2}$
    2. 6.2 Cusps and Elliptic Points
    3. 6.3 Modularity and Series Expansions at Cusps
    4. 6.4 Generalization
    5. 6.5 Summary
  7. 7 Application: Computing Algebras of Drinfeld Modular Forms

##  1\. Introduction

###  1.1. History and Motivation

The theory of modular forms in the classical number-field case has existed
since some time in the $1800$’s. One of the natural questions in the study of
any algebra is simply the dimensions of its components or graded pieces. It is
well-known that modular forms are sections of a particular line bundle on some
modular curve, and so the geometry of moduli stacks, with its tools such as
the Riemann-Roch theorem can be used to compute dimensions. Taking the idea
further, armed with such a line bundle and a space it lives, one can consider
embeddings and the associated section rings. This idea has developed into the
program of [VZB22] for computing the canonical ring of log stacky curves in
all genera and even gives minimal presentations for many such rings, that is,
explicit generators and relations.

In his $1986$ monograph [Gek86, Page $\mathrm{XIII}$] asks for a description
of algebras of Drinfeld modular forms in terms of generators and relations.
The main results of this note describe the geometry of those modular forms,
which allows one to employ techniques such as those in [VZB22] to find the
desired generators and relations by considering the geometry of the
corresponding Drinfeld modular curve. That is, we provide a means to address
Gekeler’s problem via geometric invariants.

Until now, the closest analogy in the Drinfeld setting for the isomorphism
$f\mapsto f(dz)^{\otimes k/2}$ between modular forms and sections of a line
bundle on the modular curve in the classical setting is based on e.g. [Gek86,
Page $52$] in the case of $\mathrm{GL}_{2}(\mathbb{F}_{q}[T]),$ and [BBP18,
Definition $(10.1)$]. In [BBP18, Lemma $(10.7)$] there is an isomorphism
between a ring of modular forms and a section ring of form $f(z)\mapsto
f(z)dz^{\otimes k/2}.$ Note that a manuscript version of the three preprints
by Basson-Breuer-Pink on Drinfeld modular forms of arbitrary rank is due to
appear in Memoirs of the AMS, so our citation here will soon not be the most
recent version of their theory.

There is another collection of results in this direction, which is similar to
our work in comparing modular forms for various congruence subgroups to each
other as in our second main result Theorem 6.2. By considering the Satake
compactification of Drinfeld modular varieties, Pink finds isomorphisms
between algebras of Drinfeld modular forms for open compact subgroups
$K\leq\mathrm{GL}_{r}(\hat{A})$ and normal subgroups $K^{\prime}\lhd K$ in
e.g. [Pin12, Proposition $5.5$] and he describes Drinfeld modular forms as
global sections of a line bundle in [Pin12, Section $5$]. We state such an
isomorphism and a similar geometry for Drinfeld modular forms in an elementary
way owing to our consideration exclusively of the case of moduli for rank $2$
Drinfeld modules.

There are some existing results which approach Gekeler’s problem, such as
Cornelissen’s papers [Cor97a] and [Cor97b] which deal with linear level
([Cor97b, Theorem $(3.3)$]), i.e. the algebra of modular forms for
$\Gamma(\alpha T+\beta),$ and include some results for quadratic level
([Cor97b, Proposition $(3.4)$]). Another example, [DK23, Theorem $(4.4)$],
computes the algebra of Drinfeld modular forms for $\Gamma_{0}(T).$ The best
known result for general level $N$ is [Arm08, Proposition $4.16$] which
demonstrates that for any level, $M^{2}_{2,1}(\Gamma_{0}(N)),$ the double cusp
forms of weight $2$ and type $1$ are (analytic) holomorphic differentials on a
(rigid analytic) Drinfeld modular curve
$\Gamma_{0}(N)\setminus(\Omega\cup\mathbb{P}^{1}(\mathbb{F}_{q}[T])).$ Several
ideas in [Bre16] are central to our argument, as well as being an exposition
on aspects of Gekeler’s problem in general. In particular, [Bre16] introduces
the subgroup $\Gamma_{2}$ of a given congruence subgroup
$\Gamma\leq\mathrm{GL}_{2}(A)$ and gives a moduli interpretation of the
corresponding Drinfeld modular curve. Even by the date of these most recent
papers, the generalization to the algebra of modular forms for $\Gamma_{0}(N)$
for any level $N,$ all $\Gamma_{1}(N),$ and high level (i.e. $\deg(N)\geq 2$)
$\Gamma(N)$ examples seem to be wide open.

###  1.2. Main Results

This article describes the geometry of Drinfeld modular forms: we associate to
each Drinfeld modular form a section of a particular line bundle on a
specified stacky modular curve. We also give a decomposition of the algebra of
modular forms, which allows one to compute all of the section rings in the
papers above by means of the geometric techniques of [VZB22]. This means we
have Gekeler’s elementary interpretation of the generating modular forms in
terms of Drinfeld modules viewed as points of the moduli space. We make no
restrictions on the level, and insist that our congruence subgroup in question
contains diagonal matrices mainly to simplify the proofs. We expect that this
theory works without this restriction, so we have a means to compute graded
rings of modular forms in great generality, and which makes the problem
reliant only on the geometry of the modular curve.

Throughout this article, let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a congruence
subgroup containing the diagonal matrices in
$\mathrm{GL}_{2}(\mathbb{F}_{q}[T]),$ where $q$ is a power of an odd prime and
$T$ an indeterminant. Our technique for computing the algebra of modular forms
for $\Gamma$ comes from the following observations. First, modular forms of
weight $k$ and type $l$ for $\Gamma_{2},$ the group of matrices in $\Gamma$
with square determinants, correspond to global sections of the sheaf of
differentials $\Omega^{1}(2\Delta)^{\otimes k/2}$ on the Drinfeld modular
curve associated with $\Gamma_{2},$ as in the classical identification of cusp
forms and sections of a line bundle. Second, if there are non-zero modular
forms for $\Gamma$ then the type $l$ and weight $k$ satisfy $k\equiv
2l\pmod{\\#\det\Gamma},$ and our convention will be that
$\det\Gamma=\mathbb{F}_{q}^{\times}.$ The first fact allows us to compute the
algebra of Drinfeld modular forms $M(\Gamma_{2})$ by computing the associated
section ring by techniques like those in [VZB22] and [O’D15] which use
geometric invariants of the moduli space. These invariants will depend on the
particular $\Gamma$ in question, but separate into just two cases.

Informally, our main results are:

  1. (1)

Drinfeld modular forms for $\Gamma_{2}$ are global sections of a particular
line bundle on the (stacky) Drinfeld modular curve with rigid analytic coarse
space $\Gamma_{2}\setminus(\Omega\cup\mathbb{P}^{1}(\mathbb{F}_{q}(T))),$

  2. (2)

The algebras of Drinfeld modular forms $M(\Gamma_{2})$ and $M(\Gamma)$ have
the same non-zero components $M_{k}$ when we consider their coarse grading by
only the weight $k.$ Within such a graded component, either
$M_{k,l}(\Gamma_{2})=M_{k,l}(\Gamma)$ or
$M_{k,l}(\Gamma_{2})=M_{k,l_{1}}(\Gamma)\oplus M_{k,l_{2}}(\Gamma),$ where $l$
denotes the fine grading of $M_{k}$ by type compatible with $\Gamma_{2},$ i.e.
$\displaystyle{l\equiv\frac{k}{2}\pmod{\frac{q-1}{2}}}$ and $l_{1}$ and
$l_{2}$ are two types for $\Gamma$-forms which are correspond to the two lifts
$\displaystyle{l_{1}=\frac{k}{2}}$ and
$\displaystyle{l_{2}=\frac{k}{2}+\frac{q-1}{2}}$ of $l$ to
$\mathbb{Z}/(q-1)\mathbb{Z}.$

  3. (3)

The second Theorem described above is actually a special case of a phenomenon
which occurs for more pairs of a congruence subgroup
$\Gamma\leq\mathrm{GL}_{2}(A)$ and some subgroup $\Gamma^{\prime}\leq\Gamma.$

The formal statement of these ideas is:

######  Theorem 1.1.

Let $q$ be a power of an odd prime. Let
$\Gamma\leq\mathrm{GL}_{2}(\mathbb{F}_{q}[T])$ be a congruence subgroup
containing the diagonal matrices in $\mathrm{GL}_{2}(\mathbb{F}_{q}[T])$ and
such that $\det(\gamma)\in(\mathbb{F}_{q}^{\times})^{2}$ for all
$\gamma\in\Gamma.$ Let $\Delta$ be the (Weil) divisor of cusps of the Drinfeld
modular curve $\mathscr{X}=\mathscr{X}_{\Gamma}$ with rigid analytic coarse
space
$X_{\Gamma}^{\text{an}}=\Gamma\setminus(\Omega\cup\mathbb{P}^{1}(\mathbb{F}_{q}(T))).$
There is an isomorphism of graded rings

$M(\Gamma)\cong R(\mathscr{X},\Omega^{1}_{\mathscr{X}}(2\Delta)),$

where $\Omega^{1}_{\mathscr{X}}$ is the sheaf of differentials on
$\mathscr{X}.$ The isomorphism is given by isomorphisms

$M_{k,l}(\Gamma)\to
H^{0}(\mathscr{X},\Omega^{1}_{\mathscr{X}}(2\Delta)^{\otimes k/2})$

of components given by $f\mapsto f(dz)^{\otimes k/2},$ where $k\equiv
2l\pmod{q-1}.$

######  Theorem 1.2.

Let $q$ be a power of an odd prime. Let
$\Gamma\leq\mathrm{GL}_{2}(\mathbb{F}_{q}[T])$ be a congruence subgroup
containing the diagonal matrices in $\mathrm{GL}_{2}(\mathbb{F}_{q}[T]).$ Let
$\Gamma_{2}=\\{\gamma\in\Gamma:\det(\gamma)\in(\mathbb{F}_{q}^{\times})^{2}\\}.$
Then we have $M(\Gamma)\cong M(\Gamma_{2}),$ with

$M_{k,l}(\Gamma_{2})=M_{k,l_{1}}(\Gamma)\oplus M_{k,l_{2}}(\Gamma)$

on each graded piece, where $l_{1},l_{2}$ are the two solutions to $k\equiv
2l\pmod{q-1}.$

######  Theorem 1.3.

Let $q$ be a power of an odd prime. Let
$\Gamma\leq\mathrm{GL}_{2}(\mathbb{F}_{q}[T])$ be a congruence subgroup. Let
$\Gamma_{1}=\\{\gamma\in\Gamma:\det(\gamma)=1\\}.$ Suppose that
$\Gamma_{1}\leq\Gamma^{\prime}\leq\Gamma$ for some congruence subgroup
$\Gamma^{\prime}.$ Then as algebras

$M(\Gamma)=M(\Gamma^{\prime}),$

and each component $M_{k,l}(\Gamma^{\prime})$ is some direct sum of components
$M_{k,l^{\prime}}(\Gamma)$ for some nontrivial $l^{\prime}.$

The proof of our first result is based on a comparison between invertible
sheaves on a rigid analytic space which is the analytification of a projective
moduli scheme and on the stacky modular curve whose coarse space is that
projective scheme. There are three major steps for our proof of this second
result which deal with the geometry of the coarse space of a Drinfeld modular
curve, the modularity conditions for modular forms, and finally the
coefficients of the series expansion for modular forms at cusps of the modular
curve. The third result, a generalization of the second, was pointed out by
Gebhard Böckle and is based on the proof of [Kob93, Proposition
$\mathrm{III}.28$]. We break these results and their respective proofs up into
different sections of this article.

##  2\. Background

In the classical number-field setting there is an isomorphism between the ring
of modular forms

$M=\bigoplus_{d\geq 0}M_{2d}(\Gamma)$

for $\Gamma\leq\mathrm{SL}_{2}(\mathbb{Z})$ a congruence subgroup, and the
ring of global sections of a particular line bundle, such as the sheaf of
differentials or the canonical bundle, on the corresponding modular curve. By
ring of global sections, we mean a ring of form

$R(\mathscr{X},\mathscr{L})=\bigoplus_{d\geq
0}H^{0}(\mathscr{X},\mathscr{L}^{\otimes d}),$

where $\mathscr{X}$ is a stacky curve and $\mathscr{L}$ is a line bundle on
$\mathscr{X}.$ We write $R=R(\mathscr{X},\mathscr{L})$ when the curve and line
bundle are understood. This allows one to compute algebras of modular forms
using the geometry of the moduli space.

We will briefly introduce Drinfeld modules, modular forms and modular curves.
In particular we need notation so that we can discuss series of modular forms
at cusps of the modular curve, the grading of the algebra of modular forms and
some special points on the modular curves. We also mention some of the theory
of sections rings for stacks.

###  2.1. Notation and the Drinfeld Setting

References for Drinfeld modular curves are [Gek86], [Gek01] and [MS15]; for
Drinfeld modular forms see the survey [Gek99] and the papers [GR96], [Gek88],
[Bre16], [Cor97a] and [DK23].

Let $\mathbb{F}_{q}$ be the finite field of order $q$ a power of an odd prime.
As function-field analogs of $\mathbb{Z},~{}\mathbb{Q},~{}\mathbb{R}$ and
$\mathbb{C}$ define the rings
$\displaystyle{A=\mathbb{F}_{q}[T],~{}K=\operatorname{Frac}(A)=\mathbb{F}_{q}(T),~{}K_{\infty}=\mathbb{F}_{q}\left(\\!\left(\frac{1}{T}\right)\\!\right)},$
the completion of $K$ at the place at $\infty,$ and let
$C=\widehat{\overline{K_{\infty}}}$ be the completion of the algebraic closure
of $K_{\infty}$ respectively. Then $C$ is an algebraically closed, complete,
and non-archimedean field. The Drinfeld-setting version of the upper half-
plane $\mathcal{H}\subset\mathbb{C}$ is $\Omega\overset{def}{=}C-K_{\infty}.$

Note that the group $\mathrm{GL}_{2}(A)$ acts on $\Omega$ by Möbius
transformations as $\mathrm{SL}_{2}$ acts on $\mathcal{H},$ but
$\det(\gamma)\in\mathbb{F}_{q}^{\times}$ for $\gamma\in\mathrm{GL}_{2}(A).$
Let $N\in A$ be a non-constant, monic polynomial and let $\Gamma(N)$ be the
subgroup of $\mathrm{GL}_{2}(A)$ with matrices congruent to
$\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)$ modulo $N.$ A subgroup
$\Gamma$ of $\mathrm{GL}_{2}(A)$ such that $\Gamma(N)\subseteq\Gamma$ for some
$N$ is a congruence subgroup and we call such an $N$ of the least degree the
conductor of $\Gamma.$

We establish an important assumption for this work: throughout,
$\Gamma\leq\mathrm{GL}_{2}(A)$ is some congruence subgroup such that for every
$\alpha,\alpha^{\prime}\in\mathbb{F}_{q}^{\times},~{}\Gamma$ contains the
matrices of form $\displaystyle{\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha^{\prime}\end{array}\right)},$ that is, the diagonal matrices in
$\mathrm{GL}_{2}(A).$ This means we have
$\det\Gamma=\\{\det(\gamma):\gamma\in\Gamma\\}=\mathbb{F}_{q}^{\times}.$ In
general $\det\Gamma$ is a subgroup of $\mathbb{F}_{q}^{\times}.$

Let $(\det\Gamma)^{2}$ be the set of squares of elements in $\det\Gamma:$
$(\det\Gamma)^{2}=\\{x^{2}:x\in\det\Gamma\\}.$ Let

$\Gamma_{2}\overset{def}{=}\\{\gamma\in\Gamma:\det(\gamma)\in(\det\Gamma)^{2}\\}.$

When we write $\Gamma\leq\mathrm{GL}_{2}(A),$ we mean $\Gamma$ satisfying the
conditions above, so $\det\Gamma_{2}=(\mathbb{F}_{q}^{\times})^{2}.$

The condition that $\Gamma$ has all possible determinants is simply for ease
of notation, as it is more pleasant to compute congruences modulo $q-1$ rather
than $\\#\det\Gamma.$ Our emphasis on the case when $q$ is odd is essential as
we make repeated use of the fact that $q-1$ is even.

We will make use of a kind of “parity” for congruence subgroups for which we
introduce the following terminology:

######  Definition 2.1.

We say that a congruence subgroup $\Gamma$ is square if there is some
$z\in\Omega$ such that the stabilizer $\Gamma_{z}=\\{\gamma\in\Gamma:\gamma
z=z\\}$ strictly contains
$\displaystyle{\mathbb{F}_{q}^{\times}\cong\left\\{\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha\end{array}\right):\alpha\in\mathbb{F}_{q}^{\times}\right\\}}$ and
every $\gamma\in\Gamma_{z}\setminus\mathbb{F}_{q}^{\times}$ has $\det\gamma$ a
square in $\mathbb{F}_{q}^{\times}.$ Likewise, $\Gamma$ is non-square if it
contains a stabilizer $\Gamma_{z}$ for some $z\in\Omega$ strictly larger than
$\mathbb{F}_{q}^{\times}$ and some $\gamma$ with
$\det\gamma\in\mathbb{F}_{q}^{\times}\setminus(\mathbb{F}_{q}^{\times})^{2}.$

This idea will help distinguish the geometric invariants involved in the
computations at the end of this manuscript into two cases. In our application
stabilizers are all $\mathrm{GL}_{2}(A)$-conjugate subgroups of
$\mathbb{F}_{q^{2}}^{\times}$ so that one only needs to check for a single
point $z\in\Omega$ with a stabilizer
$\Gamma_{z}\supsetneq\mathbb{F}_{q}^{\times}$ whether $\Gamma_{z}$ contains
some matrix with a non-square determinant.

###  2.2. Drinfeld modules

The theory of Drinfeld modules is rich in both algebraic and analytic
structure. Both interpretations and their equivalence are important in
understanding the moduli spaces of Drinfeld modules of a given rank. We state
only what we need for our computation of the canonical ring of certain log-
stacky moduli spaces and the corresponding algebras of Drinfeld modular forms.

####  2.2.1. Analytic Approach

We give a quick description of Drinfeld modules as lattice quotients.
Following Breuer [Bre16], we say an $A$-submodule of $C$ of form
$\Lambda=\omega_{1}A+\cdots+\omega_{r}A,$ for $\omega_{1},\cdots,\omega_{r}\in
C$ some $K_{\infty}$-linearly independent elements, is an $A$-lattice of rank
$r.$ Then the exponential function of $\Lambda$, $e_{\Lambda}:C\to C,$ defined
by

$e_{\Lambda}(z)\overset{def}{=}z\prod_{0\neq\lambda\in\Lambda}\left(1-\frac{z}{\lambda}\right)$

is holomorphic in the rigid analytic sense (see e.g. [FvdP04, Definition
$2.2.1$]), surjective, $\mathbb{F}_{q}$-linear, $\Lambda$-periodic and has
simple zeros on $\Lambda.$

Let
$C\\{X^{q}\\}\overset{def}{=}\\{a_{0}X+a_{1}X^{q}+\cdots+a_{n}X^{q^{n}}:a_{0},\cdots,a_{n}\in
C,~{}n\geq 0\\}$ denote the non-commutative polynomial ring of
$\mathbb{F}_{q}$-linear polynomials over $C,$ with the operation of
multiplication given by composition. For each $a\in A$ the exponential
satisfies the functional equation

$e_{\Lambda}(az)=\varphi_{a}^{\Lambda}(e_{\Lambda}(z)),$

where $\varphi_{a}^{\Lambda}(X)\in C\\{X^{q}\\}$ is some element of degree
$q^{r\deg a}.$ Then we say a ring homomorphism $\varphi:A\to C\\{X^{q}\\}$
given by

$a\mapsto\varphi_{a}^{\Lambda}\overset{def}{=}a_{0}(a)X+\cdots+a_{r\deg
a}(a)X^{q^{r\deg a}},$

(an $\mathbb{F}_{q}$-algebra monomorphism) is a Drinfeld module of rank $r$ if
the coefficient with largest index is non-zero.

####  2.2.2. Algebraic Approach

We recall, without any proofs, some facts concerning the algebraic theory
which corresponds to the definition above. A more complete discussion of these
next facts is found in [Pap23, Definition $3.1.4$] and [Pap23, Lemma $3.1.4$].
We are mostly interested in the notation.

First we characterize the notion of an $\mathbb{F}_{q}$-linear function.

######  Lemma 2.2.

Let $\mathbb{K}$ be a field of characteristic $p$ containing $\mathbb{F}_{q}.$
Then $f(x)\in\mathbb{K}[x]$ is $\mathbb{F}_{q}$-linear (i.e. $f(\alpha
x)=\alpha f(x)$ for all $\alpha\in\mathbb{F}_{q}$) if and only if
$\displaystyle{f(x)=\sum_{i=0}^{n}a_{i}x^{q^{i}}}.$

We state the following result so that when we define a moduli space of
Drinfeld modules, we can make sense of Drinfeld modules over an arbitrary base
scheme and therefore eventually have a well-defined category fibered in
groupoids when we consider moduli stacks later.

######  Theorem 2.3.

[Wat79, Page $65$] Let $B$ be an $A$-algebra, and let $\mathbb{G}_{a,B}$
denote the affine additive group scheme over $B$ represented by
$\operatorname{Spec}B[t].$ Then the set
$\mathrm{End}_{\mathbb{F}_{q}}(\mathbb{G}_{a,B})$ of $\mathbb{F}_{q}$-linear
endomorphisms of $\mathbb{G}_{a,B},$ is
$\mathrm{End}_{\mathbb{F}_{q}}(\mathbb{G}_{a,B})\cong B\\{X^{q}\\}.$

Finally, we can introduce algebraic Drinfeld modules over any scheme.

######  Definition 2.4.

A Drinfeld module of rank $r$ over an $A$-scheme $S$ is a pair $(E,\varphi)$
consisting of:

  * •

a $\mathbb{G}_{a}$-bundle $E$ (i.e. an additive group scheme) over $S$ such
that for all $U=\operatorname{Spec}B$ an affine open subset of $S$ for $B$ an
$A$-algebra in the Zariski topology on $S,$ there is an isomorphism
$\psi:E|_{U}\overset{\sim}{\to}\mathbb{G}_{a,B}$ of group schemes over $U$

  * •

a ring homomorphism $\varphi:A\to\mathrm{End}(E)$

such that for a family of pairs $(U_{i},\psi_{i})$ which form a trivializing
cover of $E$ (i.e. $U_{i}=\operatorname{Spec}B_{i}$ are an affine open cover
and $\psi_{i}:E_{\pi^{-1}(U_{i})}\overset{\sim}{\to}\mathbb{G}_{a,B_{i}}$ are
local isomorphisms of additive group schemes), the morphism $\varphi$
restricts to give maps $\varphi_{i}:A\to\mathrm{End}(\mathbb{G}_{a,B_{i}})$ of
form $\varphi_{i}(T)=TX+b_{1,i}X^{q}+\cdots+b_{r,i}X^{q^{r}},$ compatible with
the transition functions $\psi_{ji}=\psi_{i}\circ\psi_{j}^{-1},$ i.e.
$\varphi_{j}\circ\psi_{ij}=\psi_{ij}\circ\varphi_{i}$ on all intersections
$U_{ij}=U_{i}\cap U_{j}.$

######  Remark 2.5.

In the special case when we consider Drinfeld modules over a field, the
algebraic definition of a Drinfeld module is simpler. In particular, we have
$E=\mathbb{G}_{a},$ and we do not need any of the trivializations of our
bundle as we are working over a single affine scheme. Therefore, it suffices
to provide a ring homomorphism $\varphi:A\to\mathrm{End}(\mathbb{G}_{a}).$ We
do not make further explicit use of the algebraic definition of Drinfeld
modules in this article beyond the following examples.

######  Example 2.6.

[Car38] The Carlitz module is the rank $1$ Drinfeld module defined by

$\varphi(T)=TX+X^{q},$

and corresponds to the lattice $\overline{\pi}A\subset\Omega.$ Here,
$\overline{\pi}\in K_{\infty}(\sqrt[q-1]{-T})$ is the Carlitz period, defined
up to a $(q-1)$st root of unity. We fix one such $\overline{\pi}$ once and for
all.

As an algebraic Drinfeld module, the Carlitz module is the image of the ring
homomorphism

$\displaystyle\varphi:$ $\displaystyle A\to C\\{X^{q}\\}$ $\displaystyle
T\mapsto TX+X^{q}$

which is a rank $1$ module since $\deg\varphi=q=|T|^{1},$ over the $A$-scheme
$\operatorname{Spec}C.$ Here, $|\cdot|$ is the extension of the $\infty$-adic
absolute value to $C,$ see e.g. [Poo22, Section $(2.2)$]. We say $0\neq a\in
A$ has $|a|_{\infty}=q^{\deg a}$ and $|0|=0.$

######  Example 2.7.

Let $z\in\Omega,$ and consider the rank $2$ lattice
$\Lambda_{z}=\overline{\pi}(zA+A).$ The associated Drinfeld module of rank $2$
is

$\varphi^{z}(T)=TX+g(z)X^{q}+\Delta(z)X^{q^{2}},$

where $g$ and $\Delta$ are Drinfeld modular forms of type $0$ and weights
$q-1$ and $q^{2}-1$ respectively. We will define Drinfeld modular forms in the
next section. This is analogous to defining an elliptic curve by a short
Weierstrass equation whose coefficients are values of Eisenstein series.

Here, once again this is an algebraic Drinfeld module over an affine
$A$-scheme, we have written down in particular the image of a degree $2$ ring
homomorphism $\varphi:A\to C\\{X^{q}\\}.$ The Carlitz period $\overline{\pi}$
serves to normalize the coefficients of the series expansion of $g$ and
$\Delta$ at the cusps of $\mathrm{GL}_{2}(A)$ so that those coefficients are
elements of $A.$

###  2.3. Stacks and Section Rings

See [Alp23] for a general stacks reference; see [VZB22] for an excellent and
comprehensive reference on computing canonical rings of stacky curves and
[O’D15] for a useful generalization of [VZB22] that we need for the Drinfeld
setting. We are most interested in Deligne-Mumford stacks for this work, so
some facts and examples will be specialized to that case, but we indicate when
this occurs. We also discuss rigid analytic stacks and GAGA for rigid analytic
and algebraic stacks, but leave that theory for a later section.

It is shown in e.g. [Lau96, Corollary $1.4.3$] that the moduli space of rank
$r$ Drinfeld modules over the category of schemes of characteristic $p$ is
representable by a Deligne-Mumford algebraic stack of finite type over
$\mathbb{F}_{p}.$ One is able to compute the graded rings of global sections
of line bundles on stacks which represent the Drinfeld moduli problems by
means of geometric invariants with results that are slight variants on the
theory in [VZB22]. We will follow [VZB22] in describing this computation,
stating only select facts that we will need.

Recall from [VZB22, Definition $5.2.1$], a stacky curve $\mathscr{X}$ over a
field $\mathbb{K}$ is a smooth, proper, geometrically connected Deligne-
Mumford stack of dimension $1$ over $\mathbb{K}$ that contains a dense open
subscheme. Every stacky curve $\mathscr{X}$ over a field $\mathbb{K}$ has a
unique coarse space morphism $\pi:\mathscr{X}\to X$ with $X$ a smooth scheme
over $\mathbb{K}$ (called the coarse space). Here $\pi$ is universal for
morphisms from $\mathscr{X}$ to schemes, and the set of isomorphism classes of
$F$-points on $\mathscr{X}$ and $X$ are in bijection for any algebraically
closed field $F$ containing $\mathbb{K}.$ Note that étale locally on the
coarse space $X,$ a stacky curve $\mathscr{X}$ is the quotient of an affine
scheme by a finite (constant) group $G\leq\mathrm{Aut}(X).$ For $x\in X$ some
point, let $G_{x}$ denote the stabilizer of $x$ under the action by $G.$

Continuing the notation in the last paragraph, let $\pi:\mathscr{X}\to X$ be a
coarse space morphism. A Weil divisor is a finite formal sum of irreducible
closed substacks of codimension $1$ over $\mathbb{K}.$ On a smooth Deligne-
Mumford stack, every Weil divisor is Cartier. Any line bundle $\mathscr{L}$ on
$\mathscr{X}$ is isomorphic to $\mathcal{O}_{\mathscr{X}}(D)$ for some Cartier
divisor $D.$ Finally, there is an isomorphism of sheaves on the Zariski site
of $X:$

$\mathcal{O}_{X}(\lfloor
D\rfloor)\overset{\sim}{\to}\pi_{*}\mathcal{O}_{\mathscr{X}}(D),$

where

$\lfloor
D\rfloor=\Big{\lfloor}\sum_{i}a_{i}P_{i}\Big{\rfloor}\overset{def}{=}\sum_{i}\Big{\lfloor}\frac{a_{i}}{\\#G_{P_{i}}}\Big{\rfloor}\pi(P_{i}).$

######  Example 2.8.

Let $f:\mathscr{X}\to\mathscr{Y}$ be a morphism of stacky curves with coarse
spaces $X$ and $Y=\operatorname{Spec}k$ for $k$ some field respectively. Then
the sheaf of differentials
$\Omega^{1}_{\mathscr{X}}=\Omega^{1}_{\mathscr{X}/\operatorname{Spec}k}$ is
the sheafification (see [Alp23, Section $2.2.9$] for sheafification) of the
presheaf on $\mathscr{X}_{\text{\'{e}t}}$ given by

$(U\to\mathscr{X})\mapsto\Omega^{1}_{\mathcal{O}_{\mathscr{X}}(U)/f^{-1}\mathcal{O}_{\mathscr{Y}}(U)},$

where $\mathcal{O}_{\mathscr{X}}$ and $\mathcal{O}_{\mathscr{Y}}$ denote the
structure sheaves on $\mathscr{X}$ and $\mathscr{Y}$ respectively (see e.g.
[Alp23, Example $4.1.2$] for more details on structure sheaves for Deligne-
Mumford stacks).

In particular, we treat seriously the stackiness of the moduli when we compute
the following homogeneous coordinate rings on Drinfeld modular curves.

######  Definition 2.9.

Let $\mathscr{X}$ be a stacky curve over a field $k$ and let $\mathscr{L}$ be
an invertble sheaf on $\mathscr{X}.$ Then the section ring of $\mathscr{L}$ on
$\mathscr{X}$ is the ring

$R(\mathscr{X},\mathscr{L})=\bigoplus_{d\geq
0}H^{0}(\mathscr{X},\mathscr{L}^{\otimes d}).$

If $\mathscr{L}\cong\mathcal{O}_{\mathscr{X}}(D)$ for some Weil divisor $D$ in
particular, we can equivalently write

$R_{D}=\bigoplus_{d\geq 0}H^{0}(\mathscr{X},dD).$

Recall from [VZB22, Chapter $5.1$] that a point of a stack $\mathscr{X}$ is a
map $\operatorname{Spec}F\to\mathscr{X}$ for $F$ some field, and to a point
$x,$ we associate its stabilizer
$G_{x}\overset{def}{=}\underline{\mathrm{Isom}}(x,x),$ a functor which is a
representable by an algebraic space. If $G_{x}$ is a finite group scheme, say
that $\mathscr{X}$ is tame if $\deg G_{x}$ is not divisible by
$\operatorname{char}(F)$ for any $x\in\mathscr{X}.$ We say a point $x$ with
$G_{x}\neq\\{1\\}$ is a stacky point.

Finally, for readability of our main results, we introduce some terminology
inspired by [VZB22, Definition $5.6.2$] and [VZB22, Proposition $5.5.6$]. Let
$\mathscr{X}$ be a tame stacky curve over an algebraically closed field
$\mathbb{K}$ with coarse space $X.$ A Weil divisor $\Delta$ on $\mathscr{X}$
is a log divisor if $\Delta=\sum_{i}P_{i}$ is an effective divisor given as a
sum of distinct points (stacky or otherwise) on $\mathscr{X}.$ By [VZB22,
Proposition $5.5.6$] if $K_{\mathscr{X}}$ and $K_{X}$ are canonical divisors
on $\mathscr{X}$ and $X$ respectively, then there is a linear equivalence

$K_{\mathscr{X}}\sim K_{X}+R=K_{X}+\sum_{x}\left(1-\frac{1}{\deg
G_{x}}\right)x,$

where $G_{x}$ is the stabilizer of a closed substack $x\in\mathscr{X},$ and
the sum above is taken over closed substacks of $\mathscr{X}.$

##  3\. Drinfeld Modular Forms

In this section we introduce Drinfeld modular forms. The technical conditions
of the rigid analytic space in which we work makes it necessary to introduce
some facts about the projective line $\mathbb{P}^{1}(C)$ before we begin in
earnest on a study of modular forms. We discuss rigid analytic spaces in more
detail in the following sections.

######  Definition 3.1.

Let $\overline{\pi}\in K_{\infty}(\sqrt[q-1]{-T})$ be a fixed choice of the
Carlitz period (recall Example 2.6). Then we define a parameter at infinity

$u(z)\overset{def}{=}\frac{1}{e_{\overline{\pi}A}(\bar{\pi}z)}=\frac{1}{\bar{\pi}e_{A}(z)}=\bar{\pi}^{-1}\sum_{a\in
A}\frac{1}{z+a}.$

######  Remark 3.2.

In the Drinfeld setting, $\overline{\pi}$ plays the role of the constant $2\pi
i\in\mathbb{C}$ in the parameter $q=e^{2\pi iz}$ at infinity from the
classical setting. That is, it is a normalization factor so that the series
expansion coefficients for modular forms at cusps are elements of $A.$

One fact about this parameter which we will use later in our consideration of
modular forms is the following.

######  Lemma 3.3.

[Gek99, Page $10$] Let $\alpha\in\mathbb{F}_{q}^{\times}.$ Then
$\displaystyle{u\left(\alpha z\right)=\alpha^{-1}u(z)}.$

###### Proof.

For any $\alpha\in\mathbb{F}_{q}^{\times},$ the lattices
$\Lambda=A\omega_{1}+A\omega_{2}\subset C$ and $\alpha\cdot\Lambda$ are
similar. Furthermore, $\Lambda$ is similar to $(\omega_{1}/\omega_{2})A+A,$
where $z=\omega_{1}/\omega_{2}\in\Omega.$ So the exponential functions
$e_{\bar{\pi}\alpha A}(\bar{\pi}\alpha z)$ and $e_{A}(z)$ for $z\in\Omega$
differ by a factor of $\bar{\pi}\alpha^{-1}.$ ∎

Now we are able to define a fundamental object of study for this article.

######  Definition 3.4.

[Gek86, Definition $(3.1)$] Let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a congruence
subgroup. A modular form of weight $k\in\mathbb{Z}_{\geq 0}$ and type
$l\in\mathbb{Z}/((q-1)\mathbb{Z})$ is a holomorphic function $f:\Omega\to C$
such that

  1. (1)

$f(\gamma z)=\det(\gamma)^{-l}(cz+d)^{k}f(z)$ for all
$\displaystyle{\gamma=\left(\begin{array}[]{cc}a&b\\\
c&b\end{array}\right)\in\Gamma},$ and

  2. (2)

$f$ is holomorphic at the cusps of $\Gamma.$

######  Remark 3.5.

There are several interpretations of the second condition about holomorphy at
$\infty,$ two of which are particularly helpful for intuition and for the
proof the main theorem respectively:

  1. (1)

[Gek01, $(2.2.\mathrm{iii})$] The condition is equivalent to $f$ being bounded
on $\\{z\in\Omega:|z|_{\infty}\geq 1\\},$ where $|\cdot|_{\infty}$ is the
$\infty$-adic absolute value, in any case when $\Gamma$ has a single cusp;

  2. (2)

[Gek99, Definition $3.5.(\mathrm{iii})$] $f$ has a series expansion at cusps:

$f(z)=\sum_{n\in\mathbb{Z}}a_{n}u(z)^{n},~{}a_{n}\in C,$

where $u$ is the parameter at $\infty,$ with a positive radius of convergence.
The second condition means that $a_{n}=0$ for all $n<0.$

######  Remark 3.6.

The observation from [Gek88, Definition $(5.7)$] that if $f$ is Drinfeld
modular form, then $f(z+b)=f(z)$ for $b\in A$ means that although not
literally a Fourier series, the series expansion of a modular form at the
cusps of some congruence subgroup is “morally” the Drinfeld setting equivalent
to a Fourier series.

Finally, we mention some further terminology which is part of the definition
of a Drinfeld modular form.

######  Definition 3.7.

We say a function $f:\Omega\to C$ such that $f(\gamma
z)=\det(\gamma)^{-l}(cz+d)^{k}f(z)$ for all
$\displaystyle{\gamma=\left(\begin{array}[]{cc}a&b\\\
c&b\end{array}\right)\in\Gamma},$ where $k\in\mathbb{Z}_{\geq 0},$
$l\in\mathbb{Z}/(q-1)\mathbb{Z}$ and $\Gamma\leq\mathrm{GL}_{2}(A)$ is a
congruence subgroup, is weakly modular of weight $k$ and type $l$ for
$\Gamma.$

We introduce some terminology and notation respectively in the next
definition.

######  Definition 3.8.

Write $M_{k,l}(\Gamma)$ for the finite-dimensional $C$-vector space of
Drinfeld modular forms for $\Gamma\leq\mathrm{GL}_{2}(A)$ with weight $k$ and
type $l.$ The graded ring $M(\Gamma)$ of modular forms is

$M(\Gamma)=\bigoplus_{\begin{subarray}{c}k\geq 0\\\ l\ (\mathrm{mod}\
q-1)\end{subarray}}M_{k,l}(\Gamma)$

since $M_{k,l}\cdot M_{k^{\prime},l^{\prime}}\subset
M_{k+k^{\prime},l+l^{\prime}}.$

Now we can introduce some non-trivial facts about Drinfeld modular forms.

######  Lemma 3.9.

[Gek88, Remark $5.8.\mathrm{iii}$] Suppose $f(z)\in M_{k,l}(\Gamma)$ has a
$u$-series expansion $f(z)=\sum_{n\geq 0}a_{n}u^{n}.$ Then the coefficients
$a_{i}$ uniquely determine $f.$

###### Proof.

Although the $u$-series may not converge on all of $\Omega$ and only for
$|u(z)|$ small, since $\Omega$ is connected in the rigid analytic sense, the
result follows from having a unique $u$-series anywhere within $\Omega.$ ∎

The weight and type of Drinfeld modular forms are not independent.

######  Lemma 3.10.

[Gek88, Remark $(5.8.\mathrm{i})$] If $M_{k,l}(\Gamma)\neq 0,$ then $k\equiv
2l\pmod{q-1}.$

###### Proof.

Let $\displaystyle{\gamma=\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha\end{array}\right)}$ for some $\alpha\in\mathbb{F}_{q}^{\times}.$ By
assumption $\Gamma$ contains the matrices of form
$\displaystyle{\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha^{\prime}\end{array}\right)}$ for all
$\alpha,\alpha^{\prime}\in\mathbb{F}_{q}^{\times},$ therefore
$\gamma\in\Gamma.$ If $f$ is non-zero modular for $\Gamma$ of weight $k$ and
type $l$ then

$f(\gamma z)=f\left(\frac{\alpha
z}{\alpha}\right)=f(z)=\alpha^{k}\alpha^{-2l}f(z),$

so $\alpha^{k}=\alpha^{2l}$ in $\mathbb{F}_{q}^{\times}$ and we conclude
$k\equiv 2l\pmod{q-1}.$ ∎

######  Example 3.11.

Some famous Drinfeld modular forms are the $\mathrm{GL}_{2}(A)$-forms: $g$ of
weight $q-1$ and type $0,~{}\Delta$ of weight $q^{2}-1$ and type $0,$ and $h$
of weight $q+1$ and type $1.$ We know from Goss and Gekeler respectively, see
for example [Gek99, Theorem $(3.12)$], that

$\bigoplus_{k\geq 0}M_{k,0}(\mathrm{GL}_{2}(A))=C[g,\Delta]\quad\text{ and
}\quad\bigoplus_{\begin{subarray}{c}k\geq 0\\\ l\ (\mathrm{mod}\
q-1)\end{subarray}}M_{k,l}(\mathrm{GL}_{2}(A))=C[g,h].$

######  Example 3.12.

[Gek88] The function

$E(z)\overset{def}{=}\overline{\pi}^{-1}\sum\limits_{\begin{subarray}{c}a\in
A\\\ \text{monic}\end{subarray}}\left(\sum_{b\in A}\frac{a}{az+b}\right)$

is an analog to an Eisenstein series of weight $2$ over $\mathbb{Q},$ and we
can define a Drinfeld modular form

$E_{T}(z)\overset{def}{=}E(z)-TE(Tz)$

of weight $2$ and type $1$ for $\Gamma_{0}(T),$ the congruence subgroup of
$\mathrm{GL}_{2}(A)$ containing matrices
$\displaystyle{\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)}$ with
$c\equiv 0\ (\mathrm{mod}\ T).$

##  4\. Drinfeld Modular Curves

Let us consider the moduli space of rank $2$ Drinfeld modules, first as a
rigid analytic space, then as moduli schemes, and finally as log stacky
curves. We recall some definitions we need to discuss rigid anaytic spaces,
which are the natural means to discuss quotients of the Drinfeld “upper half-
plane” $\Omega$ by congruence subgroups. A more thorough treatment and
reference for rigid analytic geometry is [FvdP04]. We will specialize to rigid
analytic spaces over $C$ for readability.

We need the following two intermediate definitions to define a rigid analytic
space.

######  Definition 4.1.

[FvdP04, Page $46$] Let $z_{1},\cdots,z_{n}$ denote some variables. Let
$T_{n}=C\langle z_{1},\cdots,z_{n}\rangle$ be the $n$-dimensional $C$-algebra
which is the subring of the ring of formal power series
$C[[z_{1},\cdots,z_{n}]]$

$T_{n}=\left\\{\sum_{\alpha}c_{\alpha}z_{1}^{\alpha_{1}}\cdots
z_{n}^{\alpha_{n}}\in C[[z_{1},\cdots,z_{n}]]:\lim c_{\alpha}=0\right\\},$

where $\alpha=(\alpha_{1},\cdots,\alpha_{n}).$ An affinoid algebra $A$ over
$C$ is a $C$-algebra which is a finite extension of $T_{n}$ for some $n\geq
0.$

######  Definition 4.2.

[FvdP04, Definition $2.4.1$] Let $X$ be a set. A G-topology on $X$ consists of
the data:

  1. (1)

A family $\mathscr{F}$ of subsets of $X$ such that $\emptyset,X\in\mathscr{F}$
and if $U,V\in\mathscr{F},$ then $U\cap V\in\mathscr{F}$ &

  2. (2)

For each $U\in\mathscr{F}$ a set $\operatorname{Cov}(U)$ of coverings of $U$
by elements of $\mathscr{F}$

such that the following conditions are met:

  * •

$\\{U\\}\in\operatorname{Cov}(U)$

  * •

For each $U,V\in\mathscr{F}$ with $V\subset U$ and
$U\in\operatorname{Cov}(U),$ the covering $\displaystyle{U\cap
V\overset{def}{=}\\{U^{\prime}\cap V:U^{\prime}\in U\\}}$ belongs to
$\operatorname{Cov}(U)$

  * •

Let $U\in\mathscr{F},$ let $\\{U_{i}\\}_{i\in I}\in\operatorname{Cov}(U)$ and
let $\mathcal{U}_{i}\in\operatorname{Cov}(U_{i}).$ Then

$\bigcup_{i\in I}\mathcal{U}_{i}\overset{def}{=}\\{U^{\prime}:U^{\prime}\text{
belongs to some }\mathcal{U}_{i}\\}$

is an element of $\operatorname{Cov}(U).$

We say the $U\in\mathscr{F}$ are admissible sets and the elements of
$\operatorname{Cov}(U)$ are admissible coverings.

Finally, we come to the point:

######  Definition 4.3.

[FvdP04, Definition $4.3.1$] A rigid analytic space is a triple
$(X,T_{X},\mathcal{O}_{X})$ consisting of a set $X,$ a $G$-topology $T_{X}$ on
$X$ and a structure sheaf of $C$-algebras $\mathcal{O}_{X}$ on $X$ for which
there exists an admissible open covering $\\{X_{i}\\}$ of $X$ such that each
$(X_{i},T_{X_{i}},\mathcal{O}_{X_{i}})$ is an affinoid over $C$ and $U\subset
X$ belongs to $T_{X}$ if and only if $U\cap X_{i}$ belongs to $T_{X}$ for each
$i.$

For the well-definedness of Drinfeld modular curves, we recall some analytic
properties of $\Omega.$ Since
$\Omega=\mathbb{P}^{1}(C)-\mathbb{P}^{1}(K_{\infty}),$ and
$\mathbb{P}^{1}(K_{\infty})$ is compact in the rigid analytic topology, we
know from [GR96, Section $1.2$] that $\Omega$ is a rigid analytic space. The
action by $\Gamma\leq\mathrm{GL}_{2}(A)$ a congruence subgroup on $\Omega$ by
Möbius transformations has finite stabilizer for each $z\in\Omega,$ and as in
[GR96, Sections $(2.5)$ and $(2.6)$], $\Gamma\setminus\Omega$ is a rigid
analytic space.

Recall that for any scheme $S$ of locally finite type over a complete, non-
archimedian field of finite characteristic $p,$ there is a rigid analytic
space $S^{\text{an}}$ whose points coincide with those of $S$ as sets. In
fact, there is an analytification functor from the category of schemes over
$C$ to the category of rigid analytic spaces, so if $X$ is a smooth algebraic
curve over $C,$ then there is a rigid analytic space $X^{\text{an}}$ whose
points are in bijection with the $C$-points of $X.$

######  Theorem 4.4.

[Dri74] There exists a smooth, irreducible, affine algebraic curve
$Y_{\Gamma}$ over $C$ such that $\Gamma\setminus\Omega$ and the underlying
(rigid) analytic space $Y_{\Gamma}^{\text{an}}$ of $Y_{\Gamma}$ are
canonically isomorphic as rigid analytic spaces over $C.$

######  Remark 4.5.

This underlying rigid analytic space is the analytification (see [FvdP04,
Example $4.3.3$]) of $Y_{\Gamma}.$

######  Definition 4.6.

We call the affine curves $Y_{\Gamma}$ whose analytification
$Y_{\Gamma}^{\text{an}}$ are isomorphic to $\Gamma\setminus\Omega$ as rigid
analytic spaces over $C$ affine Drinfeld modular curves. Since $Y_{\Gamma}$ is
smooth and affine, it admits a smooth projective model which $X_{\Gamma}$
which is the projective Drinfeld modular curve.

######  Remark 4.7.

In the spirit of [VZB22, Section $6.2$], we say a stacky Drinfeld modular
curve $\mathscr{X}_{\Gamma}$ is the algebraization of some rigid analytic
space $\Gamma\setminus(\Omega\cup\mathbb{P}^{1}(K))=X_{\Gamma}^{\text{an}},$
whose points are in bijection with the $C$-points of the projective Drinfeld
modular curve $X_{\Gamma},$ the coarse space of $\mathscr{X}_{\Gamma}.$

Next we consider some special points on Drinfeld modular curves.

######  Definition 4.8.

Let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a congruence subgroup, let
$Y_{\Gamma}^{\text{an}}=\Gamma\setminus\Omega$ and let
$X_{\Gamma}^{\text{an}}=\Gamma\setminus(\Omega\cup\mathbb{P}^{1}(K)).$ A cusp
of $X_{\Gamma}^{\text{an}}$ is a point of
$X_{\Gamma}^{\text{an}}-Y_{\Gamma}^{\text{an}}.$

######  Remark 4.9.

As sets,
$X_{\Gamma}^{\text{an}}=\Gamma\setminus(\Omega\cup\mathbb{P}^{1}(K)),$ so
since $\mathrm{GL}_{2}(A)$ acts transitively on $\mathbb{P}^{1}(K)$ we have

$\mathcal{C}_{\Gamma}\overset{def}{=}\\{\text{cusps of
}X_{\Gamma}^{\text{an}}\\}\overset{def}{=}\Gamma\setminus\mathbb{P}^{1}(K)=\Gamma\setminus\mathrm{GL}_{2}(A)/\mathrm{GL}_{2}(A)_{\infty},$

where
$\displaystyle{\mathrm{GL}_{2}(A)_{\infty}=\\{\gamma\in\mathrm{GL}_{2}(A):\gamma(\infty)=\infty\\}=\left\\{\left(\begin{array}[]{cc}*&*\\\
0&*\end{array}\right)\right\\}}.$

######  Definition 4.10.

  1. (1)

If $e\in\Omega$ has
$(\mathrm{GL}_{2}(A))_{e}=\\{\gamma\in\mathrm{GL}_{2}(A):\gamma(e)=e\\}$
strictly larger than
$\mathbb{F}_{q}^{\times}\cong\left\\{\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha\end{array}\right)\right\\}$ then $e$ is an elliptic point on
$\Omega$. In this case,
$\mathrm{GL}_{2}(A)_{e}\cong\mathbb{F}_{q^{2}}^{\times}.$

  2. (2)

Let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a congruence subgroup. A point
$e\in\Omega$ is an elliptic point for $\Gamma$ if the stabilizer $\Gamma_{e}$
is strictly larger than
$\displaystyle{\mathbb{F}_{q}^{\times}\cong\left\\{\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha\end{array}\right):\alpha\in\mathbb{F}_{q}^{\times}\right\\}}$ (the
center of $\mathrm{GL}_{2}(\mathbb{F}_{q})$).

######  Remark 4.11.

An elliptic point $e$ on $\Omega$ is a point which is
$\mathrm{GL}_{2}(A)$-conjugate to some element of
$\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}\hookrightarrow\Omega.$ Fix once and
for all an elliptic point $e\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ on
$\Omega.$ We write

$\operatorname{Ell}(\Gamma)\overset{def}{=}\\{\text{elliptic points of
}X_{\Gamma}^{\text{an}}\\}.$

##  5\. Rigid Stacky GAGA

We need a precise notion of a rigid analytic stack for rigid stacky GAGA.
Since algebraic (stacky) Drinfeld modular curves are Deligne-Mumford stacks,
we will specialize the notion of rigid analytic Artin stacks from [EGH23,
Section $5.1.7$] to Deligne-Mumford rigid analytic stacks.

Let $\operatorname{Rig}_{C}$ denote the category of rigid analytic spaces over
$C.$ Equip $\operatorname{Rig}_{C}$ with the Tate-fpqc topology (see [CT07,
$2.1$]). The covers in this topology are generated by the admissible Tate
coverings (see [FvdP04, Section $4.2$]) and the morphisms
$\operatorname{Sp}(A)\to\operatorname{Sp}(B)$ for faithfully flat morphisms of
affinoid algebras $B\to A.$ By [Con06, Theorem $4.2.8$] all representable
functors in this topology are sheaves and coherent sheaves satisfy descent.
With this site specified, we can define rigid analytic stacks.

######  Definition 5.1.

A stack on $\operatorname{Rig}_{C}$ is a category fibered in groupoids which
satisfies descent for the Tate-fpqc topology.

Next we cite [EGH23] to define a rigid analytic Artin stack. Good references
on Artin stacks, which appear in [Art74], are [AOV07] and [AOV10].

######  Definition 5.2.

[EGH23, $5.1.10$] A rigid analytic Artin stack is a stack $\mathscr{X}$ on
$\operatorname{Rig}_{L}$ such that the diagonal
$\Delta_{\mathscr{X}}:\mathscr{X}\to\mathscr{X}\times_{L}\mathscr{X}$ is
representable by a rigid analytic space, and there exists some rigid analytic
space $U$ and a smooth surjective map $U\to\mathscr{X}.$

Now we are equipped to define the version of rigid analytic stack we will
consider in application of a rigid GAGA theorem on stacks.

######  Definition 5.3.

A rigid analytic Deligne-Mumford stack is a rigid analytic Artin stack
$\mathscr{X}$ such that the diagonal
$\Delta_{\mathscr{X}}:\mathscr{X}\to\mathscr{X}\times_{L}\mathscr{X}$ is
representable by a rigid analytic space, quasi-compact and separated for the
Tate-fpqc topology.

We will use the following rigid stacky GAGA theorem.

######  Theorem 5.4.

[PY16, $7.4$] Let $A$ be a $k$-affinoid algebra, for $k$ some non-achimedean
field. Let $\mathscr{X}$ be a proper algebraic stack over
$\operatorname{Spec}(A).$ The analytification functor on coherent sheaves
induces an equivalence of $1$-categories

$\operatorname{Coh}(\mathscr{X})\overset{\cong}{\to}\operatorname{Coh}(\mathscr{X}^{\text{an}}).$

##  6\. Proof of the Main Theorems

We prove our first main result:

######  Theorem 6.1.

Let $q$ be an odd prime and let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a congruence
subgroup containing the diagonal matrices of $\mathrm{GL}_{2}(A)$ such that
$\det(\gamma)\in(\mathbb{F}_{q}^{\times})^{2}$ for every $\gamma\in\Gamma.$
Let $\Delta$ be the (Weil) divisor of cusps of the modular curve
$\mathscr{X}_{\Gamma}$ with the rigid analytic coarse space
$X_{\Gamma}^{\text{an}}=\Gamma\setminus(\Omega\cup\mathbb{P}^{1}(K)).$ There
is an isomorphism of graded rings $M(\Gamma)\cong
R(\mathscr{X}_{\Gamma},\Omega^{1}_{\mathscr{X}_{\Gamma}}(2\Delta)),$ where
$\Omega^{1}_{\mathscr{X}_{\Gamma}}$ is the sheaf of differentials on
$\mathscr{X}_{\Gamma}.$ The isomorphism of algebras is given by the
isomorphisms of components $M_{k,l}(\Gamma)\to
H^{0}(\mathscr{X}_{\Gamma},\Omega^{1}_{\mathscr{X}_{\Gamma}}(2\Delta)^{\otimes
k/2})$ given by $f\mapsto f(dz)^{\otimes k/2}.$

###### Proof.

Suppose $f\in M_{k,l}(\Gamma).$ For any
$\displaystyle{\gamma=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\Gamma}$ we have

$f(\gamma z)d(\gamma z)^{\otimes
k/2}=(cz+d)^{k}(\det\gamma)^{-l}\frac{\det\gamma^{k/2}}{(cz+d)^{k}}f(z)dz^{\otimes
k/2},$

where $k\equiv 2l\pmod{\frac{q-1}{2}}.$ All of the factors of automorphy
cancel and

$f(\gamma z)d(\gamma z)^{\otimes k/2}=f(z)dz^{\otimes k/2},$

so the differential form $f(dz)^{\otimes k/2}\in
H^{0}(\Omega,\Omega_{\Omega}^{\otimes k/2})$ on the upper half-plane $\Omega$
is $\Gamma$-invariant. As in [GR96, Section $(2.10)$], we know $f(dz)^{\otimes
k/2}$ is holomorphic on $\Gamma\setminus\Omega.$ Since
$\displaystyle{\frac{de_{A}(z)}{dz}=1},$ we have
$\displaystyle{\frac{du}{u^{2}}=-\overline{\pi}dz,}$ so the differential $dz$
in this case has a double pole at $\infty.$ Then since $f$ is holomorphic at
the cusps of $\Gamma,$

$\operatorname{div}(f(dz)^{\otimes k/2})+k\Delta\geq 0,$

and therefore $f(dz)^{\otimes k/2}$ is a global section of the twist by
$2\Delta$ of sheaf of holomorphic differentials on the rigid analytic space
$X_{\Gamma}^{\text{an}}=\Gamma\setminus(\Omega\cup\mathbb{P}^{1}(K)).$ We
claim this is a global section of (a twist by $2\Delta$ of) the sheaf of
differentials on the algebraic stack $\mathscr{X}.$

By rigid analytic GAGA, [FvdP04, Theorem $4.10.5$], we know that the
categories of coherent sheaves on the rigid space
$\mathbb{P}^{n,\text{an}}_{C}$ and coherent sheaves on $\mathbb{P}^{n}_{C}$
are equivalent and every closed analytic subspace of
$\mathbb{P}^{n,\text{an}}_{C}$ is the analytification of some closed subspace
of $\mathbb{P}^{n}_{C}.$ So, the sheaf
$\Omega_{X_{\Gamma}^{\text{an}}}^{1}(2\Delta)$ corresponds to the sheaf
$\Omega^{1}_{X_{\Gamma}}(2\Delta)$ on the algebraic curve $X_{\Gamma}$ which
is the coarse space of $\mathscr{X}.$ Finally, by [PY16, Theorem $7.4$], we
know the sheaves $\Omega_{\mathscr{X}_{\Gamma}^{\text{an}}}^{1}(2\Delta)$ and
$\Omega^{1}_{\mathscr{X}_{\Gamma}}(2\Delta)$ on the rigid analytic stacky
curve and algebraic stacky curves $\mathscr{X}_{\Gamma}^{\text{an}}$ and
$\mathscr{X}_{\Gamma}$ respectively are equivalent.

∎

Next, we recall our second main result:

######  Theorem 6.2.

Let $q$ be a power of an odd prime. Let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a
congruence subgroup containing the diagonal matrices in $\mathrm{GL}_{2}(A).$
Let
$\Gamma_{2}=\\{\gamma\in\Gamma:\det(\gamma)\in(\mathbb{F}_{q}^{\times})^{2}\\}.$
Then $M(\Gamma)\cong M(\Gamma_{2}),$ with

$M_{k,l}(\Gamma_{2})=M_{k,l_{1}}(\Gamma)\oplus M_{k,l_{2}}(\Gamma)$

on each graded piece, where $l_{1},l_{2}$ are the two solutions to $k\equiv
2l\pmod{q-1}.$

######  Remark 6.3.

Let $\mathscr{X}_{\Gamma}$ and $\mathscr{X}_{\Gamma_{2}}$ be the (stacky)
Drinfeld modular curves which are the algebraizations of
$\Gamma\setminus(\Omega\cup\mathbb{P}^{1}(\mathbb{F}_{q}(T)))$ and
$\Gamma_{2}\setminus(\Omega\cup\mathbb{P}^{1}(\mathbb{F}_{q}(T)))$
respectively. Let $\displaystyle{D=K_{\mathscr{X}_{\Gamma}}+\Delta\sim
K_{X_{\Gamma}}+R+\Delta}$ and
$\displaystyle{D_{2}=K_{\mathscr{X}_{\Gamma_{2}}}+\Delta_{2}\sim
K_{X_{\Gamma_{2}}}+R_{2}+\Delta_{2}}$ be log canonical divisors on
$\mathscr{X}_{\Gamma_{2}}$ and $\mathscr{X}_{\Gamma_{2}},$ where
$K_{X_{\Gamma}}$ and $K_{X_{\Gamma_{2}}}$ are canonical divisors for the
coarse spaces of $\mathscr{X}_{\Gamma}$ and $\mathscr{X}_{\Gamma_{2}}$
respectively, and $\Delta$ and $\Delta_{2}$ are the log divisors of
$\mathscr{X}_{\Gamma}$ and $\mathscr{X}_{\Gamma_{2}}$ respectively.

  1. (1)

Suppose $\Gamma$ is “non-square.” Then $R=2R_{2}$ and each
$s_{2}\in\operatorname{supp}(\Delta_{2})$ has
$\\#(\Gamma_{2})_{s_{2}}=\frac{1}{2}(\\#\Gamma_{s})$ for any
$s\in\operatorname{supp}(\Delta).$

  2. (2)

Suppose $\Gamma$ is “square.” Then $M(\Gamma_{2})=M(\Gamma),$ and
$K_{\mathscr{X}_{\Gamma_{2}}}+\Delta_{2}=K_{\mathscr{X}_{\Gamma}}+\Delta.$

Since there are many intermediate lemmata involved, we break the proof of
Theorem 6.2 up into the next few parts of this section. We state and prove the
generalization afterwards.

###  6.1. Properties of $\Gamma_{2}$

We begin with some group theory and elementary number theory which inspired
our second main result and is instrumental in its proof.

######  Lemma 6.4.

Let $\Gamma\leq GL_{2}(A)$ be a congruence subgroup containing the diagonal
matrices in $\mathrm{GL}_{2}(A).$ Let
$\Gamma_{2}=\\{\gamma\in\Gamma:(\det\gamma)\in(\mathbb{F}_{q}^{\times})^{2}\\}.$
Then $\Gamma_{2}$ is a normal subgroup of $\Gamma$ with
$[\Gamma:\Gamma_{2}]=2,$ and for any
$\alpha\in\mathbb{F}_{q}^{\times}\setminus(\mathbb{F}_{q}^{\times})^{2},$ the
matrix $\left(\begin{array}[]{cc}\alpha&0\\\ 0&1\end{array}\right)$ is a
representative for the unique non-trivial left coset of $\Gamma_{2}$ in
$\Gamma.$

###### Proof.

Let $\varphi:\Gamma\to\mathbb{F}_{q}^{\times}$ be the map
$\gamma\mapsto(\det\gamma)^{(q-1)/2}.$ Then since $(\det\gamma)^{q-1}=1$ for
all $\gamma\in\Gamma,$ we see $\ker\varphi=\Gamma_{2}.$ If
$\gamma\in\Gamma\setminus\Gamma_{2}$ then
$(\det\gamma)^{(q-1)/2}\equiv-1\pmod{q-1}$ so
$\varphi(\Gamma)\cong\mathbb{Z}/2\mathbb{Z}$ as multiplicative groups and
$[\Gamma:\Gamma_{2}]=2.$

If $\gamma\in\Gamma\setminus\Gamma_{2},$ i.e.
$\det(\gamma)\in\mathbb{F}_{q}^{\times}\setminus(\mathbb{F}_{q}^{\times})^{2},$
then for any
$\alpha\in\mathbb{F}_{q}^{\times}\setminus(\mathbb{F}_{q}^{\times})^{2}$ there
is some $\gamma_{2}\in\Gamma_{2}$ with

$\gamma=\left(\begin{array}[]{cc}\alpha&0\\\ 0&1\end{array}\right)\gamma_{2}.$

∎

We recall from elementary number theory the following.

######  Lemma 6.5.

Suppose $q$ is odd. For a fixed $k,~{}l$ is such that $k\equiv 2l\pmod{q-1},$
if and only if

$l\equiv\begin{cases}\frac{k}{2}\pmod{q-1},&\text{ or }\\\
\frac{k}{2}+\frac{q-1}{2}\pmod{q-1}.\end{cases}$

###### Proof.

We know that $2l\equiv k\pmod{q-1}$ if and only if $2l-m(q-1)=k$ for some
integer $m.$ If $\gcd(2,q-1)$ does not divide $k$ then there are no solutions,
and if it does then there are exactly $\gcd(2,q-1)=2$ distinct solutions
modulo $q-1.$ ∎

###  6.2. Cusps and Elliptic Points

We wish to compare the cusps and elliptic points on the Drinfeld modular
curves for $\Gamma$ and $\Gamma_{2}.$ As our notion of elliptic point is
slightly different from Gekeler’s, so that it adapts to the notion of a stacky
Drinfeld modular curve more naturally, we discuss some of the properties of
elliptic points with the next two group-theoretic results.

######  Lemma 6.6.

Let $q$ be a power of an odd prime, let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a
congruence subgroup containing the diagonal matrices in $\mathrm{GL}_{2}(A).$
Suppose $e_{1}$ and $e_{2}$ are distinct elliptic points for $\Gamma.$ Then
the stabilizers $\Gamma_{e_{1}}$ and $\Gamma_{e_{2}}$ are
$\mathrm{GL}_{2}(A)$-conjugate.

###### Proof.

Since both $\Gamma_{e_{1}}$ and $\Gamma_{e_{2}}$ stricty contain
$\mathbb{F}_{q}^{\times}$ by definition of an elliptic point, and each
stabilizer is a subgroup of $\mathrm{GL}_{2}(A)_{e_{i}},$ where $i=1$ or $2,$
both elliptic points for $\Gamma$ are also elliptic points on $\Omega.$ Then
$e_{1}$ and $e_{2}$ are $\mathrm{GL}_{2}(A)$-equivalent to each other. If
$\gamma e_{1}=e_{2}$ for $\gamma\in\mathrm{GL}_{2}(A)$ and
$\gamma^{\prime}\in\Gamma_{e_{1}},$ then

$\begin{array}[]{rl}\gamma\gamma^{\prime}\gamma^{-1}e_{2}&=\gamma\gamma^{\prime}\gamma^{-1}(\gamma
e_{1})\\\ &=\gamma e_{1}\\\ &=e_{2}.\end{array}$

∎

######  Lemma 6.7.

Let $q$ be a power of an odd prime, let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a
congruence subgroup containing the diagonal matrices of $\mathrm{GL}_{2}(A).$
Let $\Gamma_{2}=\\{\gamma\in\Gamma:\det(\gamma)\in(\det\Gamma)^{2}\\}.$ Let
$e\in\operatorname{Ell}(\Gamma_{2}).$ Then

$[\Gamma_{e}:(\Gamma_{2})_{e}]=\begin{cases}1,&\text{if }\Gamma\text{ is
``square''}\\\ 2,&\text{if }\Gamma\text{ is ``non-square.''}\end{cases}$

###### Proof.

By definition, the stabilizer $\Gamma_{e}$ strictly contains
$\mathbb{F}_{q}^{\times}$ and as this is a subgroup of the stabilizer
$\mathrm{GL}_{2}(A)_{e},$ we see that $e$ is an elliptic point for
$\mathrm{GL}_{2}(A),$ i.e. an elliptic point on $\Omega,$ so we know
$\mathrm{GL}_{2}(A)_{e}\cong\mathbb{F}_{q^{2}}^{\times}.$ This means
$(\Gamma_{2})_{e}\unlhd\Gamma_{e}\unlhd\mathrm{GL}_{2}(A)_{e}\cong\mathbb{F}_{q^{2}}^{\times}$
and in particular since

$(\Gamma_{2})_{e}=\ker((\det)^{\frac{q-1}{2}}:\Gamma_{e}\to\mathbb{F}_{q}^{\times}),$

the result is immediate according to whether
$\displaystyle{(\det)^{\frac{q-1}{2}}}$ is surjective onto $\\{\pm 1\\}.$ By
Lemma 6.6, we need only check whether $\Gamma_{e}$ contains some $\gamma$ with
$\det\gamma\in\mathbb{F}_{q}^{\times}\setminus(\mathbb{F}_{q}^{\times})^{2}$
to determine the index of the stabilizer $(\Gamma_{2})_{e}$ for all elliptic
points $e.$ ∎

The main idea for this step of the proof of Theorem 6.2 is the following
comparison between elliptic points and cusps for $\Gamma$ and $\Gamma_{2}.$

######  Proposition 6.8.

Let $q$ be a power of an odd prime, let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a
congruence subgroup containing the diagonal matrices of $\mathrm{GL}_{2}(A).$
Let $\Gamma_{2}=\\{\gamma\in\Gamma:\det(\gamma)\in(\det\Gamma)^{2}\\}.$

  1. (1)

$\operatorname{Ell}(\Gamma)=\operatorname{Ell}(\Gamma_{2}),$

  2. (2)

$\mathcal{C}_{\Gamma}\subseteq\mathcal{C}_{\Gamma_{2}}$

###### Proof.

Suppose $e_{2}\in\operatorname{Ell}(\Gamma_{2}),$ so by definition the
stabilizer $(\Gamma_{2})_{e_{2}}$ is strictly larger than
$\mathbb{F}_{q}^{\times}.$ Since $(\Gamma_{2})_{e_{2}}$ is a subgroup of
$\Gamma_{e_{2}},$ it must be that $\Gamma_{e_{2}}$ strictly contains
$Z(\mathbb{F}_{q}),$ so $e_{2}\in\operatorname{Ell}(\Gamma).$

For the same reason, if $e\in\operatorname{Ell}(\Gamma),$ then $e$ is an
elliptic point on $\Omega,$ and we know
$\mathrm{GL}_{2}(A)_{e}\cong\mathbb{F}_{q^{2}}^{\times}.$ In particular, as
$\mathbb{F}_{q^{2}}^{\times}$ and $\mathbb{F}_{q}^{\times}$ are cyclic groups,
we know $(\Gamma_{2})_{e}$ and $\Gamma_{e}$ are cyclic and we have $1\unlhd
Z(\mathbb{F}_{q})\unlhd(\Gamma_{2})_{e}\unlhd\Gamma_{e}\unlhd\mathrm{GL}_{2}(A)_{e}\cong\mathbb{F}_{q^{2}}^{\times}.$
Since $q-1\mid\\#\Gamma_{e},$ there is some $1<n\leq q+1$ such that $n\mid
q-1$ and $\\#\Gamma_{e}=n(q-1).$ Suppose that
$\langle\gamma\rangle=\Gamma_{e}.$ Then the left cosets of
$\mathbb{F}_{q}^{\times}$ in $\Gamma_{e}$ have representatives

$\gamma^{j}\left(\begin{array}[]{cc}\frac{1}{\alpha_{i}}&0\\\
0&\frac{1}{\alpha_{i}}\end{array}\right)$

for $1\leq j\leq n(q-1)$ and $\alpha_{i}\in\mathbb{F}_{q}^{\times},$ so we can
write

$\Gamma_{e}/\mathbb{F}_{q}^{\times}\cong\mathbb{F}_{q}^{\times}\oplus\frac{\gamma}{\alpha_{0}}\mathbb{F}_{q}^{\times}\oplus\cdots\oplus\frac{\gamma}{\alpha_{q-1}}\mathbb{F}_{q}^{\times}\oplus\frac{\gamma^{2}}{\alpha_{0}}\mathbb{F}_{q}^{\times}\oplus\cdots\oplus\frac{\gamma^{(n-1)(q-1)}}{\alpha_{q-1}}\mathbb{F}_{q}^{\times}.$

We claim that if $\Gamma$ is “non-square,” the cosets with representatives
$\gamma^{j}/\alpha_{i}$ with $j$ even form a subgroup isomorphic to
$(\Gamma_{2})_{e}/\mathbb{F}_{q}^{\times}.$ If $\Gamma$ is “non-square” then
by Lemma 6.7 we know that $\Gamma_{e}$ contains some $\gamma^{\prime}$ with
$\det\gamma^{\prime}$ a non-square, so $\det\gamma$ is non-square. Otherwise
we would have $\gamma^{n}=\gamma^{\prime}$ for some $n$ and with
$\det\gamma\in(\mathbb{F}_{q}^{\times})^{2},$ we would have
$\det\gamma^{\prime}$ a square which is a clear contradiction. Then for any
even $j$ we have

$\det\left(\frac{\gamma^{j}}{\alpha}\right)=\frac{\det\gamma^{j}}{\alpha^{2}}$

is a quotient of squares so is a square. For odd $j,$ since
$\det\gamma^{j}\in\mathbb{F}_{q}^{\times}\setminus(\mathbb{F}_{q}^{\times})^{2}$
then for any $\alpha^{\prime}\in\mathbb{F}_{q}^{\times}$ non-square, there is
some $\gamma_{2}\in\Gamma_{2}$ such that
$\displaystyle{\gamma=\left(\begin{array}[]{cc}\alpha&0\\\
0&1\end{array}\right)\gamma_{2}},$ but whether a given $\alpha_{i}$ is a
square or not,

$\det(\gamma^{j}/\alpha_{i})=\frac{\alpha^{\prime}\det\gamma_{2}}{\alpha_{i}^{2}},$

which is not a square. Otherwise if $\Gamma$ is “square,” by Lemma 6.7 we have
$\Gamma_{e}=(\Gamma_{2})_{e}.$ Whether $\Gamma$ is square or not,
$(\Gamma_{e})/\mathbb{F}_{q}^{\times}$ has a nontrivial subgroup isomorphic to
$(\Gamma_{2})_{e}/\mathbb{F}_{q}^{\times},$ so the stabilizer of $e$ in
$\Gamma_{2}$ strictly contains $\mathbb{F}_{q}^{\times}$ and
$e\in\operatorname{Ell}(\Gamma_{2}).$

Let $s\in\mathbb{P}^{1}(K).$ Then $\Gamma s\supseteq\Gamma_{2}s,$ i.e. the
action of $\Gamma_{2}$ partitions $\mathbb{P}^{1}(K)$ more finely than the
action of $\Gamma.$ If $s_{1},\cdots,s_{n}$ are cusps of $\Gamma,$ we write
$\Gamma\setminus\mathbb{P}^{1}(K)=\Gamma s_{1}\sqcup\cdots\sqcup\Gamma s_{n},$
and then

$\Gamma s_{i}=\Gamma_{2}s_{i}\sqcup(\Gamma\setminus\Gamma_{2})s_{i}.$

If the points of $\mathbb{P}^{1}(K)$ in the orbits
$(\Gamma\setminus\Gamma_{2})s_{i},$ under the action by $\Gamma_{2}$ have
orbit representatives $t_{1},\cdots,t_{m}$ then we can write

$\Gamma_{2}\setminus\mathbb{P}^{1}(K)=\Gamma_{2}s_{1}\sqcup\cdots\sqcup\Gamma_{2}s_{n}\sqcup\Gamma_{2}t_{1}\sqcup\cdots\sqcup\Gamma_{2}t_{m},$

so the cusps of $\Gamma_{2}$ are
$\mathcal{C}_{\Gamma_{2}}=\\{s_{1},\cdots,s_{n},t_{1},\cdots,t_{m}\\},$ which
contains $\mathcal{C}_{\Gamma}.$

∎

######  Corollary 6.9.

Let $q$ be a power of an odd prime. Let $\Gamma\leq GL_{2}(A)$ be a congruence
subgroup. Let $\Gamma_{1}=\\{\gamma\in\Gamma:\det(\gamma)=1\\}.$ Suppose that
$\Gamma_{1}\leq\Gamma^{\prime}\leq\Gamma$ for some congruence subgroup
$\Gamma^{\prime}.$ Then
$\mathcal{C}_{\Gamma}\subseteq\mathcal{C}_{\Gamma^{\prime}},$ i.e. the cusps
of $\Gamma$ are some subset of the cusps of $\Gamma^{\prime}$

###### Proof.

The proof of the second part of Proposition 6.8 about the cusps did not make
any particular use of the special choice of $\Gamma^{\prime}=\Gamma_{2},$ and
so holds in this more general situation. ∎

###  6.3. Modularity and Series Expansions at Cusps

Our next steps in the proof of Theorem 6.2 deal with the $u$-series expansions
of modular forms.

######  Proposition 6.10.

Let $f$ be holomorphic on $\Omega$ and at the cusps of $\Gamma_{2},$ and let
$\beta=\alpha^{2}\in\mathbb{F}_{q}^{\times},$ where $\alpha$ generates
$\mathbb{F}_{q}^{\times}.$ Suppose that $f(\gamma
z)=(\det\gamma)^{-l}(cz+d)^{k}f(z)$ for
$\displaystyle{\gamma=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\Gamma_{2}}.$ Then

$f\left(\\!\left(\begin{array}[]{cc}\beta&0\\\
0&1\end{array}\right)z\\!\right)=f(\beta z)=\beta^{-k/2}f(z).$

###### Proof.

Since $\displaystyle{\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha\end{array}\right)\in\Gamma_{2}}$ we have
$\displaystyle{f\left(\frac{\alpha
z}{\alpha}\right)=f(z)=\alpha^{-2l}\alpha^{k}f(z)},$ so $\alpha^{k-2l}=1.$

Suppose that $x$ generates the cyclic group $\mathbb{F}_{q}^{\times},$ so
$\alpha=x^{n}$ for some $n.$ If $\gcd(n,q-1)=1$ (i.e. $\alpha$ is a generator)
we have $k\equiv 2l\pmod{q-1}.$ The order of $\alpha$ is

$\\#\langle\alpha\rangle=\frac{q-1}{\gcd(n,q-1)}=q-1,$

and we can write $k\equiv 2l\pmod{\\#\langle\alpha\rangle}$ so

$\gcd(n,q-1)(k-2l)\equiv 0\pmod{q-1},$

but $\gcd(n,q-1)$ is coprime to $q-1,$ from which the claim follows.

Since $\beta=\alpha^{2},$ we have$\beta^{2(k-2l)}=1,$ so

$2k=4l\ (\mathrm{mod}\ q-1).$

Then we have $k\equiv 2l\pmod{\frac{q-1}{2}},$ since if $2k=m(q-1)+4l$ for
some $m,$ we can write

$k=m\left(\frac{q-1}{2}\right)+2l.$

Then $\displaystyle{l\equiv\frac{k}{2}\pmod{\frac{q-1}{2}}},$ so

$f\left(\\!\left(\begin{array}[]{cc}\beta&0\\\
0&1\end{array}\right)z\\!\right)=f(\beta z)=\beta^{-k/2}f(z),$

since $\left(\begin{array}[]{cc}\beta&0\\\
0&1\end{array}\right)\in\Gamma_{2}.$ This matrix has square determinant by
assumption and is in $\Gamma$ by our assumption that $\Gamma$ contains all
matrices of form $\displaystyle{\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha^{\prime}\end{array}\right)},$ for any
$\alpha,\alpha^{\prime}\in\mathbb{F}_{q}^{\times}.$ ∎

We complete the proof of Theorem 6.2 with the following result.

######  Proposition 6.11.

Suppose $\Gamma$ is “non-square.” Let $f$ be a modular form of weight $k$ and
type $l$ for $\Gamma_{2}.$ Then there are two modular forms $f_{1}$ and
$f_{2}$ for $\Gamma$ of weight $k$ and types $l_{1}\equiv k/2\ (\mathrm{mod}\
q-1)$ and $l_{2}\equiv k/2+(q-1)/2\ (\mathrm{mod}\ q-1)$ respectively, such
that $f=f_{1}+f_{2}.$

###### Proof.

Suppose that $f(\gamma_{2}z)=(\det\gamma_{2})^{-l}(cz+d)^{k}f(z)$ for
$\displaystyle{\gamma_{2}=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\Gamma_{2}}.$ Write the $u$-series $f(z)=\sum_{n\geq
0}a_{n}u^{n}.$ Let $\beta=\alpha^{2}\in\mathbb{F}_{q}^{\times},$ where
$\alpha$ generates $\mathbb{F}_{q}^{\times}.$ By Proposition 6.10, $f(\beta
z)=\beta^{-k/2}f(z).$ Using this relationship, we have from Lemma 3.3

$f(\beta z)=\sum_{n\geq 0}a_{n}\beta^{-n}u^{n}=\beta^{-k/2}\left(\sum_{n\geq
0}a_{n}u^{n}\right),$

so for each non-zero $a_{n}$ we have $\beta^{-n}=\beta^{-k/2}$ or
$\alpha^{-2n}=\alpha^{-k}.$ Then $k\equiv 2n\pmod{q-1},$ so by removing the
zero summands from the $u$-series, we may write

$f(z)=\sum_{n\equiv k/2\ (\mathrm{mod}\ q-1)}a_{n}u^{n}+\sum_{n\equiv
k/2+(q-1)/2\ (\mathrm{mod}\ q-1)}a_{n}u^{n}.$

Let $\alpha\in\mathbb{F}_{q}^{\times}$ be some non-square, so by Lemma 3.3 we
have $u(\alpha z)=\alpha^{-1}u(z).$ Let

$f_{1}=\sum_{n\equiv k/2\ (\mathrm{mod}\ q-1)}a_{n}u^{n}$

and

$f_{2}=\sum_{n\equiv k/2+(q-1)/2\ (\mathrm{mod}\ q-1)}a_{n}u^{n}$

be the modular forms for $\Gamma_{2}$ uniquely determined by their $u$-series
by Lemma 3.9. Then

$f_{1}(\alpha z)=\sum_{n\equiv k/2\ (\mathrm{mod}\
q-1)}a_{n}\alpha^{-n}u^{n}=\alpha^{-l_{1}}\sum_{n\equiv k/2\ (\mathrm{mod}\
q-1)}a_{n}u^{n},$

where $\displaystyle{l_{1}\equiv\frac{k}{2}\pmod{q-1}}.$ Let
$\gamma\in\Gamma\setminus\Gamma_{2}.$ For any
$\alpha\in\mathbb{F}_{q}^{\times}\setminus(\mathbb{F}_{q}^{\times})^{2}$ there
is some $\displaystyle{\gamma_{2}=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\Gamma_{2}}$ such that

$\gamma=\left(\begin{array}[]{cc}\alpha&0\\\ 0&1\end{array}\right)\gamma_{2},$

so

$f_{1}(\gamma
z)=f_{1}(\alpha\gamma_{2}z)=\alpha^{-l}f_{1}(\gamma_{2}z)=\alpha^{-l}\det(\gamma_{2})^{-l}(cz+d)^{k}f_{1}(z)=\det(\gamma)^{-l}(cz+d)^{k}f_{1}(z)$

and $f_{1}$ is a modular form for $\Gamma.$ Likewise we have

$f_{2}(\alpha z)=\sum_{n\equiv k/2+(q-1)/2\ (\mathrm{mod}\
q-1)}a_{n}\alpha^{-n}u^{n}=\alpha^{-l_{2}}\sum_{n\equiv k/2+(q-1)/2\
(\mathrm{mod}\ q-1)}a_{n}u^{n},$

where now $\displaystyle{l_{2}\equiv\frac{k+q-1}{2}}\pmod{q-1},$ so for
$\gamma,\alpha$ and $\gamma_{2}$ as above,

$f_{2}(\gamma z)=\alpha^{-l}\det(\gamma_{2})^{-l}(cz+d)^{k}f_{2}(z)$

and $f_{2}$ is a modular form for $\Gamma.$ ∎

###  6.4. Generalization

We will show that Theorem 6.2 is actually a special case of the following
result.

######  Theorem 6.12.

Let $q$ be a power of an odd prime. Let $\Gamma\leq\mathrm{GL}_{2}(A)$ be a
congruence subgroup. Let $\Gamma_{1}=\\{\gamma\in\Gamma:\det(\gamma)=1\\}.$
Suppose that $\Gamma_{1}\leq\Gamma^{\prime}\leq\Gamma$ for some congruence
subgroup $\Gamma^{\prime}.$ Then as algebras

$M(\Gamma)=M(\Gamma^{\prime}),$

and each component $M_{k,l}(\Gamma^{\prime})$ is some direct sum of components
$M_{k,l^{\prime}}(\Gamma)$ for some nontrivial $l^{\prime}.$

######  Remark 6.13.

The subgroups $\Gamma^{\prime}$ which appear in the statement of Theorem 6.12
may be thought of as the inverse image under
$\det:\Gamma\to\mathbb{F}_{q}^{\times}$ of some subgroup of
$\mathbb{F}_{q}^{\times}.$

###### Proof.

(Theorem 6.12) Write $f|_{\gamma}$ for the (Peterson) slash operator of weight
$k$ and type $l$ for $\displaystyle{\gamma=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathrm{GL}_{2}(K)}$ defined by

$f|_{\gamma}\overset{def}{=}\det(\gamma)^{l}(cz+d)^{-k}f(\gamma z).$

If $f\in M_{k,l}(\Gamma^{\prime}),$ by normality $\Gamma^{\prime}\unlhd\Gamma$
we have that $f|_{\gamma}$ is weakly modular of weight $k$ and type $l$ for
any $\gamma\in\Gamma.$ By Corollary 6.9 we see that $f|_{\gamma}$ is
holomorphic at the cusps of $\Gamma$ since $f$ is holomorphic at the cusps of
$\Gamma^{\prime},$ indeed the $u$-series of expansions of $f|_{\gamma}$ and
$f$ agree at the cusps of $\Gamma^{\prime}.$

The action of $\Gamma^{\prime}$ is trivial, so we have an action of the finite
group

$\Gamma/\Gamma^{\prime}=\det(\Gamma)/\det(\Gamma^{\prime}),$

which has order some divisor of $q-1$ since
$\\{1\\}\leq\det\Gamma^{\prime}\leq\det\Gamma\leq\mathbb{F}_{q}^{\times}.$
Then we may describe the group ring $\mathbb{F}_{q}[\Gamma/\Gamma^{\prime}]$
via idempotents as follows. Let
$n^{\prime}\overset{def}{=}\\#(\det\Gamma^{\prime})$ and let
$n\overset{def}{=}\\#(\det\Gamma).$ Then

$\mathbb{F}_{q}[\Gamma/\Gamma^{\prime}]=\bigoplus_{i=0}^{n/n^{\prime}-1}\mathbb{F}_{q}e_{i},$

where $\Gamma$ acts on the $e_{i}$ via maps
$\gamma\mapsto(\det\gamma)^{in^{\prime}}.$ So as $\Gamma$-modules, we have

$M_{k,l}(\Gamma^{\prime})=\bigoplus_{i}M_{k,l}(\Gamma^{\prime})e_{i},$

where

$M_{k,l}(\Gamma^{\prime})e_{i}=M_{k,l+in^{\prime}}(\Gamma).$

Finally, since modular forms for $\Gamma^{\prime}$ are holomorphic at the
cusps of $\Gamma^{\prime},$ and by Corollary 6.9 the cusps of $\Gamma$ are a
subset of the cusps of $\Gamma^{\prime},$ we know $\Gamma^{\prime}$-modular
forms are holomorphic at the cusps of $\Gamma.$ ∎

######  Remark 6.14.

One can verify that the slash operators $f|_{\gamma}$ are holomorphic at the
cusps of $\Gamma$ directly by considering their $u$-series expansions at small
neighborhoods of the cusps of $\Gamma.$

######  Remark 6.15.

Theorem 6.2 is just the special case of Theorem 6.12 when
$\Gamma^{\prime}=\Gamma_{2}.$ We highlight the special case Theorem 6.2 in
this article because of its relationship with the other main result Theorem
6.1.

###  6.5. Summary

Our first result in this section, Theorem 6.1, tell us about the geometry of
Drinfeld modular forms for congruence subgroups consisting of matrices with
square determinants.

According to whether a given congruence subgroup $\Gamma$ is “square” or not,
we can decompose the algebra of Drinfeld modular forms for $\Gamma_{2}$ with
Theorem 6.2. We have shown a modular form $f$ of weight $k$ and type $l$ for
$\Gamma_{2}$ is holomorphic at the cusps of $\Gamma,$ and there are two
choices of type $l_{1}$ and $l_{2},$ the lifts of a given
$\displaystyle{l\equiv k/2\ (\mathrm{mod}\ \frac{q-1}{2})}$ to
$\mathbb{Z}/(q-1)\mathbb{Z}$ such that $f$ may be a sum of modular forms of
weight $k$ and type either $l_{1}$ or $l_{2}$ for $\Gamma,$ if $\Gamma$ is
“non-square.” Together, these conditions are the definition of a modular form,
so every modular form for $\Gamma_{2}$ is associated to a pair of $\Gamma$
modular forms in this case. Finally, we generalize this decomposition with
Theorem 6.12.

##  7\. Application: Computing Algebras of Drinfeld Modular Forms

We have seen some relationships between the modular forms for $\Gamma$ and
$\Gamma_{2}.$ Now we apply the geometry of those modular forms, in particular
in terms of stacks. We conclude the article with some examples of how we
intend to use geometric invariants to compute algebras of Drinfeld modular
forms. Our examples of Drinfeld modular curves have genus $0$ since we need
techniques for $\mathbb{Q}$-divisors such as in [O’D15] to compute the
canonical ring of a Drinfeld modular curve. In particular, the results in
[VZB22] do not apply to log curves where the cuspidal divisor $\Delta$ has
coefficients besides $1$ or is supported at stacky points. Generalizations of
both [VZB22] for all divisors with $\mathbb{Q}$-coefficients, and [O’D15] to
higher genera cases have not yet been published, but are in progress.

As a first example, we will recall the computation of canonical rings for
classical modular curves in [VZB22], and see how differs from the case of a
Drinfeld modular curve, as we demonstrate that
$M((\mathrm{GL}_{2}(A))_{2})=C[g,h].$ Since $\mathrm{GL}_{2}(A)$ is “non-
square,” we first reduce to $\mathrm{GL}_{2}(A)_{2}$ according to Theorem 6.2.
Then we use the geometry of modular forms for this smaller group, Theorem 6.1,
and geometric invariants of the modular curve for $\mathrm{GL}_{2}(A)_{2}$ to
compute a canonical ring as in [VZB22].

In the classical setting, modular curves are tame stacky curves, and likewise
in the Drinfeld setting as we will explain next. As in [Gek99, Definition
$(3.5)$], for $f$ some Drinfeld modular form, we let $v_{z}(f)$ denote the
vanishing order of $f$ at $z\in\Omega$ and $v_{\infty}(f)$ denote the
vanishing order of $f$ at $\infty.$ Then from [Gek99, Equation $(3.10)$]:

$\sideset{}{{}^{*}}{\sum}_{z\in
GL_{2}(A)\setminus\Omega}v_{z}(f)+\frac{v_{e}(f)}{q+1}+\frac{v_{\infty}(f)}{q-1}=\frac{k}{q^{2}-1},$

where $\sum^{*}$ denotes a sum over non-ellitic classes of
$\mathrm{GL}_{2}(A)\setminus\Omega.$ In particular, the characteristic $q$ of
$C$ does not divide the degree of the stabilizer of any point on a Drinfeld
modular curve, as $\deg G_{x}$ divides $q^{2}-1$ for all points $x.$

Next we turn to cusps of modular curves, where the classical and Drinfeld
settings begin to differ. From [VZB22, Definition $5.6.2$] we say a divisor
$\Delta$ on a tame stacky curve $\mathscr{X}$ is a log divisor if
$\Delta=\sum_{i}P_{i}$ is an effective divisor on $X$ given as a sum of
distinct non-stacky points on $X.$ Both in the classical and Drinfeld
settings, cusps of a modular curve are stable under Möbius transformations by
diagonal matrices. However, whereas a classical modular curve is a quotient of
the upper half-plane $\mathcal{H}=\\{z=a+bi\in\mathbb{C}:b>0\\}$ by a
congruence subgroup of $\mathrm{SL}_{2}(\mathbb{Z}),$ so the cusps of a
classical modular curve are not stacky points, in the Drinfeld setting the
divisor of cusps of a modular curve should be regarded as an effective divisor
which is a formal sum of distinct stacky points. Indeed, diagonal matrices in
$\mathrm{GL}_{2}(A)$ have determinants in $\mathbb{F}_{q}^{\times}$ as opposed
to determinant $1$ in the classical case of $\mathrm{SL}_{2}(\mathbb{Z})$
acting on the upper half-plane $\mathcal{H}$ of $\mathbb{C},$ so a log divisor
in the Drinfeld setting may have coefficients besides $1.$

We can compute section rings for general $\mathbb{Q}$-divisors on genus $0$
curves using [O’D15], so we will consider the coefficients of log canonical
divisors for Drinfeld modular curves in more detail. In particular, we compare
the stabilizers of stacky points for $\mathrm{GL}_{2}(A)$ and
$\mathrm{GL}_{2}(A)_{2},$ since we use these to write down log canonical
divisors for the stacky curves associated with these groups.

Recall the parameter at $\infty$ in the Drinfeld setting, introduced in
Definition 3.1. Since $\displaystyle{\frac{de_{A}(z)}{dz}=1},$ we have
$du=-\overline{\pi}u^{2}dz,$ so the differential $dz$ in this case has a
double pole at $\infty.$ But $\infty$ is stabilized by upper triangular
matrices in $\mathrm{GL}_{2}(A)$ as the group of upper triangular matrices is
strictly larger than
$\displaystyle{\\#\left\\{\left(\begin{array}[]{cc}\alpha&0\\\
0&\alpha\end{array}\right):\alpha\in\mathbb{F}_{q}^{\times}\right\\}}=q-1,$
the point $\infty$ is an elliptic point of $\mathrm{GL}_{2}(A)$ and hence a
stacky point for $\mathscr{X}.$ In fact, both stacky points on
$\mathscr{X}_{\mathrm{GL}_{2}(A)_{2}}$, the unique $e$ on $\Omega$ from
Definition 4.10 and $\infty$ have stabilizers half the order of their
stabilizers in $\mathrm{GL}_{2}(A),$ which comes from the double cover
$\operatorname{Spec}C[\tilde{\j}]\to\operatorname{Spec}C[j]$ (see [Bre16, Page
$7$]) which is ramified above $j=0$ and $\infty$ and the fact that
$\mathrm{GL}_{2}(A)$ is “non-square” so that
$[\mathrm{GL}_{2}(A)_{e}:(\mathrm{GL}_{2}(A)_{2})_{e}]=2$ by Lemma 6.7.

We recall from [VZB22, Proposition $5.5.6$] that for $K_{\mathscr{X}}$ a
canonical divisor on a tame stacky curve $\mathscr{X}$ with coarse space $X$
and canonical divisor $K_{X},$ if $x$ is a geometric point of $\mathscr{X}$
with non-trivial stabilizer, there is a linear equivalence

$K_{\mathscr{X}}\sim K_{X}+\sum_{x}\left(1-\frac{1}{\deg G_{x}}\right)x.$

Our main object of interest is defined in [VZB22, Defintions $5.6.1$ and
$5.6.2$] which we generalize slightly to allow for stacky points in a log
divisor: the canonical ring of a log stacky curve is the ring

$R_{D}=\bigoplus_{d\geq 0}H^{0}(\mathscr{X},dD),$

where $D=K_{\mathscr{X}}+\Delta,$ for $\Delta$ the divisor of cusps of
$\mathscr{X},$ which we call a log divisor.

We summarize the above and calculate a log canonical ring in the following
example.

######  Example 7.1.

Let $\mathscr{X}$ be the Drinfeld modular curve with coarse space $X$ whose
analytification is
$X^{\text{an}}=\mathrm{GL}_{2}(A)_{2}\setminus(\Omega\cup\mathbb{P}^{1}(K)).$
Then $\mathscr{X}$ is a stacky $\mathbb{P}^{1}$ with two stacky points:

  * •

a point $P_{e}$ with a stabilizer of order $\displaystyle{\frac{q+1}{2}}$
corresponding to the unique elliptic point of $\Omega$ (note that
$\mathrm{GL}_{2}(A)$ is “non-square”)

  * •

a cusp, denoted $\infty,$ with a stabilizer of order
$\displaystyle{\frac{q-1}{2}}.$

Let

$D=K_{\mathscr{X}}+2\Delta\sim
K_{\mathbb{P}^{1}}+\left(1-\frac{1}{\frac{q+1}{2}}\right)P_{e}+\left(1+\frac{1}{\frac{q-1}{2}}\right)\infty+2\infty$

be a log stacky canonical divisor on $\mathscr{X}.$ Then by [O’D15, Theorem
$6$] we have

$R_{D}\cong C[g,h]\cong M(\mathrm{GL}_{2}(A)_{2}).$

To conclude, we present some new example computations of algebras of Drinfeld
modular forms. The idea is to illustrate the role of our theory in this
calculation, and the limited scope of our results now indicates a clear
direction for future work.

As with our previous example, since the existing theory is most developed in
genus $0,$ we begin by seeking Drinfeld modular curves in genus $0.$ We know
from [Gek01, Theorem $8.1$] genus formulae for the modular curves associated
to $\Gamma(N),\Gamma_{1}(N)$ and $\Gamma_{0}(N),$ where we recall

$\Gamma_{1}(N)=\left\\{\left(\begin{array}[]{cc}1&*\\\
0&*\end{array}\right)\pmod{N}\right\\}\text{ and
}\Gamma_{0}(N)=\left\\{\left(\begin{array}[]{cc}*&*\\\
0&*\end{array}\right)\pmod{N}\right\\}.$

If $\deg N>1,$ then $g(X(N))>0,$ so we consider the case of linear level.
Cornelissen has two papers [Cor97a] and [Cor97b] dedicated to the Drinfeld
modular forms for $\Gamma(\alpha T+\beta),$ so we consider
$M(\Gamma_{1}(\alpha T+\beta))$ and $M(\Gamma_{0}(\alpha T+\beta)).$ In fact,
we know from [DK23, Theorem $4.4$] that for $R$ any ring such that $A\subset
R\subset C,$ the $R$-algebra of Drinfeld modular forms $M(\Gamma_{0}(T))_{R}$
is generated by $E_{T}(z)$ (from Example 3.12), and the Drinfeld modular forms

$\Delta_{T}(z)\overset{def}{=}\frac{g(Tz)-g(z)}{T^{q}-T}\text{ and
}\Delta_{W}(z)\overset{def}{=}\frac{T^{q}g(Tz)-Tg(z)}{T^{q}-T}$

for $\Gamma_{0}(T)$ (from [DK23, Equation $(4.1)$]). Furthermore, [DK23,
Theorem $4.4$] tell us that the surjective map $R[U,V,Z]\to
M(\Gamma_{0}(T))_{R}$ defined by $U\to\Delta_{W},$ $V\to\Delta_{T}$ and $Z\to
E_{T}$ induces an isomorphism

$R[U,V,Z]/(UV-Z^{q-1})\cong M(\Gamma_{0}(T))_{R}.$

Note that from [DK23, Proposition $4.3(3)$] we know that
$M_{k,l}(\Gamma_{0}(T))$ has an integral basis, i.e. a basis consisting of
modular forms with coefficients in $A.$

To apply Theorem 6.1 we need the following geometric invariants of
$\mathscr{X}(\Gamma_{1}(\alpha T+\beta))$ and $\mathscr{X}(\Gamma_{0}(\alpha
T+\beta))$:

$\begin{array}[]{c|c|c}\text{Genera}&\text{Elliptic points}&\text{Cusps}\\\
\hline\cr\begin{array}[]{c}g(\mathscr{X}(\Gamma_{1}(\alpha T+\beta)))=0\\\
\text{(by \cite[cite]{[\@@bibref{}{Gekeler-Invariants}{}{}, Thm
$8.1.(\mathrm{ii})$]})}\end{array}&\begin{array}[]{c}\\#\operatorname{Ell}(\Gamma_{1}(\alpha
T+\beta))\geq 1\\\ \text{ramified with index }q+1\text{ over }X(1)\\\
\text{(\cite[cite]{[\@@bibref{}{Gekeler-Invariants}{}{}, Proposition
$7.2$]})}\end{array}&\begin{array}[]{c}\\#\mathcal{C}_{\Gamma_{1}(\alpha
T+\beta)}=2\\\ \text{(\cite[cite]{[\@@bibref{}{Gekeler-Invariants}{}{},
Proposition $6.6.(\mathrm{i})$]})}\end{array}\\\
\hline\cr\begin{array}[]{c}g(\mathscr{X}(\Gamma_{0}(\alpha T+\beta)))=0\\\
\text{(by \cite[cite]{[\@@bibref{}{Gekeler-Invariants}{}{}, Thm
$8.1.(\mathrm{iii})$]})}\end{array}&\begin{array}[]{c}\\#\operatorname{Ell}(\Gamma_{0}(\alpha
T+\beta))\geq 1\\\ \text{ramified with index }q+1\text{ over }X(1)\\\
\text{(\cite[cite]{[\@@bibref{}{Gekeler-Invariants}{}{}, Proposition
$7.3$]})}\end{array}&\begin{array}[]{c}\\#\mathcal{C}_{\Gamma_{0}(\alpha
T+\beta)}=2\\\ \text{(\cite[cite]{[\@@bibref{}{Gekeler-Invariants}{}{},
Proposition $6.7.(\mathrm{i})$]})}\end{array}\\\ \hline\cr
g(\mathscr{X}(\Gamma_{1}(\alpha
T+\beta)_{2}))&\begin{array}[]{c}\\#\operatorname{Ell}(\Gamma_{1}(\alpha
T+\beta)_{2})=\\#\operatorname{Ell}(\Gamma_{1}(\alpha T+\beta))\\\ \text{(by
Proposition \ref{p: elliptic points and
cusps})}\end{array}&\\#\mathcal{C}_{\Gamma_{1}(\alpha T+\beta)_{2}}\\\
\hline\cr g(\mathscr{X}(\Gamma_{0}(\alpha
T+\beta)_{2}))&\begin{array}[]{c}\\#\operatorname{Ell}(\Gamma_{0}(\alpha
T+\beta)_{2})=\\#\operatorname{Ell}(\Gamma_{0}(\alpha T+\beta))\\\ \text{(by
Proposition \ref{p: elliptic points and
cusps})}\end{array}&\\#\mathcal{C}_{\Gamma_{0}(\alpha T+\beta)_{2}}\\\
\hline\cr\end{array}$

Recall that from [DK23, Section $4$] we know the only two cusps of
$\Gamma_{0}(T),$ which we write $0$ and $\infty,$ are exchanged by the matrix

$W_{T}\overset{def}{=}\left(\begin{array}[]{cc}0&-1\\\ T&0\end{array}\right).$

While [Gek01, Proposition $7.2$] and [Gek01, Proposition $7.3$] give us some
way to compute the number of elliptic points, in particular Gekeler’s
definition of an elliptic point ([Gek01, $(3.2)$] \- the class of an elliptic
point on $\Omega$ in $Y_{\Gamma}$) is slightly different from ours (recall
Definition 4.10). For our calculation to work, we must consider all points
from $\Omega$ on $X_{\Gamma_{1}(\alpha T+\beta)}$ and $X_{\Gamma_{0}(\alpha
T+\beta)}$ whose stabilizers under $\Gamma_{1}(\alpha T+\beta)$ and
$\Gamma_{0}(\alpha T+\beta)$ respectively strictly contain
$\mathbb{F}_{q}^{\times}.$ Furthermore, we need to know the order of the
stabilizers of each elliptic point for $\Gamma_{1}(\alpha T+\beta)_{2}$ and
$\Gamma_{0}(\alpha T+\beta)_{2},$ which depends on whether the congruence
subgroups $\Gamma_{1}(\alpha T+\beta)$ and $\Gamma_{0}(\alpha T+\beta)$ are
“square” by Proposition 6.8.

######  Conjecture 7.2.

Both $\Gamma_{1}(\alpha T+\beta)$ and $\Gamma_{0}(\alpha T+\beta)$ may be
“non-square” congruence subgroups according to the specific choice of level.

######  Remark 7.3.

We see this explictly for sufficiently small $q$ by means of the following
algorithm:

  1. (1)

Fix a level $N=\alpha T+\beta$ for $\alpha\in\mathbb{F}_{q}^{\times}$ and
$\beta\in\mathbb{F}_{q}.$

  2. (2)

For all $a,b,c,d\in\mathbb{F}_{q},$ compute the determinant $(aN+1)d-bcN.$ If
this is an element of $\mathbb{F}_{q}^{\times},$ then
$\displaystyle{\gamma\overset{def}{=}\left(\begin{array}[]{cc}aN+1&b\\\
cN&d\end{array}\right)\in\Gamma_{1}(\alpha T+\beta)}$ (so
$\gamma\in\Gamma_{0}(\alpha T+\beta)$ as well).

  3. (3)

If $c\neq 0$ and the polynomial
$\displaystyle{z^{2}+\frac{d-(aN+1)}{cN}z-\frac{b}{cN}}$ is irreducible over
$K,$ then we know $\gamma\in\Gamma_{i}(\alpha
T+\beta)_{e}\setminus\mathbb{F}_{q}^{\times}$ is a non-trivial stabilizer of
the elliptic point.

This irreducibility condition follows from the fact that for
$z\in\Omega=C-K_{\infty},$ and we have

$\frac{az+b}{cz+d}=z\iff\begin{cases}a=d\text{ and }b=0=c,&\text{ or }\\\
cz^{2}+(d-a)z-b=0&\text{ irreducible,}\end{cases}$

since if this polynomial in $A[z]$ had a solution it would be an element of
$K,$ which is a contradiction to our definition of $z.$

We see that by brute-force consideration of only the constants in $A,$ as long
as $q\leq 25,$ that we can not only find such irreducibles, but we can do so
in such a way that every element of $\mathbb{F}_{q}^{\times}$ is the
determinant of some matrix in the stabilizer $\Gamma_{1}(\alpha T+\beta)_{e},$
and hence also $\Gamma_{0}(\alpha T+\beta)_{e}.$ Therefore, it seems
reasonable to believe that by enumerating all polynomials in $A,$ and for
larger characteristic, that these conclusions will remain the same.

######  Example 7.4.

For completeness, we illustrate a non-trivial, non-square matrix in
$\Gamma_{1}(4T+3)_{e}$ in the case $q=7.$ We have
$\displaystyle{\gamma=\left(\begin{array}[]{cc}4T+4&1\\\
5T+2&3\end{array}\right)}$ since $4T+4\equiv 1\pmod{4T+3}$ and $5T+2\equiv
0\pmod{4T+3}$ (since $5T+2=3(4T+3)$) and Sage tells us the corresponding
polynomial $\displaystyle{z^{2}+\frac{2T+4}{T+6}z+\frac{4}{T+6}}$ is
irreducible. Note that $\det\gamma=3,$ which is not a square in
$\mathbb{F}_{7}.$

Gekeler explains multiple techniques for computing the genus of a Drinfeld
modular curve in [Gek01, Section $8$], but to avoid repeating too much of his
notation, we describe just two:

  1. (1)

Compute fibers of the ramified graph coverings $\Gamma\setminus\mathscr{T}\to
GL_{2}(A)\setminus\mathscr{T},$ where $\mathscr{T}$ is the Bruhat-Tits tree

  2. (2)

Riemann-Hurwitz and [Gek01, Proposition $8.3$].

The content of [Gek01, Proposition $8.3$] is that canonical coverings of
Drinfeld modular curves have the least cuspidal ramification allowed by the
group structure of the stabilizers of cusps and the only ramification possible
is at elliptic points or cusps. This theory applies to the covers

$\mathscr{X}(\Gamma_{1}(\alpha T+\beta)_{2}))\to\mathscr{X}(\Gamma_{1}(\alpha
T+\beta))\text{ and }\mathscr{X}(\Gamma_{0}(\alpha
T+\beta)_{2})\to\mathscr{X}(\Gamma_{0}(\alpha T+\beta)),$

which are canonical in the sense that by the universal property of pull-backs
there are maps

$\psi_{i}:\mathscr{X}(\Gamma_{i}(\alpha
T+\beta)_{2})\to\mathscr{X}(\Gamma_{i}(\alpha
T+\beta))\times_{\mathscr{X}(\mathrm{GL}_{2}(A))}\mathscr{X}(\mathrm{GL}_{2}(A)_{2}),$

for $i=0,1$ and if we compose $\psi_{i}$ with the canonical projection from
the fiber-product onto $\mathscr{X}(\Gamma_{i}(\alpha T+\beta)),$ we have a
cover. Finally, all of the cusps of $X(N)$ are
$\mathrm{Gal}(X(N)/\mathrm{GL}_{2}(A))$-conjugate, so if we consider
$x=\infty$ in particular, and denote its stabilizer

$G_{\infty}=\Gamma_{i}(\alpha T+\beta)_{\infty}/\mathbb{F}_{q}^{\times},$

the first ramification group $G_{\infty,1}$ is its $p$-Sylow subgroup
$U_{i}(\alpha T+\beta)\cdot\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times},$
where

$U_{i}(\alpha T+\beta)=\left\\{\left(\begin{array}[]{cc}1&b\\\
0&1\end{array}\right)\in\Gamma_{i}(\alpha T+\beta)\right\\}.$

######  Example 7.5.

We use Riemann-Hurwitz to provide formulas for the genera of the square-
determinant Drinfeld modular curves $\mathscr{X}(\Gamma_{1}(\alpha
T+\beta)_{2})$ and $\mathscr{X}(\Gamma_{0}(\alpha T+\beta)_{2}):$

$g(\mathscr{X}(\Gamma_{i}(\alpha
T+\beta)_{2}))=-1+\frac{1}{2}\left(\sum_{P_{e}\in\operatorname{Ell}(\Gamma_{i}(\alpha
T+\beta)_{2})}(e_{P_{e}}-1)\right)+\frac{1}{2}\left(\sum_{P\in\mathcal{C}_{\Gamma_{i}(\alpha
T+\beta)_{2}}}(e_{P}-1)\right),$

where $e_{P}$ is the ramification index of the point $P$ on $\Gamma_{i}(\alpha
T+\beta)_{2}\setminus(\Omega\cup\mathbb{P}^{1}(K))$ over the corresponding
point on $\Gamma_{i}(\alpha T+\beta)\setminus(\Omega\cup\mathbb{P}^{1}(K)),$
and $i=0$ or $1.$

To finish these calculations, which let us to decide an appropriate technique
for computing the log canonical ring $R(\mathscr{X}(\Gamma_{1}(\alpha
T+\beta)_{2}),K_{\mathscr{X}(\Gamma_{1}(\alpha T+\beta)_{2})}(2\Delta_{1}))$
we need a stronger comparision between the cusps of the pairs
$\Gamma_{1}(\alpha T+\beta)$ and $\Gamma_{1}(\alpha T+\beta)_{2}$ than
Proposition 6.8.

We conclude with one final example, where we will use
$M(\Gamma_{0}(T))=C[U,V,Z]/(UV-Z^{2})$ from [DK23, Theorem $4.4$] to make sure
that the log stacky canonical ring of the corresponding Drinfeld modular curve
$\mathscr{X}_{\Gamma_{0}(T)_{2}}$ does in fact compute this algebra of
Drinfeld modular forms for $\Gamma_{0}(T)_{2}.$

######  Example 7.6.

Since $UV-Z^{2}$ describes a conic, we know that the curve
$C[U,V,Z]/(UV-Z^{2})\subset\mathbb{P}^{2}_{C}$ is rational, and all rational
curves have genus $0.$ There are $2$ cusps, say $0$ and $\infty$ for
$\mathscr{X}_{\Gamma_{0}(T)}$ so there are at least the same cusps on
$\mathscr{X}_{\Gamma_{0}(T)_{2}}$ and hence there are $2$ elliptic points.

Let $\overline{\Gamma_{0}(T)_{2}}$ denote the image of $\Gamma_{0}(T)_{2}$ in
$\mathrm{GL}_{2}(A/T)\cong\mathrm{GL}_{2}(\mathbb{F}_{q}).$ As in [Gek01,
Section $3$], let $(A/T)^{2}_{\text{prim}}$ denote the primitive vectors in
$A/T\times A/T,$ i.e. those vectors which span a non-zero direct summand. Then
from [Gek01, Section $3$] we know

$\\{\text{cusps of
}X_{\Gamma_{0}(T)_{2}}\\}\cong\overline{\Gamma_{0}(T)_{2}}\setminus(A/T)^{2}_{\text{prim}}/\mathbb{F}_{q}^{\times},$

so the cusps of $X_{\Gamma_{0}(T)_{2}}$ are precisely the
$\Gamma_{0}(T)_{2}$-orbits of $0$ and $\infty$ which correspond to the
primitve vectors $(1,0)$ and $(0,1).$ So, there are exactly these two cusps
and no further elliptic points. Let $\alpha\in\mathbb{Q}$ be such that

$\frac{2k-2l-kq}{k(q-1)}\leq\alpha<\frac{2k-2l-kq}{k(q-1)}+1$

and the number $r$ of best lower approximations to $\alpha$ with denominator
strictly greater than $1$ is $r=2.$ Then let

$\displaystyle D\overset{def}{=}K_{\mathscr{X}_{\Gamma_{0}(T)_{2}}}+2\Delta$
$\displaystyle\sim K_{\mathbb{P}^{1}}+\alpha(0)+\alpha(\infty)+2(0+\infty)$
$\displaystyle=\alpha(\infty)+(\alpha+2)(0),$

since $K_{\mathbb{P}^{1}}=-2\infty.$ We see that

$\displaystyle h^{0}\left(\frac{k}{2}D\right)$
$\displaystyle=2\Big{\lfloor}\frac{k}{2}(\alpha)\Big{\rfloor}+k+1$
$\displaystyle=k\left(\frac{2k-2l-kq}{k(q-1)}\right)+k+1$
$\displaystyle=1+\frac{k-2l}{q-1}$
$\displaystyle=\dim_{C}(M_{k,l}(\Gamma_{0}(T))),$

where we know this dimension from [DK23, Proposition $4.1$].

Finally, we see from [O’D15, Theorem $6$] that the canonical ring $R_{D},$
i.e. the log stacky canonical ring for $\mathscr{X}_{\Gamma_{0}(T)_{2}},$ is
generated by $3$ functions, $\Delta_{T},$ $\Delta_{W}$ and $E_{T}$
corresponding to $U,V$ and $Z$ respectively, and has a single relation
$UV-Z^{2}.$ We include a rough sketch of the monoid
$M\overset{def}{=}\\{(d,c)\in\mathbb{Z}^{2}:-d(\alpha+2)\leq c\leq d\alpha\\}$
from [O’D15], where generators for $R_{D}$ correspond to shaded-in lattice
points in degrees $2$ and $q-1$:

$q-1$

As this theory develops we expect results along these lines for congruence
subgroups with conductor of higher degree, and for modular curves with greater
genera will become explicit.

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